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    \textbf{GEO006+5.ax} Rules of construction






$\forall a,b{:}\;( ( \mbox{point}(a)\mbox{ and }\mbox{point}(b)\mbox{ and }\mbox{distinct\_points}(a,b) ) \ \Rightarrow \ \mbox{line}(\mbox{line\_connecting}(a,b)) )\qquad\mbox{fof}(\mbox{con}_1,\mbox{axiom})$

$\forall l,m{:}\;( ( \mbox{line}(l)\mbox{ and }\mbox{line}(m)\mbox{ and }\mbox{convergent\_lines}(l,m) ) \ \Rightarrow \ \mbox{point}(\mbox{intersection\_point}(l,m)) )\qquad\mbox{fof}(\mbox{int}_1,\mbox{axiom})$

$\forall l,a{:}\;( ( \mbox{line}(l)\mbox{ and }\mbox{point}(a) ) \ \Rightarrow \ \mbox{line}(\mbox{parallel\_through\_point}(l,a)) )\qquad\mbox{fof}(\mbox{par}_1,\mbox{axiom})$

$\forall l,a{:}\;( ( \mbox{line}(l)\mbox{ and }\mbox{point}(a) ) \ \Rightarrow \ \mbox{line}(\mbox{orthogonal\_through\_point}(l,a)) )\qquad\mbox{fof}(\mbox{orth}_1,\mbox{axiom})$

\medskip\textbf{GEO004+1.ax} Betweenness for simple curves






$\forall c,p,q,r{:}\;( \mbox{between\_c}(c,p,q,r)\iff ( p\ne r\mbox{ and }\exists \mbox{cpp}{:}\;( \mbox{part\_of}(\mbox{cpp},c)\mbox{ and }\mbox{end\_point}(p,\mbox{cpp})\mbox{ and }\mbox{end\_point}(r,\mbox{cpp})\mbox{ and }\mbox{inner\_point}(q,\mbox{cpp}) ) ) )\qquad\mbox{fof}(\mbox{between\_c\_defn},\mbox{axiom})$

\medskip\textbf{GEO004+3.ax} Trajectories






$\forall x,y,p{:}\;( \mbox{connect}(x,y,p)\iff \mbox{once}(\mbox{at\_the\_same\_time}(\mbox{at}(x,p),\mbox{at}(y,p))) )\qquad\mbox{fof}(\mbox{connect\_defn},\mbox{axiom})$

$\forall a,b{:}\;( \mbox{once}(\mbox{at\_the\_same\_time}(a,b))\iff \mbox{once}(\mbox{at\_the\_same\_time}(b,a)) )\qquad\mbox{fof}(\mbox{symmetry\_of\_at\_the\_same\_time},\mbox{axiom})$

$\forall a,b,c{:}\;( \mbox{once}(\mbox{at\_the\_same\_time}(\mbox{at\_the\_same\_time}(a,b),c))\iff \mbox{once}(\mbox{at\_the\_same\_time}(a,\mbox{at\_the\_same\_time}(b,c))) )\qquad\mbox{fof}(\mbox{assciativity\_of\_at\_the\_same\_time},\mbox{axiom})$

$\forall a{:}\;( \mbox{once}(a) \ \Rightarrow \ \mbox{once}(\mbox{at\_the\_same\_time}(a,a)) )\qquad\mbox{fof}(\mbox{idempotence\_of\_at\_the\_same\_time},\mbox{axiom})$

$\forall a,b{:}\;( \mbox{once}(\mbox{at\_the\_same\_time}(a,b)) \ \Rightarrow \ ( \mbox{once}(a)\mbox{ and }\mbox{once}(b) ) )\qquad\mbox{fof}(\mbox{conjunction\_at\_the\_same\_time},\mbox{axiom})$

$\forall x,p{:}\;( \mbox{once}(\mbox{at}(x,p))\iff \mbox{incident\_o}(p,\mbox{trajectory\_of}(x)) )\qquad\mbox{fof}(\mbox{at\_on\_trajectory},\mbox{axiom})$

$\forall x{:}\;\exists o{:}\;\mbox{trajectory\_of}(x) = o\qquad\mbox{fof}(\mbox{trajectories\_are\_oriented\_curves},\mbox{axiom})$

$\forall p_1,p_2,q_1,q_2,x,y{:}\;( ( \mbox{once}(\mbox{at\_the\_same\_time}(\mbox{at}(x,p_1),\mbox{at}(y,p_2)))\mbox{ and }\mbox{once}(\mbox{at\_the\_same\_time}(\mbox{at}(x,q_1),\mbox{at}(y,q_2))) ) \ \Rightarrow \ \neg\,\mbox{ordered\_by}(\mbox{trajectory\_of}(x),p_1,q_1)\mbox{ and }\mbox{ordered\_by}(\mbox{trajectory\_of}(y),q_2,p_2) )\qquad\mbox{fof}(\mbox{homogeneous\_behaviour},\mbox{axiom})$

$\forall a{:}\;( \mbox{once}(a) \ \Rightarrow \ \forall x{:}\;\exists p{:}\;\mbox{once}(\mbox{at\_the\_same\_time}(a,\mbox{at}(x,p))) )\qquad\mbox{fof}(\mbox{localization},\mbox{axiom})$

\medskip\textbf{GEO002-1.ax} Colinearity axioms for the GEO002 geometry axioms






$\mbox{between}(x,y,z) \ \Rightarrow \ \mbox{colinear}(x,y,z)\qquad\mbox{cnf}(\mbox{colinearity}_1,\mbox{axiom})$

$\mbox{between}(y,z,x) \ \Rightarrow \ \mbox{colinear}(x,y,z)\qquad\mbox{cnf}(\mbox{colinearity}_2,\mbox{axiom})$

$\mbox{between}(z,x,y) \ \Rightarrow \ \mbox{colinear}(x,y,z)\qquad\mbox{cnf}(\mbox{colinearity}_3,\mbox{axiom})$

$\mbox{colinear}(x,y,z) \ \Rightarrow \ ( \mbox{between}(x,y,z)\mbox{ or }\mbox{between}(y,z,x)\mbox{ or }\mbox{between}(z,x,y) )\qquad\mbox{cnf}(\mbox{colinearity}_4,\mbox{axiom})$

\medskip\textbf{GEO008+0.ax} Apartness geometry






$\forall x{:}\;\neg\,\mbox{distinct\_points}(x,x)\qquad\mbox{fof}(\mbox{apart}_1,\mbox{axiom})$

$\forall x{:}\;\neg\,\mbox{distinct\_lines}(x,x)\qquad\mbox{fof}(\mbox{apart}_2,\mbox{axiom})$

$\forall x{:}\;\neg\,\mbox{convergent\_lines}(x,x)\qquad\mbox{fof}(\mbox{apart}_3,\mbox{axiom})$

$\forall x,y,z{:}\;( \mbox{distinct\_points}(x,y) \ \Rightarrow \ ( \mbox{distinct\_points}(x,z)\mbox{ or }\mbox{distinct\_points}(y,z) ) )\qquad\mbox{fof}(\mbox{apart}_4,\mbox{axiom})$

$\forall x,y,z{:}\;( \mbox{distinct\_lines}(x,y) \ \Rightarrow \ ( \mbox{distinct\_lines}(x,z)\mbox{ or }\mbox{distinct\_lines}(y,z) ) )\qquad\mbox{fof}(\mbox{apart}_5,\mbox{axiom})$

$\forall x,y,z{:}\;( \mbox{convergent\_lines}(x,y) \ \Rightarrow \ ( \mbox{convergent\_lines}(x,z)\mbox{ or }\mbox{convergent\_lines}(y,z) ) )\qquad\mbox{fof}(\mbox{apart}_6,\mbox{axiom})$

$\forall x,y,z{:}\;( \mbox{distinct\_points}(x,y) \ \Rightarrow \ ( \mbox{apart\_point\_and\_line}(z,\mbox{line\_connecting}(x,y)) \ \Rightarrow \ ( \mbox{distinct\_points}(z,x)\mbox{ and }\mbox{distinct\_points}(z,y) ) ) )\qquad\mbox{fof}(\mbox{con}_1,\mbox{axiom})$

$\forall x,y,z{:}\;( \mbox{convergent\_lines}(x,y) \ \Rightarrow \ ( ( \mbox{apart\_point\_and\_line}(z,x)\mbox{ or }\mbox{apart\_point\_and\_line}(z,y) ) \ \Rightarrow \ \mbox{distinct\_points}(z,\mbox{intersection\_point}(x,y)) ) )\qquad\mbox{fof}(\mbox{con}_2,\mbox{axiom})$

$\forall x,y,u,v{:}\;( ( \mbox{distinct\_points}(x,y)\mbox{ and }\mbox{distinct\_lines}(u,v) ) \ \Rightarrow \ ( \mbox{apart\_point\_and\_line}(x,u)\mbox{ or }\mbox{apart\_point\_and\_line}(x,v)\mbox{ or }\mbox{apart\_point\_and\_line}(y,u)\mbox{ or }\mbox{apart\_point\_and\_line}(y,v) ) )\qquad\mbox{fof}(\mbox{cu}_1,\mbox{axiom})$

$\forall x,y,z{:}\;( \mbox{apart\_point\_and\_line}(x,y) \ \Rightarrow \ ( \mbox{distinct\_points}(x,z)\mbox{ or }\mbox{apart\_point\_and\_line}(z,y) ) )\qquad\mbox{fof}(\mbox{ceq}_1,\mbox{axiom})$

$\forall x,y,z{:}\;( \mbox{apart\_point\_and\_line}(x,y) \ \Rightarrow \ ( \mbox{distinct\_lines}(y,z)\mbox{ or }\mbox{apart\_point\_and\_line}(x,z) ) )\qquad\mbox{fof}(\mbox{ceq}_2,\mbox{axiom})$

$\forall x,y{:}\;( \mbox{convergent\_lines}(x,y) \ \Rightarrow \ \mbox{distinct\_lines}(x,y) )\qquad\mbox{fof}(\mbox{ceq}_3,\mbox{axiom})$

\medskip\textbf{GEO004+0.ax} Simple curve axioms






$\forall c,c_1{:}\;( \mbox{part\_of}(c_1,c)\iff \forall p{:}\;( \mbox{incident\_c}(p,c_1) \ \Rightarrow \ \mbox{incident\_c}(p,c) ) )\qquad\mbox{fof}(\mbox{part\_of\_defn},\mbox{axiom})$

$\forall c,c_1,c_2{:}\;( c = c_1 + c_2\iff \forall q{:}\;( \mbox{incident\_c}(q,c)\iff ( \mbox{incident\_c}(q,c_1)\mbox{ or }\mbox{incident\_c}(q,c_2) ) ) )\qquad\mbox{fof}(\mbox{sum\_defn},\mbox{axiom})$

$\forall p,c{:}\;( \mbox{end\_point}(p,c)\iff ( \mbox{incident\_c}(p,c)\mbox{ and }\forall c_1,c_2{:}\;( ( \mbox{part\_of}(c_1,c)\mbox{ and }\mbox{part\_of}(c_2,c)\mbox{ and }\mbox{incident\_c}(p,c_1)\mbox{ and }\mbox{incident\_c}(p,c_2) ) \ \Rightarrow \ ( \mbox{part\_of}(c_1,c_2)\mbox{ or }\mbox{part\_of}(c_2,c_1) ) ) ) )\qquad\mbox{fof}(\mbox{end\_point\_defn},\mbox{axiom})$

$\forall p,c{:}\;( \mbox{inner\_point}(p,c)\iff ( \mbox{incident\_c}(p,c)\mbox{ and }\neg\,\mbox{end\_point}(p,c) ) )\qquad\mbox{fof}(\mbox{inner\_point\_defn},\mbox{axiom})$

$\forall p,c,c_1{:}\;( p\wedge c{=}c_1\iff ( \mbox{incident\_c}(p,c)\mbox{ and }\mbox{incident\_c}(p,c_1)\mbox{ and }\forall q{:}\;( ( \mbox{incident\_c}(q,c)\mbox{ and }\mbox{incident\_c}(q,c_1) ) \ \Rightarrow \ ( \mbox{end\_point}(q,c)\mbox{ and }\mbox{end\_point}(q,c_1) ) ) ) )\qquad\mbox{fof}(\mbox{meet\_defn},\mbox{axiom})$

$\forall c{:}\;( \mbox{closed}(c)\iff \neg\,\exists p{:}\;\mbox{end\_point}(p,c) )\qquad\mbox{fof}(\mbox{closed\_defn},\mbox{axiom})$

$\forall c{:}\;( \mbox{open}(c)\iff \exists p{:}\;\mbox{end\_point}(p,c) )\qquad\mbox{fof}(\mbox{open\_defn},\mbox{axiom})$

$\forall c,c_1{:}\;( ( \mbox{part\_of}(c_1,c)\mbox{ and }c_1\ne c ) \ \Rightarrow \ \mbox{open}(c_1) )\qquad\mbox{fof}(c_1,\mbox{axiom})$

$\forall c,c_1,c_2,c_3{:}\;( ( \mbox{part\_of}(c_1,c)\mbox{ and }\mbox{part\_of}(c_2,c)\mbox{ and }\mbox{part\_of}(c_3,c)\mbox{ and }\exists p{:}\;( \mbox{end\_point}(p,c_1)\mbox{ and }\mbox{end\_point}(p,c_2)\mbox{ and }\mbox{end\_point}(p,c_3) ) ) \ \Rightarrow \ ( \mbox{part\_of}(c_2,c_3)\mbox{ or }\mbox{part\_of}(c_3,c_2)\mbox{ or }\mbox{part\_of}(c_1,c_2)\mbox{ or }\mbox{part\_of}(c_2,c_1)\mbox{ or }\mbox{part\_of}(c_1,c_3)\mbox{ or }\mbox{part\_of}(c_3,c_1) ) )\qquad\mbox{fof}(c_2,\mbox{axiom})$

$\forall c{:}\;\exists p{:}\;\mbox{inner\_point}(p,c)\qquad\mbox{fof}(c_3,\mbox{axiom})$

$\forall c,p{:}\;( \mbox{inner\_point}(p,c) \ \Rightarrow \ \exists c_1,c_2{:}\;( p\wedge c_1{=}c_2\mbox{ and }c = c_1 + c_2 ) )\qquad\mbox{fof}(c_4,\mbox{axiom})$

$\forall c,p,q,r{:}\;( ( \mbox{end\_point}(p,c)\mbox{ and }\mbox{end\_point}(q,c)\mbox{ and }\mbox{end\_point}(r,c) ) \ \Rightarrow \ ( p = q\mbox{ or }p = r\mbox{ or }q = r ) )\qquad\mbox{fof}(c_5,\mbox{axiom})$

$\forall c,p{:}\;( \mbox{end\_point}(p,c) \ \Rightarrow \ \exists q{:}\;( \mbox{end\_point}(q,c)\mbox{ and }p\ne q ) )\qquad\mbox{fof}(c_6,\mbox{axiom})$

$\forall c,c_1,c_2,p{:}\;( ( \mbox{closed}(c)\mbox{ and }p\wedge c_1{=}c_2\mbox{ and }c = c_1 + c_2 ) \ \Rightarrow \ \forall q{:}\;( \mbox{end\_point}(q,c_1) \ \Rightarrow \ q\wedge c_1{=}c_2 ) )\qquad\mbox{fof}(c_7,\mbox{axiom})$

$\forall c_1,c_2{:}\;( \exists p{:}\;p\wedge c_1{=}c_2 \ \Rightarrow \ \exists c{:}\;c = c_1 + c_2 )\qquad\mbox{fof}(c_8,\mbox{axiom})$

$\forall c,c_1{:}\;( \forall p{:}\;( \mbox{incident\_c}(p,c)\iff \mbox{incident\_c}(p,c_1) ) \ \Rightarrow \ c = c_1 )\qquad\mbox{fof}(c_9,\mbox{axiom})$

\medskip\textbf{GEO007+1.ax} Principles of a classical calculus of directed lines






$\forall l{:}\;\neg\,\mbox{unequally\_directed\_lines}(l,l)\qquad\mbox{fof}(\mbox{cld}_1,\mbox{axiom})$

$\forall l,m,n{:}\;( ( \neg\,\mbox{unequally\_directed\_lines}(l,m)\mbox{ and }\neg\,\mbox{unequally\_directed\_lines}(l,n) ) \ \Rightarrow \ \neg\,\mbox{unequally\_directed\_lines}(m,n) )\qquad\mbox{fof}(\mbox{cld}_2,\mbox{axiom})$

$\forall a,b,l,m{:}\;( \neg\,\mbox{unequally\_directed\_lines}(l,m)\mbox{ and }\mbox{unequally\_directed\_lines}(l,\mbox{reverse\_line}(m))\iff ( \neg\,\mbox{unequally\_directed\_lines}(l,m)\mbox{ or }\neg\,\mbox{unequally\_directed\_lines}(l,\mbox{reverse\_line}(m)) ) )\qquad\mbox{fof}(\mbox{cld}_3,\mbox{axiom})$

$\forall l,m,n{:}\;( ( ( \neg\,\mbox{unequally\_directed\_lines}(l,m)\mbox{ or }\neg\,\mbox{unequally\_directed\_lines}(l,\mbox{reverse\_line}(m)) )\mbox{ and }( \neg\,\mbox{unequally\_directed\_lines}(l,n)\mbox{ or }\neg\,\mbox{unequally\_directed\_lines}(l,\mbox{reverse\_line}(n)) ) ) \ \Rightarrow \ ( \neg\,\mbox{unequally\_directed\_lines}(m,n)\mbox{ or }\neg\,\mbox{unequally\_directed\_lines}(m,\mbox{reverse\_line}(n)) ) )\qquad\mbox{fof}(\mbox{cld3a},\mbox{axiom})$

$\forall l,m{:}\;( ( \mbox{line}(l)\mbox{ and }\mbox{line}(m) ) \ \Rightarrow \ \neg\,\neg\,\mbox{unequally\_directed\_lines}(l,m)\mbox{ and }\neg\,\mbox{unequally\_directed\_lines}(l,\mbox{reverse\_line}(m)) )\qquad\mbox{fof}(\mbox{cld}_4,\mbox{axiom})$

$\forall l,m,n{:}\;( \neg\,\mbox{unequally\_directed\_lines}(l,\mbox{reverse\_line}(m))\mbox{ and }( \neg\,\mbox{unequally\_directed\_lines}(l,\mbox{reverse\_line}(n)) \ \Rightarrow \ \neg\,\mbox{unequally\_directed\_lines}(m,n) ) )\qquad\mbox{fof}(\mbox{cld}_5,\mbox{axiom})$

\medskip\textbf{GEO002-2.ax} Reflection axioms for the GEO002 geometry axioms






$\mbox{reflection}(u,v) = \mbox{extension}(u,v,u,v)\qquad\mbox{cnf}(\mbox{reflection},\mbox{axiom})$

\medskip\textbf{GEO006+0.ax} Apartness geometry






$\forall x{:}\;\neg\,\mbox{distinct\_points}(x,x)\qquad\mbox{fof}(\mbox{apart}_1,\mbox{axiom})$

$\forall x{:}\;\neg\,\mbox{distinct\_lines}(x,x)\qquad\mbox{fof}(\mbox{apart}_2,\mbox{axiom})$

$\forall x{:}\;\neg\,\mbox{convergent\_lines}(x,x)\qquad\mbox{fof}(\mbox{apart}_3,\mbox{axiom})$

$\forall x,y,z{:}\;( \mbox{distinct\_points}(x,y) \ \Rightarrow \ ( \mbox{distinct\_points}(x,z)\mbox{ or }\mbox{distinct\_points}(y,z) ) )\qquad\mbox{fof}(\mbox{apart}_4,\mbox{axiom})$

$\forall x,y,z{:}\;( \mbox{distinct\_lines}(x,y) \ \Rightarrow \ ( \mbox{distinct\_lines}(x,z)\mbox{ or }\mbox{distinct\_lines}(y,z) ) )\qquad\mbox{fof}(\mbox{apart}_5,\mbox{axiom})$

$\forall x,y,z{:}\;( \mbox{convergent\_lines}(x,y) \ \Rightarrow \ ( \mbox{convergent\_lines}(x,z)\mbox{ or }\mbox{convergent\_lines}(y,z) ) )\qquad\mbox{fof}(\mbox{ax}_6,\mbox{axiom})$

$\forall x,y{:}\;( \mbox{distinct\_points}(x,y) \ \Rightarrow \ \neg\,\mbox{apart\_point\_and\_line}(x,\mbox{line\_connecting}(x,y)) )\qquad\mbox{fof}(\mbox{ci}_1,\mbox{axiom})$

$\forall x,y{:}\;( \mbox{distinct\_points}(x,y) \ \Rightarrow \ \neg\,\mbox{apart\_point\_and\_line}(y,\mbox{line\_connecting}(x,y)) )\qquad\mbox{fof}(\mbox{ci}_2,\mbox{axiom})$

$\forall x,y{:}\;( \mbox{convergent\_lines}(x,y) \ \Rightarrow \ \neg\,\mbox{apart\_point\_and\_line}(\mbox{intersection\_point}(x,y),x) )\qquad\mbox{fof}(\mbox{ci}_3,\mbox{axiom})$

$\forall x,y{:}\;( \mbox{convergent\_lines}(x,y) \ \Rightarrow \ \neg\,\mbox{apart\_point\_and\_line}(\mbox{intersection\_point}(x,y),y) )\qquad\mbox{fof}(\mbox{ci}_4,\mbox{axiom})$

$\forall x,y,u,v{:}\;( ( \mbox{distinct\_points}(x,y)\mbox{ and }\mbox{distinct\_lines}(u,v) ) \ \Rightarrow \ ( \mbox{apart\_point\_and\_line}(x,u)\mbox{ or }\mbox{apart\_point\_and\_line}(x,v)\mbox{ or }\mbox{apart\_point\_and\_line}(y,u)\mbox{ or }\mbox{apart\_point\_and\_line}(y,v) ) )\qquad\mbox{fof}(\mbox{cu}_1,\mbox{axiom})$

$\forall x,y,z{:}\;( \mbox{apart\_point\_and\_line}(x,y) \ \Rightarrow \ ( \mbox{distinct\_points}(x,z)\mbox{ or }\mbox{apart\_point\_and\_line}(z,y) ) )\qquad\mbox{fof}(\mbox{ceq}_1,\mbox{axiom})$

$\forall x,y,z{:}\;( \mbox{apart\_point\_and\_line}(x,y) \ \Rightarrow \ ( \mbox{distinct\_lines}(y,z)\mbox{ or }\mbox{apart\_point\_and\_line}(x,z) ) )\qquad\mbox{fof}(\mbox{ceq}_2,\mbox{axiom})$

$\forall x,y,z{:}\;( \mbox{convergent\_lines}(x,y) \ \Rightarrow \ ( \mbox{distinct\_lines}(y,z)\mbox{ or }\mbox{convergent\_lines}(x,z) ) )\qquad\mbox{fof}(\mbox{ceq}_3,\mbox{axiom})$

\medskip\textbf{GEO002-0.ax} Tarski geometry axioms






$\mbox{equidistant}(x,y,y,x)\qquad\mbox{cnf}(\mbox{reflexivity\_for\_equidistance},\mbox{axiom})$

$( \mbox{equidistant}(x,y,z,v)\mbox{ and }\mbox{equidistant}(x,y,v_2,w) ) \ \Rightarrow \ \mbox{equidistant}(z,v,v_2,w)\qquad\mbox{cnf}(\mbox{transitivity\_for\_equidistance},\mbox{axiom})$

$\mbox{equidistant}(x,y,z,z) \ \Rightarrow \ x = y\qquad\mbox{cnf}(\mbox{identity\_for\_equidistance},\mbox{axiom})$

$\mbox{between}(x,y,\mbox{extension}(x,y,w,v))\qquad\mbox{cnf}(\mbox{segment\_construction}_1,\mbox{axiom})$

$\mbox{equidistant}(y,\mbox{extension}(x,y,w,v),w,v)\qquad\mbox{cnf}(\mbox{segment\_construction}_2,\mbox{axiom})$

$( \mbox{equidistant}(x,y,x_1,y_1)\mbox{ and }\mbox{equidistant}(y,z,y_1,z_1)\mbox{ and }\mbox{equidistant}(x,v,x_1,v_1)\mbox{ and }\mbox{equidistant}(y,v,y_1,v_1)\mbox{ and }\mbox{between}(x,y,z)\mbox{ and }\mbox{between}(x_1,y_1,z_1) ) \ \Rightarrow \ ( x = y\mbox{ or }\mbox{equidistant}(z,v,z_1,v_1) )\qquad\mbox{cnf}(\mbox{outer\_five\_segment},\mbox{axiom})$

$\mbox{between}(x,y,x) \ \Rightarrow \ x = y\qquad\mbox{cnf}(\mbox{identity\_for\_betweeness},\mbox{axiom})$

$( \mbox{between}(u,v,w)\mbox{ and }\mbox{between}(y,x,w) ) \ \Rightarrow \ \mbox{between}(v,\mbox{inner\_pasch}(u,v,w,x,y),y)\qquad\mbox{cnf}(\mbox{inner\_pasch}_1,\mbox{axiom})$

$( \mbox{between}(u,v,w)\mbox{ and }\mbox{between}(y,x,w) ) \ \Rightarrow \ \mbox{between}(x,\mbox{inner\_pasch}(u,v,w,x,y),u)\qquad\mbox{cnf}(\mbox{inner\_pasch}_2,\mbox{axiom})$

$\neg\,\mbox{between}(\mbox{lower\_dimension\_point}_1,\mbox{lower\_dimension\_point}_2,\mbox{lower\_dimension\_point}_3)\qquad\mbox{cnf}(\mbox{lower\_dimension}_1,\mbox{axiom})$

$\neg\,\mbox{between}(\mbox{lower\_dimension\_point}_2,\mbox{lower\_dimension\_point}_3,\mbox{lower\_dimension\_point}_1)\qquad\mbox{cnf}(\mbox{lower\_dimension}_2,\mbox{axiom})$

$\neg\,\mbox{between}(\mbox{lower\_dimension\_point}_3,\mbox{lower\_dimension\_point}_1,\mbox{lower\_dimension\_point}_2)\qquad\mbox{cnf}(\mbox{lower\_dimension}_3,\mbox{axiom})$

$( \mbox{equidistant}(x,w,x,v)\mbox{ and }\mbox{equidistant}(y,w,y,v)\mbox{ and }\mbox{equidistant}(z,w,z,v) ) \ \Rightarrow \ ( \mbox{between}(x,y,z)\mbox{ or }\mbox{between}(y,z,x)\mbox{ or }\mbox{between}(z,x,y)\mbox{ or }w = v )\qquad\mbox{cnf}(\mbox{upper\_dimension},\mbox{axiom})$

$( \mbox{between}(u,w,y)\mbox{ and }\mbox{between}(v,w,x) ) \ \Rightarrow \ ( u = w\mbox{ or }\mbox{between}(u,v,\mbox{euclid}_1(u,v,w,x,y)) )\qquad\mbox{cnf}(\mbox{euclid}_1,\mbox{axiom})$

$( \mbox{between}(u,w,y)\mbox{ and }\mbox{between}(v,w,x) ) \ \Rightarrow \ ( u = w\mbox{ or }\mbox{between}(u,x,\mbox{euclid}_2(u,v,w,x,y)) )\qquad\mbox{cnf}(\mbox{euclid}_2,\mbox{axiom})$

$( \mbox{between}(u,w,y)\mbox{ and }\mbox{between}(v,w,x) ) \ \Rightarrow \ ( u = w\mbox{ or }\mbox{between}(\mbox{euclid}_1(u,v,w,x,y),y,\mbox{euclid}_2(u,v,w,x,y)) )\qquad\mbox{cnf}(\mbox{euclid}_3,\mbox{axiom})$

$( \mbox{equidistant}(u,v,u,v_1)\mbox{ and }\mbox{equidistant}(u,x,u,x_1)\mbox{ and }\mbox{between}(u,v,x)\mbox{ and }\mbox{between}(v,w,x) ) \ \Rightarrow \ \mbox{between}(v_1,\mbox{continuous}(u,v,v_1,w,x,x_1),x_1)\qquad\mbox{cnf}(\mbox{continuity}_1,\mbox{axiom})$

$( \mbox{equidistant}(u,v,u,v_1)\mbox{ and }\mbox{equidistant}(u,x,u,x_1)\mbox{ and }\mbox{between}(u,v,x)\mbox{ and }\mbox{between}(v,w,x) ) \ \Rightarrow \ \mbox{equidistant}(u,w,u,\mbox{continuous}(u,v,v_1,w,x,x_1))\qquad\mbox{cnf}(\mbox{continuity}_2,\mbox{axiom})$

\medskip\textbf{GEO006+2.ax} Affine geometry






$\forall x,y{:}\;\neg\,\mbox{convergent\_lines}(\mbox{parallel\_through\_point}(y,x),y)\qquad\mbox{fof}(\mbox{cp}_1,\mbox{axiom})$

$\forall x,y{:}\;\neg\,\mbox{apart\_point\_and\_line}(x,\mbox{parallel\_through\_point}(y,x))\qquad\mbox{fof}(\mbox{cp}_2,\mbox{axiom})$

$\forall x,y,z{:}\;( \mbox{distinct\_lines}(y,z) \ \Rightarrow \ ( \mbox{apart\_point\_and\_line}(x,y)\mbox{ or }\mbox{apart\_point\_and\_line}(x,z)\mbox{ or }\mbox{convergent\_lines}(y,z) ) )\qquad\mbox{fof}(\mbox{cup}_1,\mbox{axiom})$

\medskip\textbf{GEO002-3.ax} Insertion axioms for the GEO002 geometry axioms






$\mbox{insertion}(u_1,w_1,u,v) = \mbox{extension}(\mbox{extension}(w_1,u_1,\mbox{lower\_dimension\_point}_1,\mbox{lower\_dimension\_point}_2),u_1,u,v)\qquad\mbox{cnf}(\mbox{insertion},\mbox{axiom})$

\medskip\textbf{GEO006+3.ax} Orthogonality






$\forall l,m{:}\;( \mbox{convergent\_lines}(l,m)\mbox{ or }\mbox{unorthogonal\_lines}(l,m) )\qquad\mbox{fof}(\mbox{occu}_1,\mbox{axiom})$

$\forall l,m,n{:}\;( ( \mbox{convergent\_lines}(l,m)\mbox{ and }\mbox{unorthogonal\_lines}(l,m) ) \ \Rightarrow \ ( ( \mbox{convergent\_lines}(l,n)\mbox{ and }\mbox{unorthogonal\_lines}(l,n) )\mbox{ or }( \mbox{convergent\_lines}(m,n)\mbox{ and }\mbox{unorthogonal\_lines}(m,n) ) ) )\qquad\mbox{fof}(\mbox{oac}_1,\mbox{axiom})$

$\forall a,l{:}\;\neg\,\mbox{unorthogonal\_lines}(\mbox{orthogonal\_through\_point}(l,a),l)\qquad\mbox{fof}(\mbox{ooc}_1,\mbox{axiom})$

$\forall a,l{:}\;\neg\,\mbox{apart\_point\_and\_line}(a,\mbox{orthogonal\_through\_point}(l,a))\qquad\mbox{fof}(\mbox{ooc}_2,\mbox{axiom})$

$\forall a,l,m,n{:}\;( \mbox{distinct\_lines}(l,m) \ \Rightarrow \ ( \mbox{apart\_point\_and\_line}(a,l)\mbox{ or }\mbox{apart\_point\_and\_line}(a,m)\mbox{ or }\mbox{unorthogonal\_lines}(l,n)\mbox{ or }\mbox{unorthogonal\_lines}(m,n) ) )\qquad\mbox{fof}(\mbox{ouo}_1,\mbox{axiom})$

\medskip\textbf{GEO004-3.ax} Trajectories






$\mbox{connect}(a,b,c) \ \Rightarrow \ \mbox{once}(\mbox{at\_the\_same\_time}(\mbox{at}(a,c),\mbox{at}(b,c)))\qquad\mbox{cnf}(\mbox{connect\_defn}_1,\mbox{axiom})$

$\mbox{once}(\mbox{at\_the\_same\_time}(\mbox{at}(a,b),\mbox{at}(c,b))) \ \Rightarrow \ \mbox{connect}(a,c,b)\qquad\mbox{cnf}(\mbox{connect\_defn}_2,\mbox{axiom})$

$\mbox{once}(\mbox{at\_the\_same\_time}(a,b)) \ \Rightarrow \ \mbox{once}(\mbox{at\_the\_same\_time}(b,a))\qquad\mbox{cnf}(\mbox{symmetry\_of\_at\_the\_same\_time}_3,\mbox{axiom})$

$\mbox{once}(\mbox{at\_the\_same\_time}(a,b)) \ \Rightarrow \ \mbox{once}(\mbox{at\_the\_same\_time}(b,a))\qquad\mbox{cnf}(\mbox{symmetry\_of\_at\_the\_same\_time}_4,\mbox{axiom})$

$\mbox{once}(\mbox{at\_the\_same\_time}(\mbox{at\_the\_same\_time}(a,b),c)) \ \Rightarrow \ \mbox{once}(\mbox{at\_the\_same\_time}(a,\mbox{at\_the\_same\_time}(b,c)))\qquad\mbox{cnf}(\mbox{assciativity\_of\_at\_the\_same\_time}_5,\mbox{axiom})$

$\mbox{once}(\mbox{at\_the\_same\_time}(a,\mbox{at\_the\_same\_time}(b,c))) \ \Rightarrow \ \mbox{once}(\mbox{at\_the\_same\_time}(\mbox{at\_the\_same\_time}(a,b),c))\qquad\mbox{cnf}(\mbox{assciativity\_of\_at\_the\_same\_time}_6,\mbox{axiom})$

$\mbox{once}(a) \ \Rightarrow \ \mbox{once}(\mbox{at\_the\_same\_time}(a,a))\qquad\mbox{cnf}(\mbox{idempotence\_of\_at\_the\_same\_time}_7,\mbox{axiom})$

$\mbox{once}(\mbox{at\_the\_same\_time}(a,b)) \ \Rightarrow \ \mbox{once}(a)\qquad\mbox{cnf}(\mbox{conjunction\_at\_the\_same\_time}_8,\mbox{axiom})$

$\mbox{once}(\mbox{at\_the\_same\_time}(a,b)) \ \Rightarrow \ \mbox{once}(b)\qquad\mbox{cnf}(\mbox{conjunction\_at\_the\_same\_time}_9,\mbox{axiom})$

$\mbox{once}(\mbox{at}(a,b)) \ \Rightarrow \ \mbox{incident\_o}(b,\mbox{trajectory\_of}(a))\qquad\mbox{cnf}(\mbox{at\_on\_trajectory}_{10},\mbox{axiom})$

$\mbox{incident\_o}(a,\mbox{trajectory\_of}(b)) \ \Rightarrow \ \mbox{once}(\mbox{at}(b,a))\qquad\mbox{cnf}(\mbox{at\_on\_trajectory}_{11},\mbox{axiom})$

$\mbox{trajectory\_of}(a) = \mbox{ax3\_sk}_1(a)\qquad\mbox{cnf}(\mbox{trajectories\_are\_oriented\_curves}_{12},\mbox{axiom})$

$( \mbox{once}(\mbox{at\_the\_same\_time}(\mbox{at}(a,b),\mbox{at}(c,d)))\mbox{ and }\mbox{once}(\mbox{at\_the\_same\_time}(\mbox{at}(a,e),\mbox{at}(c,f)))\mbox{ and }\mbox{ordered\_by}(\mbox{trajectory\_of}(a),b,e) ) \ \Rightarrow \ \neg\,\mbox{ordered\_by}(\mbox{trajectory\_of}(c),f,d)\qquad\mbox{cnf}(\mbox{homogeneous\_behaviour}_{13},\mbox{axiom})$

$\mbox{once}(a) \ \Rightarrow \ \mbox{once}(\mbox{at\_the\_same\_time}(a,\mbox{at}(b,\mbox{ax3\_sk}_2(b,a))))\qquad\mbox{cnf}(\mbox{localization}_{14},\mbox{axiom})$

\medskip\textbf{GEO001-1.ax} Colinearity axioms for the GEO001 geometry axioms






$\mbox{colinear}(x,y,z) \ \Rightarrow \ ( \mbox{between}(x,y,z)\mbox{ or }\mbox{between}(y,x,z)\mbox{ or }\mbox{between}(x,z,y) )\qquad\mbox{cnf}(\mbox{colinearity}_1,\mbox{axiom})$

$\mbox{between}(x,y,z) \ \Rightarrow \ \mbox{colinear}(x,y,z)\qquad\mbox{cnf}(\mbox{colinearity}_2,\mbox{axiom})$

$\mbox{between}(y,x,z) \ \Rightarrow \ \mbox{colinear}(x,y,z)\qquad\mbox{cnf}(\mbox{colinearity}_3,\mbox{axiom})$

$\mbox{between}(x,z,y) \ \Rightarrow \ \mbox{colinear}(x,y,z)\qquad\mbox{cnf}(\mbox{colinearity}_4,\mbox{axiom})$

\medskip\textbf{GEO006+1.ax} Projective geometry






$\forall x,y{:}\;( \mbox{distinct\_lines}(x,y) \ \Rightarrow \ \mbox{convergent\_lines}(x,y) )\qquad\mbox{fof}(p_1,\mbox{axiom})$

\medskip\textbf{GEO004+2.ax} Oriented curves






$\forall o,p,q,r{:}\;( \mbox{between\_o}(o,p,q,r)\iff ( ( \mbox{ordered\_by}(o,p,q)\mbox{ and }\mbox{ordered\_by}(o,q,r) )\mbox{ or }( \mbox{ordered\_by}(o,r,q)\mbox{ and }\mbox{ordered\_by}(o,q,p) ) ) )\qquad\mbox{fof}(\mbox{between\_o\_defn},\mbox{axiom})$

$\forall p,o{:}\;( \mbox{start\_point}(p,o)\iff ( \mbox{incident\_o}(p,o)\mbox{ and }\forall q{:}\;( ( p\ne q\mbox{ and }\mbox{incident\_o}(q,o) ) \ \Rightarrow \ \mbox{ordered\_by}(o,p,q) ) ) )\qquad\mbox{fof}(\mbox{start\_point\_defn},\mbox{axiom})$

$\forall p,o{:}\;( \mbox{finish\_point}(p,o)\iff ( \mbox{incident\_o}(p,o)\mbox{ and }\forall q{:}\;( ( p\ne q\mbox{ and }\mbox{incident\_o}(q,o) ) \ \Rightarrow \ \mbox{ordered\_by}(o,q,p) ) ) )\qquad\mbox{fof}(\mbox{finish\_point\_defn},\mbox{axiom})$

$\forall o,p,q{:}\;( \mbox{ordered\_by}(o,p,q) \ \Rightarrow \ ( \mbox{incident\_o}(p,o)\mbox{ and }\mbox{incident\_o}(q,o) ) )\qquad\mbox{fof}(o_1,\mbox{axiom})$

$\forall o{:}\;\exists c{:}\;( \mbox{open}(c)\mbox{ and }\forall p{:}\;( \mbox{incident\_o}(p,o)\iff \mbox{incident\_c}(p,c) ) )\qquad\mbox{fof}(o_2,\mbox{axiom})$

$\forall p,q,r,o{:}\;( \mbox{between\_o}(o,p,q,r)\iff \exists c{:}\;( \forall p{:}\;( \mbox{incident\_o}(p,o)\iff \mbox{incident\_c}(p,c) )\mbox{ and }\mbox{between\_c}(c,p,q,r) ) )\qquad\mbox{fof}(o_3,\mbox{axiom})$

$\forall o{:}\;\exists p{:}\;\mbox{start\_point}(p,o)\qquad\mbox{fof}(o_4,\mbox{axiom})$

$\forall p,q,c{:}\;( ( \mbox{open}(c)\mbox{ and }p\ne q\mbox{ and }\mbox{incident\_c}(p,c)\mbox{ and }\mbox{incident\_c}(q,c) ) \ \Rightarrow \ \exists o{:}\;( \forall r{:}\;( \mbox{incident\_o}(r,o)\iff \mbox{incident\_c}(r,c) )\mbox{ and }\mbox{ordered\_by}(o,p,q) ) )\qquad\mbox{fof}(o_5,\mbox{axiom})$

$\forall o_1,o_2{:}\;( \forall p,q{:}\;( \mbox{ordered\_by}(o_1,p,q)\iff \mbox{ordered\_by}(o_2,p,q) ) \ \Rightarrow \ o_1 = o_2 )\qquad\mbox{fof}(o_6,\mbox{axiom})$

$\forall c,o{:}\;( c = \mbox{underlying\_curve}(o)\iff \forall p{:}\;( \mbox{incident\_o}(p,o)\iff \mbox{incident\_c}(p,c) ) )\qquad\mbox{fof}(\mbox{underlying\_curve\_defn},\mbox{axiom})$

\medskip\textbf{GEO001-0.ax} Tarski geometry axioms






$\mbox{between}(x,y,x) \ \Rightarrow \ x = y\qquad\mbox{cnf}(\mbox{identity\_for\_betweeness},\mbox{axiom})$

$( \mbox{between}(x,y,v)\mbox{ and }\mbox{between}(y,z,v) ) \ \Rightarrow \ \mbox{between}(x,y,z)\qquad\mbox{cnf}(\mbox{transitivity\_for\_betweeness},\mbox{axiom})$

$( \mbox{between}(x,y,z)\mbox{ and }\mbox{between}(x,y,v) ) \ \Rightarrow \ ( x = y\mbox{ or }\mbox{between}(x,z,v)\mbox{ or }\mbox{between}(x,v,z) )\qquad\mbox{cnf}(\mbox{connectivity\_for\_betweeness},\mbox{axiom})$

$\mbox{equidistant}(x,y,y,x)\qquad\mbox{cnf}(\mbox{reflexivity\_for\_equidistance},\mbox{axiom})$

$\mbox{equidistant}(x,y,z,z) \ \Rightarrow \ x = y\qquad\mbox{cnf}(\mbox{identity\_for\_equidistance},\mbox{axiom})$

$( \mbox{equidistant}(x,y,z,v)\mbox{ and }\mbox{equidistant}(x,y,v_2,w) ) \ \Rightarrow \ \mbox{equidistant}(z,v,v_2,w)\qquad\mbox{cnf}(\mbox{transitivity\_for\_equidistance},\mbox{axiom})$

$( \mbox{between}(x,w,v)\mbox{ and }\mbox{between}(y,v,z) ) \ \Rightarrow \ \mbox{between}(x,\mbox{outer\_pasch}(w,x,y,z,v),y)\qquad\mbox{cnf}(\mbox{outer\_pasch}_1,\mbox{axiom})$

$( \mbox{between}(x,w,v)\mbox{ and }\mbox{between}(y,v,z) ) \ \Rightarrow \ \mbox{between}(z,w,\mbox{outer\_pasch}(w,x,y,z,v))\qquad\mbox{cnf}(\mbox{outer\_pasch}_2,\mbox{axiom})$

$( \mbox{between}(x,v,w)\mbox{ and }\mbox{between}(y,v,z) ) \ \Rightarrow \ ( x = v\mbox{ or }\mbox{between}(x,z,\mbox{euclid}_1(w,x,y,z,v)) )\qquad\mbox{cnf}(\mbox{euclid}_1,\mbox{axiom})$

$( \mbox{between}(x,v,w)\mbox{ and }\mbox{between}(y,v,z) ) \ \Rightarrow \ ( x = v\mbox{ or }\mbox{between}(x,y,\mbox{euclid}_2(w,x,y,z,v)) )\qquad\mbox{cnf}(\mbox{euclid}_2,\mbox{axiom})$

$( \mbox{between}(x,v,w)\mbox{ and }\mbox{between}(y,v,z) ) \ \Rightarrow \ ( x = v\mbox{ or }\mbox{between}(\mbox{euclid}_1(w,x,y,z,v),w,\mbox{euclid}_2(w,x,y,z,v)) )\qquad\mbox{cnf}(\mbox{euclid}_3,\mbox{axiom})$

$( \mbox{equidistant}(x,y,x_1,y_1)\mbox{ and }\mbox{equidistant}(y,z,y_1,z_1)\mbox{ and }\mbox{equidistant}(x,v,x_1,v_1)\mbox{ and }\mbox{equidistant}(y,v,y_1,v_1)\mbox{ and }\mbox{between}(x,y,z)\mbox{ and }\mbox{between}(x_1,y_1,z_1) ) \ \Rightarrow \ ( x = y\mbox{ or }\mbox{equidistant}(z,v,z_1,v_1) )\qquad\mbox{cnf}(\mbox{outer\_five\_segment},\mbox{axiom})$

$\mbox{between}(x,y,\mbox{extension}(x,y,w,v))\qquad\mbox{cnf}(\mbox{segment\_construction}_1,\mbox{axiom})$

$\mbox{equidistant}(y,\mbox{extension}(x,y,w,v),w,v)\qquad\mbox{cnf}(\mbox{segment\_construction}_2,\mbox{axiom})$

$\neg\,\mbox{between}(\mbox{lower\_dimension\_point}_1,\mbox{lower\_dimension\_point}_2,\mbox{lower\_dimension\_point}_3)\qquad\mbox{cnf}(\mbox{lower\_dimension}_1,\mbox{axiom})$

$\neg\,\mbox{between}(\mbox{lower\_dimension\_point}_2,\mbox{lower\_dimension\_point}_3,\mbox{lower\_dimension\_point}_1)\qquad\mbox{cnf}(\mbox{lower\_dimension}_2,\mbox{axiom})$

$\neg\,\mbox{between}(\mbox{lower\_dimension\_point}_3,\mbox{lower\_dimension\_point}_1,\mbox{lower\_dimension\_point}_2)\qquad\mbox{cnf}(\mbox{lower\_dimension}_3,\mbox{axiom})$

$( \mbox{equidistant}(x,w,x,v)\mbox{ and }\mbox{equidistant}(y,w,y,v)\mbox{ and }\mbox{equidistant}(z,w,z,v) ) \ \Rightarrow \ ( \mbox{between}(x,y,z)\mbox{ or }\mbox{between}(y,z,x)\mbox{ or }\mbox{between}(z,x,y)\mbox{ or }w = v )\qquad\mbox{cnf}(\mbox{upper\_dimension},\mbox{axiom})$

$( \mbox{equidistant}(v,x,v,x_1)\mbox{ and }\mbox{equidistant}(v,z,v,z_1)\mbox{ and }\mbox{between}(v,x,z)\mbox{ and }\mbox{between}(x,y,z) ) \ \Rightarrow \ \mbox{equidistant}(v,y,v,\mbox{continuous}(x,y,z,x_1,z_1,v))\qquad\mbox{cnf}(\mbox{continuity}_1,\mbox{axiom})$

$( \mbox{equidistant}(v,x,v,x_1)\mbox{ and }\mbox{equidistant}(v,z,v,z_1)\mbox{ and }\mbox{between}(v,x,z)\mbox{ and }\mbox{between}(x,y,z) ) \ \Rightarrow \ \mbox{between}(x_1,\mbox{continuous}(x,y,z,x_1,z_1,v),z_1)\qquad\mbox{cnf}(\mbox{continuity}_2,\mbox{axiom})$

\medskip\textbf{GEO006+4.ax} Classical orthogonality






$\forall l,m{:}\;\neg\,\neg\,\mbox{convergent\_lines}(l,m)\mbox{ and }\neg\,\mbox{unorthogonal\_lines}(l,m)\qquad\mbox{fof}(\mbox{coipo}_1,\mbox{axiom})$

$\forall l,m,n{:}\;( ( ( \neg\,\mbox{convergent\_lines}(l,m)\mbox{ or }\neg\,\mbox{unorthogonal\_lines}(l,m) )\mbox{ and }( \neg\,\mbox{convergent\_lines}(l,n)\mbox{ or }\neg\,\mbox{unorthogonal\_lines}(l,n) ) ) \ \Rightarrow \ ( \neg\,\mbox{convergent\_lines}(m,n)\mbox{ or }\neg\,\mbox{unorthogonal\_lines}(m,n) ) )\qquad\mbox{fof}(\mbox{cotno}_1,\mbox{axiom})$

$\forall l,m,n{:}\;( ( \neg\,\mbox{unorthogonal\_lines}(l,m)\mbox{ and }\neg\,\mbox{unorthogonal\_lines}(l,n) ) \ \Rightarrow \ \neg\,\mbox{convergent\_lines}(m,n) )\qquad\mbox{fof}(\mbox{couo}_1,\mbox{axiom})$

\medskip\textbf{GEO004-1.ax} Betweenness for simple curves






$\mbox{between\_c}(a,b,c,d) \ \Rightarrow \ b\ne d\qquad\mbox{cnf}(\mbox{between\_c\_defn}_1,\mbox{axiom})$

$\mbox{between\_c}(a,b,c,d) \ \Rightarrow \ \mbox{part\_of}(\mbox{ax1\_sk}_1(d,c,b,a),a)\qquad\mbox{cnf}(\mbox{between\_c\_defn}_2,\mbox{axiom})$

$\mbox{between\_c}(a,b,c,d) \ \Rightarrow \ \mbox{end\_point}(b,\mbox{ax1\_sk}_1(d,c,b,a))\qquad\mbox{cnf}(\mbox{between\_c\_defn}_3,\mbox{axiom})$

$\mbox{between\_c}(a,b,c,d) \ \Rightarrow \ \mbox{end\_point}(d,\mbox{ax1\_sk}_1(d,c,b,a))\qquad\mbox{cnf}(\mbox{between\_c\_defn}_4,\mbox{axiom})$

$\mbox{between\_c}(a,b,c,d) \ \Rightarrow \ \mbox{inner\_point}(c,\mbox{ax1\_sk}_1(d,c,b,a))\qquad\mbox{cnf}(\mbox{between\_c\_defn}_5,\mbox{axiom})$

$( \mbox{part\_of}(c,d)\mbox{ and }\mbox{end\_point}(a,c)\mbox{ and }\mbox{end\_point}(b,c)\mbox{ and }\mbox{inner\_point}(e,c) ) \ \Rightarrow \ ( a = b\mbox{ or }\mbox{between\_c}(d,a,e,b) )\qquad\mbox{cnf}(\mbox{between\_c\_defn}_6,\mbox{axiom})$

\medskip\textbf{GEO006+6.ax} Geometry definitions






$\forall x,y{:}\;( \mbox{equal\_points}(x,y)\iff \neg\,\mbox{distinct\_points}(x,y) )\qquad\mbox{fof}(\mbox{ax}_1,\mbox{axiom})$

$\forall x,y{:}\;( \mbox{equal\_lines}(x,y)\iff \neg\,\mbox{distinct\_lines}(x,y) )\qquad\mbox{fof}(\mbox{ax}_2,\mbox{axiom})$

$\forall x,y{:}\;( \mbox{parallel\_lines}(x,y)\iff \neg\,\mbox{convergent\_lines}(x,y) )\qquad\mbox{fof}(a_3,\mbox{axiom})$

$\forall x,y{:}\;( \mbox{incident\_point\_and\_line}(x,y)\iff \neg\,\mbox{apart\_point\_and\_line}(x,y) )\qquad\mbox{fof}(a_4,\mbox{axiom})$

$\forall x,y{:}\;( \mbox{orthogonal\_lines}(x,y)\iff \neg\,\mbox{unorthogonal\_lines}(x,y) )\qquad\mbox{fof}(a_5,\mbox{axiom})$

\medskip\textbf{GEO007-2.p} Betweenness for 4 points on a line

For all pairs of distinct points y and z, if w and x are
 on the line yz to the left of y (i.e. not between y and z),
 then either w is betweeen x and y or x is between w and y.




$\mbox{include}(\mbox{'Axioms/GEO002-0.ax'})$

$a\ne c\qquad\mbox{cnf}(\mbox{a\_not\_c},\mbox{hypothesis})$

$\mbox{between}(d,a,c)\qquad\mbox{cnf}(\mbox{a\_between\_a\_and\_c},\mbox{hypothesis})$

$\mbox{between}(e,a,c)\qquad\mbox{cnf}(\mbox{a\_between\_a\_and\_e},\mbox{hypothesis})$

$\neg\,\mbox{between}(d,e,a)\qquad\mbox{cnf}(\mbox{e\_not\_between\_d\_and\_a},\mbox{hypothesis})$

$\neg\,\mbox{between}(e,d,a)\qquad\mbox{cnf}(\mbox{prove\_d\_between\_e\_and\_a},\mbox{negated\_conjecture})$

\medskip\textbf{GEO010-1.p} Collinearity is invariant

For all points x, y, and z, if x, y, and z are collinear
 in one order, they are collinear in any order.




$\mbox{include}(\mbox{'Axioms/GEO001-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO001-1.ax'})$

$\mbox{colinear}(a,b,c)\qquad\mbox{cnf}(\mbox{abc\_colinear},\mbox{hypothesis})$

$( \mbox{colinear}(a,c,b)\mbox{ and }\mbox{colinear}(b,a,c)\mbox{ and }\mbox{colinear}(b,c,a)\mbox{ and }\mbox{colinear}(c,a,b) ) \ \Rightarrow \ \neg\,\mbox{colinear}(c,b,a)\qquad\mbox{cnf}(\mbox{prove\_colinear\_in\_all\_orders},\mbox{negated\_conjecture})$

\medskip\textbf{GEO232+3.p} A line is not oppositely directed to itself






$\mbox{include}(\mbox{'Axioms/GEO009+0.ax'})$

$\forall l{:}\;\mbox{unequally\_directed\_opposite\_lines}(l,l)\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO026-3.p} Extension is unique






$\mbox{include}(\mbox{'Axioms/GEO002-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO002-2.ax'})$

$\mbox{equidistant}(u,v,u,v)\qquad\mbox{cnf}(d_1,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(w,x,u,v)\qquad\mbox{cnf}(d_2,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(v,u,w,x)\qquad\mbox{cnf}(d_3,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(u,v,x,w)\qquad\mbox{cnf}(\mbox{d4}_1,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(v,u,x,w)\qquad\mbox{cnf}(\mbox{d4}_2,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(w,x,v,u)\qquad\mbox{cnf}(\mbox{d4}_3,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(x,w,u,v)\qquad\mbox{cnf}(\mbox{d4}_4,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(x,w,v,u)\qquad\mbox{cnf}(\mbox{d4}_5,\mbox{axiom})$

$( \mbox{equidistant}(u,v,w,x)\mbox{ and }\mbox{equidistant}(w,x,y,z) ) \ \Rightarrow \ \mbox{equidistant}(u,v,y,z)\qquad\mbox{cnf}(d_5,\mbox{axiom})$

$v = \mbox{extension}(u,v,w,w)\qquad\mbox{cnf}(e_1,\mbox{axiom})$

$y = \mbox{extension}(u,v,w,x) \ \Rightarrow \ \mbox{between}(u,v,y)\qquad\mbox{cnf}(b_0,\mbox{axiom})$

$\mbox{between}(u,v,\mbox{reflection}(u,v))\qquad\mbox{cnf}(\mbox{r2}_1,\mbox{axiom})$

$\mbox{equidistant}(v,\mbox{reflection}(u,v),u,v)\qquad\mbox{cnf}(\mbox{r2}_2,\mbox{axiom})$

$u = v \ \Rightarrow \ v = \mbox{reflection}(u,v)\qquad\mbox{cnf}(\mbox{r3}_1,\mbox{axiom})$

$u = \mbox{reflection}(u,u)\qquad\mbox{cnf}(\mbox{r3}_2,\mbox{axiom})$

$v = \mbox{reflection}(u,v) \ \Rightarrow \ u = v\qquad\mbox{cnf}(r_4,\mbox{axiom})$

$\mbox{equidistant}(u,u,v,v)\qquad\mbox{cnf}(d_7,\mbox{axiom})$

$( \mbox{equidistant}(u,v,u_1,v_1)\mbox{ and }\mbox{equidistant}(v,w,v_1,w_1)\mbox{ and }\mbox{between}(u,v,w)\mbox{ and }\mbox{between}(u_1,v_1,w_1) ) \ \Rightarrow \ \mbox{equidistant}(u,w,u_1,w_1)\qquad\mbox{cnf}(d_8,\mbox{axiom})$

$\mbox{between}(u,v,w)\qquad\mbox{cnf}(\mbox{v\_between\_u\_and\_w},\mbox{hypothesis})$

$\mbox{between}(u,v,x)\qquad\mbox{cnf}(\mbox{v\_between\_u\_and\_x},\mbox{hypothesis})$

$\mbox{equidistant}(v,w,v,x)\qquad\mbox{cnf}(\mbox{v\_to\_w\_equals\_v\_to\_x},\mbox{hypothesis})$

$u\ne v\qquad\mbox{cnf}(\mbox{u\_not\_v},\mbox{hypothesis})$

$w\ne x\qquad\mbox{cnf}(\mbox{prove\_w\_is\_x},\mbox{negated\_conjecture})$

\medskip\textbf{GEO146-1.p} Symmetry of connect






$\mbox{include}(\mbox{'Axioms/GEO004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO004-2.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO004-3.ax'})$

$\mbox{connect}(\mbox{sk}_{27},\mbox{sk}_{28},\mbox{sk}_{29})\mbox{ or }\mbox{connect}(\mbox{sk}_{28},\mbox{sk}_{27},\mbox{sk}_{29})\qquad\mbox{cnf}(\mbox{t12}_{156},\mbox{negated\_conjecture})$

$\mbox{connect}(\mbox{sk}_{27},\mbox{sk}_{28},\mbox{sk}_{29}) \ \Rightarrow \ \mbox{connect}(\mbox{sk}_{27},\mbox{sk}_{28},\mbox{sk}_{29})\qquad\mbox{cnf}(\mbox{t12}_{157},\mbox{negated\_conjecture})$

$\mbox{connect}(\mbox{sk}_{28},\mbox{sk}_{27},\mbox{sk}_{29}) \ \Rightarrow \ \mbox{connect}(\mbox{sk}_{28},\mbox{sk}_{27},\mbox{sk}_{29})\qquad\mbox{cnf}(\mbox{t12}_{158},\mbox{negated\_conjecture})$

$\mbox{connect}(\mbox{sk}_{28},\mbox{sk}_{27},\mbox{sk}_{29}) \ \Rightarrow \ \neg\,\mbox{connect}(\mbox{sk}_{27},\mbox{sk}_{28},\mbox{sk}_{29})\qquad\mbox{cnf}(\mbox{t12}_{159},\mbox{negated\_conjecture})$

\medskip\textbf{GEO191+2.p} Symmetry of apartness

If the lines X and Y are convergent, U and V are convergent,
 and the intersection point of X and Y is apart from U and V,
 then the intersection point of U and V is apart from X and Y.




$\mbox{include}(\mbox{'Axioms/GEO008+0.ax'})$

$\forall x,y,u,v{:}\;( ( \mbox{convergent\_lines}(x,y)\mbox{ and }\mbox{convergent\_lines}(u,v)\mbox{ and }( \mbox{apart\_point\_and\_line}(\mbox{intersection\_point}(x,y),u)\mbox{ or }\mbox{apart\_point\_and\_line}(\mbox{intersection\_point}(x,y),v) ) ) \ \Rightarrow \ ( \mbox{apart\_point\_and\_line}(\mbox{intersection\_point}(u,v),x)\mbox{ or }\mbox{apart\_point\_and\_line}(\mbox{intersection\_point}(u,v),y) ) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO019-3.p} Corollary 3 to symmetries of equidistance

Show that if the distance from A to B equals the distance
 from C to D, then the distance from C to D equals the
 distance from B to A.




$\mbox{include}(\mbox{'Axioms/GEO002-0.ax'})$

$\mbox{equidistant}(u,v,u,v)\qquad\mbox{cnf}(d_1,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(w,x,u,v)\qquad\mbox{cnf}(d_2,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(v,u,w,x)\qquad\mbox{cnf}(d_3,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x)\qquad\mbox{cnf}(\mbox{u\_to\_v\_equals\_w\_to\_x},\mbox{hypothesis})$

$\neg\,\mbox{equidistant}(w,x,v,u)\qquad\mbox{cnf}(\mbox{prove\_symmetry},\mbox{negated\_conjecture})$

\medskip\textbf{GEO180+3.p} Triangle axiom 1

If X and Y are distinct points and Z is apart from the line
 connecting X and Y, then X is apart from the line connecting
 Z and Y.




$\mbox{include}(\mbox{'Axioms/GEO006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+1.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+2.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+3.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+4.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+5.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+6.ax'})$

$\forall x,y,z{:}\;( \mbox{distinct\_points}(x,y) \ \Rightarrow \ ( \mbox{apart\_point\_and\_line}(z,\mbox{line\_connecting}(x,y)) \ \Rightarrow \ \mbox{apart\_point\_and\_line}(x,\mbox{line\_connecting}(z,y)) ) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO181+1.p} Triangle axiom 2

If X and Y are distinct points and Z is apart from the line
 connecting X and Y, then Y is apart from the line connecting
 X and Z.




$\mbox{include}(\mbox{'Axioms/GEO006+0.ax'})$

$\forall x,y,z{:}\;( \mbox{distinct\_points}(x,y) \ \Rightarrow \ ( \mbox{apart\_point\_and\_line}(z,\mbox{line\_connecting}(x,y)) \ \Rightarrow \ \mbox{apart\_point\_and\_line}(y,\mbox{line\_connecting}(x,z)) ) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO214+2.p} Corollary to non-orthogonal lines and a third line

If the line L is not orthogonal to M, then M is orthogonal to
 L.




$\mbox{include}(\mbox{'Axioms/GEO008+0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+2.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+3.ax'})$

$\forall l,m{:}\;( \mbox{unorthogonal\_lines}(l,m) \ \Rightarrow \ \mbox{unorthogonal\_lines}(m,l) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO084-1.p} Sum is monotone, part 2






$\mbox{include}(\mbox{'Axioms/GEO004-0.ax'})$

$\mbox{part\_of}(\mbox{sk}_{14},\mbox{sk}_{16})\qquad\mbox{cnf}(\mbox{corollary\_2\_6\_2}_{67},\mbox{negated\_conjecture})$

$\mbox{part\_of}(\mbox{sk}_{15},\mbox{sk}_{16})\qquad\mbox{cnf}(\mbox{corollary\_2\_6\_2}_{68},\mbox{negated\_conjecture})$

$\mbox{sk}_{17}\wedge \mbox{sk}_{14}{=}\mbox{sk}_{15}\qquad\mbox{cnf}(\mbox{corollary\_2\_6\_2}_{69},\mbox{negated\_conjecture})$

$\neg\,\mbox{part\_of}(\mbox{sk}_{14} + \mbox{sk}_{15},\mbox{sk}_{16})\qquad\mbox{cnf}(\mbox{corollary\_2\_6\_2}_{70},\mbox{negated\_conjecture})$

\medskip\textbf{GEO203+1.p} Equal lines from points

If the lines X and Y are convergent, and X and Z are convergent,
 the intersection point of X and Y, and the intersection point
 of X and Z are distinct, then the line connecting these points
 is equal to X.




$\mbox{include}(\mbox{'Axioms/GEO006+0.ax'})$

$\forall x,y,z{:}\;( ( \mbox{convergent\_lines}(x,y)\mbox{ and }\mbox{convergent\_lines}(x,z)\mbox{ and }\mbox{distinct\_points}(\mbox{intersection\_point}(x,y),\mbox{intersection\_point}(x,z)) ) \ \Rightarrow \ \neg\,\mbox{distinct\_lines}(\mbox{line\_connecting}(\mbox{intersection\_point}(x,y),\mbox{intersection\_point}(x,z)),x) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO230+3.p} Reversed lines are equal and conversely directed






$\mbox{include}(\mbox{'Axioms/GEO009+0.ax'})$

$\forall a,b{:}\;( \mbox{equal\_lines}(\mbox{line\_connecting}(a,b),\mbox{line\_connecting}(b,a))\mbox{ and }\mbox{equally\_directed\_lines}(\mbox{line\_connecting}(b,a),\mbox{reverse\_line}(\mbox{line\_connecting}(a,b))) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO227+1.p} Lines not directed and opposite






$\mbox{include}(\mbox{'Axioms/GEO007+0.ax'})$

$\forall l,m{:}\;( ( \mbox{line}(l)\mbox{ and }\mbox{line}(m) ) \ \Rightarrow \ \neg\,\neg\,\mbox{unequally\_directed\_lines}(l,m)\mbox{ and }\neg\,\mbox{unequally\_directed\_lines}(l,\mbox{reverse\_line}(m)) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO086+1.p} Every sub-curve of an open curve is open






$\mbox{include}(\mbox{'Axioms/GEO004+0.ax'})$

$\forall c,\mbox{cpp}{:}\;( ( \mbox{open}(c)\mbox{ and }\mbox{part\_of}(\mbox{cpp},c) ) \ \Rightarrow \ \mbox{open}(\mbox{cpp}) )\qquad\mbox{fof}(\mbox{theorem\_2\_7}_2,\mbox{conjecture})$

\medskip\textbf{GEO092+1.p} Common point of open sum is the meeting point

If two curves meet and their sum is open, then the only point
 they have in common is their meeting-point.




$\mbox{include}(\mbox{'Axioms/GEO004+0.ax'})$

$\forall c_1,c_2,p{:}\;( ( p\wedge c_1{=}c_2\mbox{ and }\mbox{open}(c_1 + c_2) ) \ \Rightarrow \ \forall q{:}\;( q\ne p \ \Rightarrow \ \neg\,\mbox{incident\_c}(q,c_1)\mbox{ and }\mbox{incident\_c}(q,c_2) ) )\qquad\mbox{fof}(\mbox{proposition\_2\_14}_1,\mbox{conjecture})$

\medskip\textbf{GEO200+3.p} Line equals its converse

If the points X and Y are distinct, then the line connecting
 X and Y is equal to the line connecting Y and X.




$\mbox{include}(\mbox{'Axioms/GEO006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+1.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+2.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+3.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+4.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+5.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+6.ax'})$

$\forall x,y{:}\;( \mbox{distinct\_points}(x,y) \ \Rightarrow \ \mbox{equal\_lines}(\mbox{line\_connecting}(x,y),\mbox{line\_connecting}(y,x)) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO143-1.p} Unique oppositely oriented curve 4

For every oriented line there is exactly one uniquely determined
 oriented line with the same underlying curve that orders the
 points in the opposite way.




$\mbox{include}(\mbox{'Axioms/GEO004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO004-2.ax'})$

$\mbox{ordered\_by}(\mbox{sk}_{25},\mbox{sk}_{26},\mbox{sk}_{27})\qquad\mbox{cnf}(\mbox{theorem\_4\_23\_4}_{133},\mbox{negated\_conjecture})$

$\mbox{ordered\_by}(\mbox{sk}_{25},\mbox{sk}_{27},\mbox{sk}_{28})\mbox{ or }\mbox{between}(\mbox{sk}_{25},\mbox{sk}_{26},\mbox{sk}_{27},\mbox{sk}_{28})\qquad\mbox{cnf}(\mbox{theorem\_4\_23\_4}_{134},\mbox{negated\_conjecture})$

$\mbox{ordered\_by}(\mbox{sk}_{25},\mbox{sk}_{27},\mbox{sk}_{28}) \ \Rightarrow \ \mbox{ordered\_by}(\mbox{sk}_{25},\mbox{sk}_{27},\mbox{sk}_{28})\qquad\mbox{cnf}(\mbox{theorem\_4\_23\_4}_{135},\mbox{negated\_conjecture})$

$\mbox{between}(\mbox{sk}_{25},\mbox{sk}_{26},\mbox{sk}_{27},\mbox{sk}_{28}) \ \Rightarrow \ \mbox{between}(\mbox{sk}_{25},\mbox{sk}_{26},\mbox{sk}_{27},\mbox{sk}_{28})\qquad\mbox{cnf}(\mbox{theorem\_4\_23\_4}_{136},\mbox{negated\_conjecture})$

$\mbox{between}(\mbox{sk}_{25},\mbox{sk}_{26},\mbox{sk}_{27},\mbox{sk}_{28}) \ \Rightarrow \ \neg\,\mbox{ordered\_by}(\mbox{sk}_{25},\mbox{sk}_{27},\mbox{sk}_{28})\qquad\mbox{cnf}(\mbox{theorem\_4\_23\_4}_{137},\mbox{negated\_conjecture})$

\medskip\textbf{GEO138+1.p} Curve and ordered points determine oriented curve

A curve and a ordered pair of points uniquely determine an
 oriented curve.




$\mbox{include}(\mbox{'Axioms/GEO004+0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO004+1.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO004+2.ax'})$

$\forall o_1,o_2{:}\;( ( \mbox{underlying\_curve}(o_1) = \mbox{underlying\_curve}(o_2)\mbox{ and }\exists p,q{:}\;( \mbox{ordered\_by}(o_1,p,q)\mbox{ and }\mbox{ordered\_by}(o_2,p,q) ) ) \ \Rightarrow \ o_1 = o_2 )\qquad\mbox{fof}(\mbox{corollary\_4}_{21},\mbox{conjecture})$

\medskip\textbf{GEO093-1.p} Sum of meeting open curves is open

If two open sub-curves of an open curve meet, then their sum
 is also open.




$\mbox{include}(\mbox{'Axioms/GEO004-0.ax'})$

$\mbox{open}(\mbox{sk}_{14})\qquad\mbox{cnf}(\mbox{proposition\_2\_14\_2}_{67},\mbox{negated\_conjecture})$

$\mbox{part\_of}(\mbox{sk}_{15},\mbox{sk}_{14})\qquad\mbox{cnf}(\mbox{proposition\_2\_14\_2}_{68},\mbox{negated\_conjecture})$

$\mbox{part\_of}(\mbox{sk}_{16},\mbox{sk}_{14})\qquad\mbox{cnf}(\mbox{proposition\_2\_14\_2}_{69},\mbox{negated\_conjecture})$

$\mbox{sk}_{17}\wedge \mbox{sk}_{15}{=}\mbox{sk}_{16}\qquad\mbox{cnf}(\mbox{proposition\_2\_14\_2}_{70},\mbox{negated\_conjecture})$

$\neg\,\mbox{open}(\mbox{sk}_{15} + \mbox{sk}_{16})\qquad\mbox{cnf}(\mbox{proposition\_2\_14\_2}_{71},\mbox{negated\_conjecture})$

\medskip\textbf{GEO193+3.p} Corollary to symmetry of apartness

If the lines X, Y, and Z are pairwise convergent, and the
 intersection point of X and Y is apart from a line Z, then
 the intersection point of Y and Z is apart from X.




$\mbox{include}(\mbox{'Axioms/GEO006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+1.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+2.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+3.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+4.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+5.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+6.ax'})$

$\forall x,y,z{:}\;( ( \mbox{convergent\_lines}(x,y)\mbox{ and }\mbox{convergent\_lines}(z,y)\mbox{ and }\mbox{convergent\_lines}(x,z) ) \ \Rightarrow \ ( \mbox{apart\_point\_and\_line}(\mbox{intersection\_point}(x,y),z) \ \Rightarrow \ \mbox{apart\_point\_and\_line}(\mbox{intersection\_point}(z,y),x) ) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO225+2.p} Existence of line joining distinct points

Assume orthogonal geometry. Given a point and a line, to find
 a point incident with the line.




$\mbox{include}(\mbox{'Axioms/GEO008+0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+5.ax'})$

$\forall a,b{:}\;( ( \mbox{point}(a)\mbox{ and }\mbox{point}(b)\mbox{ and }\mbox{distinct\_points}(a,b) ) \ \Rightarrow \ \exists x{:}\;( \mbox{line}(x) \ \Rightarrow \ ( \neg\,\mbox{apart\_point\_and\_line}(a,x)\mbox{ and }\neg\,\mbox{apart\_point\_and\_line}(b,x) ) ) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO129-1.p} Precedence on an oriented curve is a transitive relation






$\mbox{include}(\mbox{'Axioms/GEO004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO004-2.ax'})$

$\mbox{ordered\_by}(\mbox{sk}_{25},\mbox{sk}_{26},\mbox{sk}_{27})\qquad\mbox{cnf}(\mbox{theorem\_4\_14}_{133},\mbox{negated\_conjecture})$

$\mbox{ordered\_by}(\mbox{sk}_{25},\mbox{sk}_{27},\mbox{sk}_{28})\qquad\mbox{cnf}(\mbox{theorem\_4\_14}_{134},\mbox{negated\_conjecture})$

$\neg\,\mbox{ordered\_by}(\mbox{sk}_{25},\mbox{sk}_{26},\mbox{sk}_{28})\qquad\mbox{cnf}(\mbox{theorem\_4\_14}_{135},\mbox{negated\_conjecture})$

\medskip\textbf{GEO244+3.p} Configurations in terms of apartness






$\mbox{include}(\mbox{'Axioms/GEO009+0.ax'})$

$\forall a,b,c{:}\;( \mbox{distinct\_points}(a,b) \ \Rightarrow \ ( \mbox{right\_apart\_point}(c,\mbox{line\_connecting}(a,b)) \ \Rightarrow \ \mbox{left\_apart\_point}(c,\mbox{line\_connecting}(b,a)) ) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO112-1.p} Basic property of orderings on linear structures 2

If Q is between P and R wrt. c, then Q is between R and P wrt. c




$\mbox{include}(\mbox{'Axioms/GEO004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO004-1.ax'})$

$\mbox{between\_c}(\mbox{sk}_{15},\mbox{sk}_{16},\mbox{sk}_{17},\mbox{sk}_{18})\qquad\mbox{cnf}(\mbox{theorem\_3\_8\_2}_{77},\mbox{negated\_conjecture})$

$\neg\,\mbox{between\_c}(\mbox{sk}_{15},\mbox{sk}_{18},\mbox{sk}_{17},\mbox{sk}_{16})\qquad\mbox{cnf}(\mbox{theorem\_3\_8\_2}_{78},\mbox{negated\_conjecture})$

\medskip\textbf{GEO125-1.p} Every oriented curve has at most one finishing point






$\mbox{include}(\mbox{'Axioms/GEO004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO004-2.ax'})$

$\mbox{finish\_point}(\mbox{sk}_{26},\mbox{sk}_{25})\qquad\mbox{cnf}(\mbox{corollary\_4\_10\_2}_{133},\mbox{negated\_conjecture})$

$\mbox{finish\_point}(\mbox{sk}_{27},\mbox{sk}_{25})\qquad\mbox{cnf}(\mbox{corollary\_4\_10\_2}_{134},\mbox{negated\_conjecture})$

$\mbox{sk}_{26}\ne \mbox{sk}_{27}\qquad\mbox{cnf}(\mbox{corollary\_4\_10\_2}_{135},\mbox{negated\_conjecture})$

\medskip\textbf{GEO086-1.p} Every sub-curve of an open curve is open






$\mbox{include}(\mbox{'Axioms/GEO004-0.ax'})$

$\mbox{open}(\mbox{sk}_{14})\qquad\mbox{cnf}(\mbox{theorem\_2\_7\_2}_{67},\mbox{negated\_conjecture})$

$\mbox{part\_of}(\mbox{sk}_{15},\mbox{sk}_{14})\qquad\mbox{cnf}(\mbox{theorem\_2\_7\_2}_{68},\mbox{negated\_conjecture})$

$\neg\,\mbox{open}(\mbox{sk}_{15})\qquad\mbox{cnf}(\mbox{theorem\_2\_7\_2}_{69},\mbox{negated\_conjecture})$

\medskip\textbf{GEO222+3.p} Parallel to orthogonal to orthogonal

A line L is parallel to the line, that is orthogonal to the
 orthogonal to L through A, and goes through A as well.




$\mbox{include}(\mbox{'Axioms/GEO006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+1.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+2.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+3.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+4.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+5.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+6.ax'})$

$\forall a,l{:}\;\mbox{parallel\_lines}(l,\mbox{orthogonal\_through\_point}(\mbox{orthogonal\_through\_point}(l,a),a))\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO251+3.p} A point in each region formed by parallel lines






$\mbox{include}(\mbox{'Axioms/GEO009+0.ax'})$

$\forall a,b,l{:}\;( \mbox{right\_apart\_point}(b,\mbox{parallel\_through\_point}(l,a)) \ \Rightarrow \ \mbox{left\_apart\_point}(a,\mbox{parallel\_through\_point}(l,b)) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO209+2.p} Pont and three parallel lines

If the point A is apart from the line L, but incident with the
 lines M and N, and L is parallel to M and N, then M and N are
 equal.




$\mbox{include}(\mbox{'Axioms/GEO008+0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+2.ax'})$

$\forall a,l,m,n{:}\;( ( \mbox{apart\_point\_and\_line}(a,l)\mbox{ and }\neg\,\mbox{apart\_point\_and\_line}(a,m)\mbox{ and }\neg\,\mbox{apart\_point\_and\_line}(a,n)\mbox{ and }\neg\,\mbox{convergent\_lines}(m,l)\mbox{ and }\neg\,\mbox{convergent\_lines}(n,l) ) \ \Rightarrow \ \neg\,\mbox{distinct\_lines}(m,n) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO146+1.p} Symmetry of connect






$\mbox{include}(\mbox{'Axioms/GEO004+0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO004+1.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO004+2.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO004+3.ax'})$

$\forall x,y,p{:}\;( \mbox{connect}(x,y,p)\iff \mbox{connect}(y,x,p) )\qquad\mbox{fof}(t_{12},\mbox{conjecture})$

\medskip\textbf{GEO009-2.p} First inner connectivity property of betweenness

For all points x, y, z, and w, if y and w are between x and
 z, then either y is between x and w or w is between x and y.




$\mbox{include}(\mbox{'Axioms/GEO002-0.ax'})$

$\mbox{between}(a,c,e)\qquad\mbox{cnf}(\mbox{c\_between\_a\_and\_e},\mbox{hypothesis})$

$\mbox{between}(a,d,e)\qquad\mbox{cnf}(\mbox{d\_between\_a\_and\_e},\mbox{hypothesis})$

$\neg\,\mbox{between}(a,c,d)\qquad\mbox{cnf}(\mbox{c\_not\_between\_a\_and\_d},\mbox{hypothesis})$

$\neg\,\mbox{between}(a,d,c)\qquad\mbox{cnf}(\mbox{prove\_d\_between\_a\_and\_c},\mbox{negated\_conjecture})$

\medskip\textbf{GEO022-3.p} Ordinary transitivity of equidistance

This form of transitivity is different from that expressed
 in the axioms.




$\mbox{include}(\mbox{'Axioms/GEO002-0.ax'})$

$\mbox{equidistant}(u,v,u,v)\qquad\mbox{cnf}(d_1,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(w,x,u,v)\qquad\mbox{cnf}(d_2,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(v,u,w,x)\qquad\mbox{cnf}(d_3,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(u,v,x,w)\qquad\mbox{cnf}(\mbox{d4}_1,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(v,u,x,w)\qquad\mbox{cnf}(\mbox{d4}_2,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(w,x,v,u)\qquad\mbox{cnf}(\mbox{d4}_3,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(x,w,u,v)\qquad\mbox{cnf}(\mbox{d4}_4,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(x,w,v,u)\qquad\mbox{cnf}(\mbox{d4}_5,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x)\qquad\mbox{cnf}(\mbox{u\_to\_v\_equals\_w\_to\_x},\mbox{hypothesis})$

$\mbox{equidistant}(w,x,y,z)\qquad\mbox{cnf}(\mbox{w\_to\_x\_equals\_y\_to\_z},\mbox{hypothesis})$

$\neg\,\mbox{equidistant}(u,v,y,z)\qquad\mbox{cnf}(\mbox{prove\_transitivity},\mbox{negated\_conjecture})$

\medskip\textbf{GEO052-2.p} Second inner connectivity property of betweenness






$\mbox{include}(\mbox{'Axioms/GEO002-0.ax'})$

$\mbox{between}(u,v,x)\qquad\mbox{cnf}(\mbox{v\_between\_u\_and\_x},\mbox{hypothesis})$

$\mbox{between}(u,w,x)\qquad\mbox{cnf}(\mbox{w\_between\_u\_and\_x},\mbox{hypothesis})$

$\neg\,\mbox{between}(v,w,x)\qquad\mbox{cnf}(\mbox{w\_not\_between\_v\_and\_x},\mbox{hypothesis})$

$\neg\,\mbox{between}(w,v,x)\qquad\mbox{cnf}(\mbox{prove\_v\_between\_w\_and\_x},\mbox{negated\_conjecture})$

\medskip\textbf{GEO193+2.p} Corollary to symmetry of apartness

If the lines X, Y, and Z are pairwise convergent, and the
 intersection point of X and Y is apart from a line Z, then
 the intersection point of Y and Z is apart from X.




$\mbox{include}(\mbox{'Axioms/GEO008+0.ax'})$

$\forall x,y,z{:}\;( ( \mbox{convergent\_lines}(x,y)\mbox{ and }\mbox{convergent\_lines}(z,y)\mbox{ and }\mbox{convergent\_lines}(x,z) ) \ \Rightarrow \ ( \mbox{apart\_point\_and\_line}(\mbox{intersection\_point}(x,y),z) \ \Rightarrow \ \mbox{apart\_point\_and\_line}(\mbox{intersection\_point}(z,y),x) ) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO058-3.p} U is the only fixed point of reflection(U,V)






$\mbox{include}(\mbox{'Axioms/GEO002-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO002-2.ax'})$

$\mbox{equidistant}(u,v,u,v)\qquad\mbox{cnf}(d_1,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(w,x,u,v)\qquad\mbox{cnf}(d_2,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(v,u,w,x)\qquad\mbox{cnf}(d_3,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(u,v,x,w)\qquad\mbox{cnf}(\mbox{d4}_1,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(v,u,x,w)\qquad\mbox{cnf}(\mbox{d4}_2,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(w,x,v,u)\qquad\mbox{cnf}(\mbox{d4}_3,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(x,w,u,v)\qquad\mbox{cnf}(\mbox{d4}_4,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(x,w,v,u)\qquad\mbox{cnf}(\mbox{d4}_5,\mbox{axiom})$

$( \mbox{equidistant}(u,v,w,x)\mbox{ and }\mbox{equidistant}(w,x,y,z) ) \ \Rightarrow \ \mbox{equidistant}(u,v,y,z)\qquad\mbox{cnf}(d_5,\mbox{axiom})$

$v = \mbox{extension}(u,v,w,w)\qquad\mbox{cnf}(e_1,\mbox{axiom})$

$y = \mbox{extension}(u,v,w,x) \ \Rightarrow \ \mbox{between}(u,v,y)\qquad\mbox{cnf}(b_0,\mbox{axiom})$

$\mbox{between}(u,v,\mbox{reflection}(u,v))\qquad\mbox{cnf}(\mbox{r2}_1,\mbox{axiom})$

$\mbox{equidistant}(v,\mbox{reflection}(u,v),u,v)\qquad\mbox{cnf}(\mbox{r2}_2,\mbox{axiom})$

$u = v \ \Rightarrow \ v = \mbox{reflection}(u,v)\qquad\mbox{cnf}(\mbox{r3}_1,\mbox{axiom})$

$u = \mbox{reflection}(u,u)\qquad\mbox{cnf}(\mbox{r3}_2,\mbox{axiom})$

$v = \mbox{reflection}(u,v)\qquad\mbox{cnf}(\mbox{v\_equals\_reflection},\mbox{hypothesis})$

$u\ne v\qquad\mbox{cnf}(\mbox{prove\_u\_equals\_v},\mbox{negated\_conjecture})$

\medskip\textbf{GEO134+1.p} Betweenness and precedence property 3






$\mbox{include}(\mbox{'Axioms/GEO004+0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO004+1.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO004+2.ax'})$

$\forall o,p,q,r,s{:}\;( ( \mbox{ordered\_by}(o,p,q)\mbox{ and }p\ne r\mbox{ and }\neg\,\mbox{between}(o,r,p,q) ) \ \Rightarrow \ ( \mbox{ordered\_by}(o,r,s)\iff \mbox{between}(o,p,r,s) ) )\qquad\mbox{fof}(\mbox{theorem\_4\_17}_3,\mbox{conjecture})$

\medskip\textbf{GEO210+3.p} Uniqueness of orthogonality

If the point A is incident with line L, and the line L is
 orthogonal to M, then L is equal to the orthogonal to M
 through A.




$\mbox{include}(\mbox{'Axioms/GEO006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+1.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+2.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+3.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+4.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+5.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+6.ax'})$

$\forall a,l,m{:}\;( ( \mbox{incident\_point\_and\_line}(a,l)\mbox{ and }\mbox{orthogonal\_lines}(l,m) ) \ \Rightarrow \ \mbox{equal\_lines}(l,\mbox{orthogonal\_through\_point}(m,a)) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO068-2.p} Theorem of similar situations for collinear U, V, W






$\mbox{include}(\mbox{'Axioms/GEO002-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO002-1.ax'})$

$\mbox{equidistant}(u,v,u_1,v_1)\qquad\mbox{cnf}(\mbox{u\_to\_v\_equals\_u1\_to\_v}_1,\mbox{hypothesis})$

$\mbox{equidistant}(v,w,v_1,w_1)\qquad\mbox{cnf}(\mbox{v\_to\_w\_equals\_v1\_to\_w}_1,\mbox{hypothesis})$

$\mbox{equidistant}(u,w,u_1,w_1)\qquad\mbox{cnf}(\mbox{u\_to\_w\_equals\_u1\_to\_w}_1,\mbox{hypothesis})$

$\mbox{colinear}(u,v,w)\qquad\mbox{cnf}(\mbox{uvw\_colinear},\mbox{hypothesis})$

$\neg\,\mbox{colinear}(u_1,v_1,w_1)\qquad\mbox{cnf}(\mbox{prove\_u1v1w1\_colinear},\mbox{negated\_conjecture})$

\medskip\textbf{GEO042-2.p} First inner transitivity property of betweenness






$\mbox{include}(\mbox{'Axioms/GEO002-0.ax'})$

$\mbox{between}(u,v,x)\qquad\mbox{cnf}(\mbox{v\_between\_u\_and\_x},\mbox{hypothesis})$

$\mbox{between}(v,w,x)\qquad\mbox{cnf}(\mbox{w\_between\_v\_and\_x},\mbox{hypothesis})$

$\neg\,\mbox{between}(u,v,w)\qquad\mbox{cnf}(\mbox{prove\_v\_between\_u\_and\_w},\mbox{negated\_conjecture})$

\medskip\textbf{GEO191+1.p} Symmetry of apartness

If the lines X and Y are convergent, U and V are convergent,
 and the intersection point of X and Y is apart from U and V,
 then the intersection point of U and V is apart from X and Y.




$\mbox{include}(\mbox{'Axioms/GEO006+0.ax'})$

$\forall x,y,u,v{:}\;( ( \mbox{convergent\_lines}(x,y)\mbox{ and }\mbox{convergent\_lines}(u,v)\mbox{ and }( \mbox{apart\_point\_and\_line}(\mbox{intersection\_point}(x,y),u)\mbox{ or }\mbox{apart\_point\_and\_line}(\mbox{intersection\_point}(x,y),v) ) ) \ \Rightarrow \ ( \mbox{apart\_point\_and\_line}(\mbox{intersection\_point}(u,v),x)\mbox{ or }\mbox{apart\_point\_and\_line}(\mbox{intersection\_point}(u,v),y) ) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO028-3.p} Corollary 2 to unique extension






$\mbox{include}(\mbox{'Axioms/GEO002-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO002-2.ax'})$

$\mbox{equidistant}(u,v,u,v)\qquad\mbox{cnf}(d_1,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(w,x,u,v)\qquad\mbox{cnf}(d_2,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(v,u,w,x)\qquad\mbox{cnf}(d_3,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(u,v,x,w)\qquad\mbox{cnf}(\mbox{d4}_1,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(v,u,x,w)\qquad\mbox{cnf}(\mbox{d4}_2,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(w,x,v,u)\qquad\mbox{cnf}(\mbox{d4}_3,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(x,w,u,v)\qquad\mbox{cnf}(\mbox{d4}_4,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(x,w,v,u)\qquad\mbox{cnf}(\mbox{d4}_5,\mbox{axiom})$

$( \mbox{equidistant}(u,v,w,x)\mbox{ and }\mbox{equidistant}(w,x,y,z) ) \ \Rightarrow \ \mbox{equidistant}(u,v,y,z)\qquad\mbox{cnf}(d_5,\mbox{axiom})$

$v = \mbox{extension}(u,v,w,w)\qquad\mbox{cnf}(e_1,\mbox{axiom})$

$y = \mbox{extension}(u,v,w,x) \ \Rightarrow \ \mbox{between}(u,v,y)\qquad\mbox{cnf}(b_0,\mbox{axiom})$

$\mbox{between}(u,v,\mbox{reflection}(u,v))\qquad\mbox{cnf}(\mbox{r2}_1,\mbox{axiom})$

$\mbox{equidistant}(v,\mbox{reflection}(u,v),u,v)\qquad\mbox{cnf}(\mbox{r2}_2,\mbox{axiom})$

$u = v \ \Rightarrow \ v = \mbox{reflection}(u,v)\qquad\mbox{cnf}(\mbox{r3}_1,\mbox{axiom})$

$u = \mbox{reflection}(u,u)\qquad\mbox{cnf}(\mbox{r3}_2,\mbox{axiom})$

$v = \mbox{reflection}(u,v) \ \Rightarrow \ u = v\qquad\mbox{cnf}(r_4,\mbox{axiom})$

$\mbox{equidistant}(u,u,v,v)\qquad\mbox{cnf}(d_7,\mbox{axiom})$

$( \mbox{equidistant}(u,v,u_1,v_1)\mbox{ and }\mbox{equidistant}(v,w,v_1,w_1)\mbox{ and }\mbox{between}(u,v,w)\mbox{ and }\mbox{between}(u_1,v_1,w_1) ) \ \Rightarrow \ \mbox{equidistant}(u,w,u_1,w_1)\qquad\mbox{cnf}(d_8,\mbox{axiom})$

$( \mbox{between}(u,v,w)\mbox{ and }\mbox{between}(u,v,x)\mbox{ and }\mbox{equidistant}(v,w,v,x) ) \ \Rightarrow \ ( u = v\mbox{ or }w = x )\qquad\mbox{cnf}(d_9,\mbox{axiom})$

$\mbox{equidistant}(w,x,y,z)\qquad\mbox{cnf}(\mbox{w\_to\_x\_equals\_y\_to\_z},\mbox{hypothesis})$

$u\ne v\qquad\mbox{cnf}(\mbox{u\_not\_v},\mbox{hypothesis})$

$\mbox{extension}(u,v,w,x)\ne \mbox{extension}(u,v,y,z)\qquad\mbox{cnf}(\mbox{prove\_equal\_extensions},\mbox{negated\_conjecture})$

\medskip\textbf{GEO196+1.p} Symmetry of incidence

If the lines X and Y are convergent, U and V are convergent,
 and the intersection point of X and Y is incident with U and V,
 then the intersection point of U and V is incident with X and Y.




$\mbox{include}(\mbox{'Axioms/GEO006+0.ax'})$

$\forall x,y,u,v{:}\;( ( \mbox{convergent\_lines}(x,y)\mbox{ and }\mbox{convergent\_lines}(u,v)\mbox{ and }\neg\,\mbox{apart\_point\_and\_line}(\mbox{intersection\_point}(x,y),u)\mbox{ and }\neg\,\mbox{apart\_point\_and\_line}(\mbox{intersection\_point}(x,y),v) ) \ \Rightarrow \ ( \neg\,\mbox{apart\_point\_and\_line}(\mbox{intersection\_point}(u,v),x)\mbox{ and }\neg\,\mbox{apart\_point\_and\_line}(\mbox{intersection\_point}(u,v),y) ) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO021-3.p} Corollary 5 to symmetries of equidistance

Show that if the distance from A to B equals the distance
 from C to D, then the distance from D to C equals the
 distance from B to A.




$\mbox{include}(\mbox{'Axioms/GEO002-0.ax'})$

$\mbox{equidistant}(u,v,u,v)\qquad\mbox{cnf}(d_1,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(w,x,u,v)\qquad\mbox{cnf}(d_2,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(v,u,w,x)\qquad\mbox{cnf}(d_3,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x)\qquad\mbox{cnf}(\mbox{u\_to\_v\_equals\_w\_to\_x},\mbox{hypothesis})$

$\neg\,\mbox{equidistant}(x,w,v,u)\qquad\mbox{cnf}(\mbox{prove\_symmetry},\mbox{negated\_conjecture})$

\medskip\textbf{GEO176+1.p} Lemma on symmetry and apartness

If two lines are convergent and a point is apart from at least
 one of these lines, then this point is distinct from the
 intersection point of these lines.




$\mbox{include}(\mbox{'Axioms/GEO006+0.ax'})$

$\forall x,y,u,v{:}\;( ( \mbox{distinct\_points}(x,y)\mbox{ and }\mbox{convergent\_lines}(u,v)\mbox{ and }( \mbox{apart\_point\_and\_line}(x,u)\mbox{ or }\mbox{apart\_point\_and\_line}(x,v) ) ) \ \Rightarrow \ \mbox{distinct\_points}(x,\mbox{intersection\_point}(u,v)) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO034-2.p} Corollary to the first inner five-segment theorem






$\mbox{include}(\mbox{'Axioms/GEO002-0.ax'})$

$\mbox{between}(u,v,w)\qquad\mbox{cnf}(\mbox{v\_between\_u\_and\_w},\mbox{hypothesis})$

$\mbox{equidistant}(u,v,u,x)\qquad\mbox{cnf}(\mbox{u\_to\_v\_equals\_u\_to\_x},\mbox{hypothesis})$

$\mbox{equidistant}(w,v,w,x)\qquad\mbox{cnf}(\mbox{w\_to\_v\_equals\_w\_to\_x},\mbox{hypothesis})$

$v\ne x\qquad\mbox{cnf}(\mbox{prove\_v\_is\_x},\mbox{negated\_conjecture})$

\medskip\textbf{GEO005-1.p} Isosceles triangle based on line segment

For any line segment, there exists an isoceles triangle with
 that line segment as its base.




$\mbox{include}(\mbox{'Axioms/GEO001-0.ax'})$

$a\ne b\qquad\mbox{cnf}(\mbox{a\_not\_b},\mbox{hypothesis})$

$\mbox{equidistant}(a,x,b,x) \ \Rightarrow \ \mbox{between}(a,x,b)\qquad\mbox{cnf}(\mbox{prove\_apex},\mbox{negated\_conjecture})$

\medskip\textbf{GEO095+1.p} Endpoints of open sum are endpoints of curves

If two curves meet and their sum is open, then the endpoints of
 the two curves that are not the meeting-point are also the
 endpoints of the sum of these curves.




$\mbox{include}(\mbox{'Axioms/GEO004+0.ax'})$

$\forall c_1,c_2,p{:}\;( ( p\wedge c_1{=}c_2\mbox{ and }\mbox{open}(c_1 + c_2) ) \ \Rightarrow \ \exists q,r{:}\;( p\ne q\mbox{ and }q\ne r\mbox{ and }p\ne r\mbox{ and }\mbox{end\_point}(q,c_1 + c_2)\mbox{ and }\mbox{end\_point}(q,c_1)\mbox{ and }\mbox{end\_point}(r,c_1 + c_2)\mbox{ and }\mbox{end\_point}(r,c_2) ) )\qquad\mbox{fof}(\mbox{proposition\_2\_14}_4,\mbox{conjecture})$

\medskip\textbf{GEO026-2.p} Extension is unique






$\mbox{include}(\mbox{'Axioms/GEO002-0.ax'})$

$\mbox{between}(u,v,w)\qquad\mbox{cnf}(\mbox{v\_between\_u\_and\_w},\mbox{hypothesis})$

$\mbox{between}(u,v,x)\qquad\mbox{cnf}(\mbox{v\_between\_u\_and\_x},\mbox{hypothesis})$

$\mbox{equidistant}(v,w,v,x)\qquad\mbox{cnf}(\mbox{v\_to\_w\_equals\_v\_to\_x},\mbox{hypothesis})$

$u\ne v\qquad\mbox{cnf}(\mbox{u\_not\_v},\mbox{hypothesis})$

$w\ne x\qquad\mbox{cnf}(\mbox{prove\_w\_is\_x},\mbox{negated\_conjecture})$

\medskip\textbf{GEO246+1.p} Configurations in terms of apartness






$\mbox{include}(\mbox{'Axioms/GEO007+0.ax'})$

$\forall a,b,c{:}\;( \mbox{distinct\_points}(a,b) \ \Rightarrow \ ( \mbox{left\_apart\_point}(c,\mbox{reverse\_line}(\mbox{line\_connecting}(a,b))) \ \Rightarrow \ \mbox{left\_apart\_point}(b,\mbox{line\_connecting}(a,c)) ) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO197+1.p} Corollary to symmetry of incidence

If the lines X, Y, and Z are pairwise convergent, and the
 intersection point of X and Y is incident with Z, then the
 intersection point of Z and Y is incident with X.




$\mbox{include}(\mbox{'Axioms/GEO006+0.ax'})$

$\forall x,y,z{:}\;( ( \mbox{convergent\_lines}(x,y)\mbox{ and }\mbox{convergent\_lines}(z,y)\mbox{ and }\mbox{convergent\_lines}(x,z)\mbox{ and }\neg\,\mbox{apart\_point\_and\_line}(\mbox{intersection\_point}(x,y),z) ) \ \Rightarrow \ \neg\,\mbox{apart\_point\_and\_line}(\mbox{intersection\_point}(z,y),x) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO171+2.p} Uniqueness of constructed lines






$\mbox{include}(\mbox{'Axioms/GEO008+0.ax'})$

$\forall x,y{:}\;( \mbox{convergent\_lines}(x,y) \ \Rightarrow \ \mbox{distinct\_lines}(x,y) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO178+2.p} Lemma on symmetry and apartness

If two points X and Y are distinct and a point Z is apart from
 the line connecting X and Y, then Z and X are distinct, and
 Z and Y are distinct.




$\mbox{include}(\mbox{'Axioms/GEO008+0.ax'})$

$\forall x,y,z{:}\;( ( \mbox{distinct\_points}(x,y)\mbox{ and }\mbox{apart\_point\_and\_line}(z,\mbox{line\_connecting}(x,y)) ) \ \Rightarrow \ ( \mbox{distinct\_points}(z,x)\mbox{ and }\mbox{distinct\_points}(z,y) ) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO061-2.p} Theorem of point insertion






$\mbox{include}(\mbox{'Axioms/GEO002-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO002-3.ax'})$

$\mbox{equidistant}(u,v,u_1,\mbox{insertion}(u_1,w_1,u,v)) \ \Rightarrow \ \mbox{between}(u,v,w)\qquad\mbox{cnf}(\mbox{part}_1,\mbox{negated\_conjecture})$

$\mbox{equidistant}(u,v,u_1,\mbox{insertion}(u_1,w_1,u,v)) \ \Rightarrow \ \mbox{equidistant}(u,w,u_1,w_1)\qquad\mbox{cnf}(\mbox{part}_2,\mbox{negated\_conjecture})$

$( \mbox{equidistant}(u,v,u_1,\mbox{insertion}(u_1,w_1,u,v))\mbox{ and }\mbox{between}(u_1,\mbox{insertion}(u_1,w_1,u,v),w_1) ) \ \Rightarrow \ \neg\,\mbox{equidistant}(v,w,\mbox{insertion}(u_1,w_1,u,v),w_1)\qquad\mbox{cnf}(\mbox{part}_3,\mbox{negated\_conjecture})$

\medskip\textbf{GEO197+3.p} Corollary to symmetry of incidence

If the lines X, Y, and Z are pairwise convergent, and the
 intersection point of X and Y is incident with Z, then the
 intersection point of Z and Y is incident with X.




$\mbox{include}(\mbox{'Axioms/GEO006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+1.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+2.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+3.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+4.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+5.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+6.ax'})$

$\forall x,y,z{:}\;( ( \mbox{convergent\_lines}(x,y)\mbox{ and }\mbox{convergent\_lines}(z,y)\mbox{ and }\mbox{convergent\_lines}(x,z)\mbox{ and }\mbox{incident\_point\_and\_line}(\mbox{intersection\_point}(x,y),z) ) \ \Rightarrow \ \mbox{incident\_point\_and\_line}(\mbox{intersection\_point}(z,y),x) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO046-2.p} Second inner transitivity property of betweenness






$\mbox{include}(\mbox{'Axioms/GEO002-0.ax'})$

$\mbox{between}(u,v,x)\qquad\mbox{cnf}(\mbox{v\_between\_u\_and\_x},\mbox{hypothesis})$

$\mbox{between}(v,w,x)\qquad\mbox{cnf}(\mbox{w\_between\_v\_and\_x},\mbox{hypothesis})$

$\neg\,\mbox{between}(u,w,x)\qquad\mbox{cnf}(\mbox{prove\_w\_between\_u\_and\_x},\mbox{negated\_conjecture})$

\medskip\textbf{GEO172+3.p} Uniqueness of constructed points

If two lines are convergent and there is a point that is
 incident with both lines, then this point is equivalent to the
 intersection point of these lines.




$\mbox{include}(\mbox{'Axioms/GEO006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+1.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+2.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+3.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+4.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+5.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+6.ax'})$

$\forall x,y,z{:}\;( ( \mbox{convergent\_lines}(x,y)\mbox{ and }\mbox{incident\_point\_and\_line}(z,x)\mbox{ and }\mbox{incident\_point\_and\_line}(z,y) ) \ \Rightarrow \ \mbox{equal\_points}(z,\mbox{intersection\_point}(x,y)) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO095-1.p} Endpoints of open sum are endpoints of curves

If two curves meet and their sum is open, then the endpoints of
 the two curves that are not the meeting-point are also the
 endpoints of the sum of these curves.




$\mbox{include}(\mbox{'Axioms/GEO004-0.ax'})$

$\mbox{sk}_{16}\wedge \mbox{sk}_{14}{=}\mbox{sk}_{15}\qquad\mbox{cnf}(\mbox{proposition\_2\_14\_4}_{67},\mbox{negated\_conjecture})$

$\mbox{open}(\mbox{sk}_{14} + \mbox{sk}_{15})\qquad\mbox{cnf}(\mbox{proposition\_2\_14\_4}_{68},\mbox{negated\_conjecture})$

$( \mbox{end\_point}(a,\mbox{sk}_{14} + \mbox{sk}_{15})\mbox{ and }\mbox{end\_point}(a,\mbox{sk}_{14})\mbox{ and }\mbox{end\_point}(b,\mbox{sk}_{14} + \mbox{sk}_{15})\mbox{ and }\mbox{end\_point}(b,\mbox{sk}_{15}) ) \ \Rightarrow \ ( \mbox{sk}_{16} = a\mbox{ or }a = b\mbox{ or }\mbox{sk}_{16} = b )\qquad\mbox{cnf}(\mbox{proposition\_2\_14\_4}_{69},\mbox{negated\_conjecture})$

\medskip\textbf{GEO016-3.p} Equidistance is symmetric within its argument pairs

Show that if the distance from A to B equals the distance
 from C to D, then the distance from B to A equals the
 distance from C to D.




$\mbox{include}(\mbox{'Axioms/GEO002-0.ax'})$

$\mbox{equidistant}(u,v,u,v)\qquad\mbox{cnf}(d_1,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(w,x,u,v)\qquad\mbox{cnf}(d_2,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x)\qquad\mbox{cnf}(\mbox{u\_to\_v\_equals\_w\_to\_x},\mbox{hypothesis})$

$\neg\,\mbox{equidistant}(v,u,w,x)\qquad\mbox{cnf}(\mbox{prove\_symmetry},\mbox{negated\_conjecture})$

\medskip\textbf{GEO092-1.p} Common point of open sum is the meeting point

If two curves meet and their sum is open, then the only point
 they have in common is their meeting-point.




$\mbox{include}(\mbox{'Axioms/GEO004-0.ax'})$

$\mbox{sk}_{16}\wedge \mbox{sk}_{14}{=}\mbox{sk}_{15}\qquad\mbox{cnf}(\mbox{proposition\_2\_14\_1}_{67},\mbox{negated\_conjecture})$

$\mbox{open}(\mbox{sk}_{14} + \mbox{sk}_{15})\qquad\mbox{cnf}(\mbox{proposition\_2\_14\_1}_{68},\mbox{negated\_conjecture})$

$\mbox{sk}_{17}\ne \mbox{sk}_{16}\qquad\mbox{cnf}(\mbox{proposition\_2\_14\_1}_{69},\mbox{negated\_conjecture})$

$\mbox{incident\_c}(\mbox{sk}_{17},\mbox{sk}_{14})\qquad\mbox{cnf}(\mbox{proposition\_2\_14\_1}_{70},\mbox{negated\_conjecture})$

$\mbox{incident\_c}(\mbox{sk}_{17},\mbox{sk}_{15})\qquad\mbox{cnf}(\mbox{proposition\_2\_14\_1}_{71},\mbox{negated\_conjecture})$

\medskip\textbf{GEO010-2.p} Collinearity is invariant

For all points x, y, and z, if x, y, and z are collinear
 in one order, they are collinear in any order.




$\mbox{include}(\mbox{'Axioms/GEO002-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO002-1.ax'})$

$\mbox{colinear}(a,b,c)\qquad\mbox{cnf}(\mbox{abc\_colinear},\mbox{hypothesis})$

$( \mbox{colinear}(a,c,b)\mbox{ and }\mbox{colinear}(b,a,c)\mbox{ and }\mbox{colinear}(b,c,a)\mbox{ and }\mbox{colinear}(c,a,b) ) \ \Rightarrow \ \neg\,\mbox{colinear}(c,b,a)\qquad\mbox{cnf}(\mbox{prove\_colinear\_in\_all\_orders},\mbox{negated\_conjecture})$

\medskip\textbf{GEO054-2.p} Corollary 2 to the segment construction axiom






$\mbox{include}(\mbox{'Axioms/GEO002-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO002-2.ax'})$

$\neg\,\mbox{between}(u,v,\mbox{reflection}(u,v))\qquad\mbox{cnf}(\mbox{prove\_v\_between\_u\_and\_reflection},\mbox{negated\_conjecture})$

\medskip\textbf{GEO207+1.p} Irreflexivity of line convergence






$\mbox{include}(\mbox{'Axioms/GEO006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+2.ax'})$

$\forall x{:}\;\neg\,\mbox{convergent\_lines}(x,x)\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO194+3.p} Corollary to symmetry of apartness

If the lines X, Y, and Z are pairwise convergent, and the
 intersection point of X and Y is apart from a line Z, then
 the intersection point of X and Z is apart from Y.




$\mbox{include}(\mbox{'Axioms/GEO006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+1.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+2.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+3.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+4.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+5.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+6.ax'})$

$\forall x,y,z{:}\;( ( \mbox{convergent\_lines}(x,y)\mbox{ and }\mbox{convergent\_lines}(z,y)\mbox{ and }\mbox{convergent\_lines}(x,z) ) \ \Rightarrow \ ( \mbox{apart\_point\_and\_line}(\mbox{intersection\_point}(x,y),z) \ \Rightarrow \ \mbox{apart\_point\_and\_line}(\mbox{intersection\_point}(x,z),y) ) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO205+1.p} Convergent lines and equal points

If the lines X and Y are convergent, and Y and Z are
 equivalent, then X and Z are convergent, and the intersection
 point of X and Y, and the intersection point of X and Z are
 equal.




$\mbox{include}(\mbox{'Axioms/GEO006+0.ax'})$

$\forall x,y,z{:}\;( ( \mbox{convergent\_lines}(x,y)\mbox{ and }\neg\,\mbox{distinct\_lines}(y,z) ) \ \Rightarrow \ ( \mbox{convergent\_lines}(x,z)\mbox{ and }\neg\,\mbox{distinct\_points}(\mbox{intersection\_point}(x,y),\mbox{intersection\_point}(x,z)) ) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO054-3.p} Corollary 2 to the segment construction axiom






$\mbox{include}(\mbox{'Axioms/GEO002-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO002-2.ax'})$

$\mbox{equidistant}(u,v,u,v)\qquad\mbox{cnf}(d_1,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(w,x,u,v)\qquad\mbox{cnf}(d_2,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(v,u,w,x)\qquad\mbox{cnf}(d_3,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(u,v,x,w)\qquad\mbox{cnf}(\mbox{d4}_1,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(v,u,x,w)\qquad\mbox{cnf}(\mbox{d4}_2,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(w,x,v,u)\qquad\mbox{cnf}(\mbox{d4}_3,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(x,w,u,v)\qquad\mbox{cnf}(\mbox{d4}_4,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(x,w,v,u)\qquad\mbox{cnf}(\mbox{d4}_5,\mbox{axiom})$

$( \mbox{equidistant}(u,v,w,x)\mbox{ and }\mbox{equidistant}(w,x,y,z) ) \ \Rightarrow \ \mbox{equidistant}(u,v,y,z)\qquad\mbox{cnf}(d_5,\mbox{axiom})$

$v = \mbox{extension}(u,v,w,w)\qquad\mbox{cnf}(e_1,\mbox{axiom})$

$y = \mbox{extension}(u,v,w,x) \ \Rightarrow \ \mbox{between}(u,v,y)\qquad\mbox{cnf}(b_0,\mbox{axiom})$

$\neg\,\mbox{between}(u,v,\mbox{reflection}(u,v))\qquad\mbox{cnf}(\mbox{prove\_v\_between\_u\_and\_reflection},\mbox{negated\_conjecture})$

\medskip\textbf{GEO002-1.p} For all points x and y, x is between x and y






$\mbox{include}(\mbox{'Axioms/GEO001-0.ax'})$

$\neg\,\mbox{between}(a,a,b)\qquad\mbox{cnf}(\mbox{prove\_a\_between\_a\_and\_b},\mbox{negated\_conjecture})$

\medskip\textbf{GEO214+1.p} Corollary to non-orthogonal lines and a third line

If the line L is not orthogonal to M, then M is orthogonal to L.




$\mbox{include}(\mbox{'Axioms/GEO006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+2.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+3.ax'})$

$\forall l,m{:}\;( \mbox{unorthogonal\_lines}(l,m) \ \Rightarrow \ \mbox{unorthogonal\_lines}(m,l) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO025-3.p} Addition of equal segments






$\mbox{include}(\mbox{'Axioms/GEO002-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO002-2.ax'})$

$\mbox{equidistant}(u,v,u,v)\qquad\mbox{cnf}(d_1,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(w,x,u,v)\qquad\mbox{cnf}(d_2,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(v,u,w,x)\qquad\mbox{cnf}(d_3,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(u,v,x,w)\qquad\mbox{cnf}(\mbox{d4}_1,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(v,u,x,w)\qquad\mbox{cnf}(\mbox{d4}_2,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(w,x,v,u)\qquad\mbox{cnf}(\mbox{d4}_3,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(x,w,u,v)\qquad\mbox{cnf}(\mbox{d4}_4,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(x,w,v,u)\qquad\mbox{cnf}(\mbox{d4}_5,\mbox{axiom})$

$( \mbox{equidistant}(u,v,w,x)\mbox{ and }\mbox{equidistant}(w,x,y,z) ) \ \Rightarrow \ \mbox{equidistant}(u,v,y,z)\qquad\mbox{cnf}(d_5,\mbox{axiom})$

$v = \mbox{extension}(u,v,w,w)\qquad\mbox{cnf}(e_1,\mbox{axiom})$

$y = \mbox{extension}(u,v,w,x) \ \Rightarrow \ \mbox{between}(u,v,y)\qquad\mbox{cnf}(b_0,\mbox{axiom})$

$\mbox{between}(u,v,\mbox{reflection}(u,v))\qquad\mbox{cnf}(\mbox{r2}_1,\mbox{axiom})$

$\mbox{equidistant}(v,\mbox{reflection}(u,v),u,v)\qquad\mbox{cnf}(\mbox{r2}_2,\mbox{axiom})$

$u = v \ \Rightarrow \ v = \mbox{reflection}(u,v)\qquad\mbox{cnf}(\mbox{r3}_1,\mbox{axiom})$

$u = \mbox{reflection}(u,u)\qquad\mbox{cnf}(\mbox{r3}_2,\mbox{axiom})$

$v = \mbox{reflection}(u,v) \ \Rightarrow \ u = v\qquad\mbox{cnf}(r_4,\mbox{axiom})$

$\mbox{equidistant}(u,u,v,v)\qquad\mbox{cnf}(d_7,\mbox{axiom})$

$\mbox{equidistant}(u,v,u_1,v_1)\qquad\mbox{cnf}(\mbox{u\_to\_v\_equals\_u1\_to\_v}_1,\mbox{hypothesis})$

$\mbox{equidistant}(v,w,v_1,w_1)\qquad\mbox{cnf}(\mbox{v\_to\_w\_equals\_v1\_to\_w}_1,\mbox{hypothesis})$

$\mbox{between}(u,v,w)\qquad\mbox{cnf}(\mbox{v\_between\_u\_and\_w},\mbox{hypothesis})$

$\mbox{between}(u_1,v_1,w_1)\qquad\mbox{cnf}(\mbox{v1\_between\_u1\_and\_w}_1,\mbox{hypothesis})$

$\neg\,\mbox{equidistant}(u,w,u_1,w_1)\qquad\mbox{cnf}(\mbox{prove\_equal\_sums},\mbox{negated\_conjecture})$

\medskip\textbf{GEO006-2.p} Betweenness for 3 points on a line

For any three distinct points x, y, and z, if y is between
 x and z, then both x is not between y and z and z is not
 between x and y.




$\mbox{include}(\mbox{'Axioms/GEO002-0.ax'})$

$a\ne c\qquad\mbox{cnf}(\mbox{a\_not\_c},\mbox{hypothesis})$

$a\ne d\qquad\mbox{cnf}(\mbox{a\_not\_d},\mbox{hypothesis})$

$c\ne d\qquad\mbox{cnf}(\mbox{c\_not\_d},\mbox{hypothesis})$

$\mbox{between}(a,c,d)\qquad\mbox{cnf}(\mbox{c\_between\_a\_and\_d},\mbox{hypothesis})$

$\mbox{between}(c,a,d)\mbox{ or }\mbox{between}(a,d,c)\qquad\mbox{cnf}(\mbox{prove\_not\_between\_others},\mbox{negated\_conjecture})$

\medskip\textbf{GEO261+1.p} Lemma for parallel projection preserves or reverses order






$\mbox{include}(\mbox{'Axioms/GEO007+0.ax'})$

$\forall l,m,a,b,c{:}\;( ( \mbox{between\_on\_line}(l,a,b,c)\mbox{ and }\mbox{convergent\_lines}(l,m)\mbox{ and }\neg\,\mbox{apart\_point\_and\_line}(b,m) ) \ \Rightarrow \ \mbox{divides\_points}(m,a,c) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO145-1.p} Starting point and precedence

If R is the starting point of o, then P precedes Q wrt. o, iff
 P is identical with R and Q is on o but different from R or P
 is between R and Q on o.




$\mbox{include}(\mbox{'Axioms/GEO004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO004-2.ax'})$

$\mbox{start\_point}(\mbox{sk}_{26},\mbox{sk}_{25})\qquad\mbox{cnf}(\mbox{theorem\_4\_24}_{133},\mbox{negated\_conjecture})$

$\mbox{ordered\_by}(\mbox{sk}_{25},\mbox{sk}_{27},\mbox{sk}_{28})\mbox{ or }\mbox{sk}_{27} = \mbox{sk}_{26}\mbox{ or }\mbox{between}(\mbox{sk}_{25},\mbox{sk}_{26},\mbox{sk}_{27},\mbox{sk}_{28})\qquad\mbox{cnf}(\mbox{theorem\_4\_24}_{134},\mbox{negated\_conjecture})$

$\mbox{sk}_{28} = \mbox{sk}_{26} \ \Rightarrow \ ( \mbox{ordered\_by}(\mbox{sk}_{25},\mbox{sk}_{27},\mbox{sk}_{28})\mbox{ or }\mbox{between}(\mbox{sk}_{25},\mbox{sk}_{26},\mbox{sk}_{27},\mbox{sk}_{28}) )\qquad\mbox{cnf}(\mbox{theorem\_4\_24}_{135},\mbox{negated\_conjecture})$

$\mbox{ordered\_by}(\mbox{sk}_{25},\mbox{sk}_{27},\mbox{sk}_{28})\mbox{ or }\mbox{incident\_o}(\mbox{sk}_{28},\mbox{sk}_{25})\mbox{ or }\mbox{between}(\mbox{sk}_{25},\mbox{sk}_{26},\mbox{sk}_{27},\mbox{sk}_{28})\qquad\mbox{cnf}(\mbox{theorem\_4\_24}_{136},\mbox{negated\_conjecture})$

$\mbox{ordered\_by}(\mbox{sk}_{25},\mbox{sk}_{27},\mbox{sk}_{28}) \ \Rightarrow \ \mbox{ordered\_by}(\mbox{sk}_{25},\mbox{sk}_{27},\mbox{sk}_{28})\qquad\mbox{cnf}(\mbox{theorem\_4\_24}_{137},\mbox{negated\_conjecture})$

$( \mbox{sk}_{27} = \mbox{sk}_{26}\mbox{ and }\mbox{incident\_o}(\mbox{sk}_{28},\mbox{sk}_{25}) ) \ \Rightarrow \ ( \mbox{sk}_{28} = \mbox{sk}_{26}\mbox{ or }\mbox{sk}_{27} = \mbox{sk}_{26}\mbox{ or }\mbox{between}(\mbox{sk}_{25},\mbox{sk}_{26},\mbox{sk}_{27},\mbox{sk}_{28}) )\qquad\mbox{cnf}(\mbox{theorem\_4\_24}_{138},\mbox{negated\_conjecture})$

$( \mbox{sk}_{27} = \mbox{sk}_{26}\mbox{ and }\mbox{incident\_o}(\mbox{sk}_{28},\mbox{sk}_{25})\mbox{ and }\mbox{sk}_{28} = \mbox{sk}_{26} ) \ \Rightarrow \ ( \mbox{sk}_{28} = \mbox{sk}_{26}\mbox{ or }\mbox{between}(\mbox{sk}_{25},\mbox{sk}_{26},\mbox{sk}_{27},\mbox{sk}_{28}) )\qquad\mbox{cnf}(\mbox{theorem\_4\_24}_{139},\mbox{negated\_conjecture})$

$( \mbox{sk}_{27} = \mbox{sk}_{26}\mbox{ and }\mbox{incident\_o}(\mbox{sk}_{28},\mbox{sk}_{25}) ) \ \Rightarrow \ ( \mbox{sk}_{28} = \mbox{sk}_{26}\mbox{ or }\mbox{incident\_o}(\mbox{sk}_{28},\mbox{sk}_{25})\mbox{ or }\mbox{between}(\mbox{sk}_{25},\mbox{sk}_{26},\mbox{sk}_{27},\mbox{sk}_{28}) )\qquad\mbox{cnf}(\mbox{theorem\_4\_24}_{140},\mbox{negated\_conjecture})$

$\mbox{between}(\mbox{sk}_{25},\mbox{sk}_{26},\mbox{sk}_{27},\mbox{sk}_{28}) \ \Rightarrow \ ( \mbox{sk}_{27} = \mbox{sk}_{26}\mbox{ or }\mbox{between}(\mbox{sk}_{25},\mbox{sk}_{26},\mbox{sk}_{27},\mbox{sk}_{28}) )\qquad\mbox{cnf}(\mbox{theorem\_4\_24}_{141},\mbox{negated\_conjecture})$

$( \mbox{between}(\mbox{sk}_{25},\mbox{sk}_{26},\mbox{sk}_{27},\mbox{sk}_{28})\mbox{ and }\mbox{sk}_{28} = \mbox{sk}_{26} ) \ \Rightarrow \ \mbox{between}(\mbox{sk}_{25},\mbox{sk}_{26},\mbox{sk}_{27},\mbox{sk}_{28})\qquad\mbox{cnf}(\mbox{theorem\_4\_24}_{142},\mbox{negated\_conjecture})$

$\mbox{between}(\mbox{sk}_{25},\mbox{sk}_{26},\mbox{sk}_{27},\mbox{sk}_{28}) \ \Rightarrow \ ( \mbox{incident\_o}(\mbox{sk}_{28},\mbox{sk}_{25})\mbox{ or }\mbox{between}(\mbox{sk}_{25},\mbox{sk}_{26},\mbox{sk}_{27},\mbox{sk}_{28}) )\qquad\mbox{cnf}(\mbox{theorem\_4\_24}_{143},\mbox{negated\_conjecture})$

$( \mbox{sk}_{27} = \mbox{sk}_{26}\mbox{ and }\mbox{incident\_o}(\mbox{sk}_{28},\mbox{sk}_{25})\mbox{ and }\mbox{ordered\_by}(\mbox{sk}_{25},\mbox{sk}_{27},\mbox{sk}_{28}) ) \ \Rightarrow \ \mbox{sk}_{28} = \mbox{sk}_{26}\qquad\mbox{cnf}(\mbox{theorem\_4\_24}_{144},\mbox{negated\_conjecture})$

$\mbox{between}(\mbox{sk}_{25},\mbox{sk}_{26},\mbox{sk}_{27},\mbox{sk}_{28}) \ \Rightarrow \ \neg\,\mbox{ordered\_by}(\mbox{sk}_{25},\mbox{sk}_{27},\mbox{sk}_{28})\qquad\mbox{cnf}(\mbox{theorem\_4\_24}_{145},\mbox{negated\_conjecture})$

\medskip\textbf{GEO228+1.p} Obervability of equal or opposite direction






$\mbox{include}(\mbox{'Axioms/GEO007+0.ax'})$

$\forall l,m{:}\;( \neg\,\mbox{convergent\_lines}(l,m)\iff ( \neg\,\mbox{unequally\_directed\_lines}(l,m)\mbox{ or }\neg\,\mbox{unequally\_directed\_lines}(l,\mbox{reverse\_line}(m)) ) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO267+3.p} Possible unequally directed opposite lines






$\mbox{include}(\mbox{'Axioms/GEO009+0.ax'})$

$\forall l,m,n{:}\;( \mbox{unequally\_directed\_opposite\_lines}(l,m) \ \Rightarrow \ ( \mbox{unequally\_directed\_opposite\_lines}(m,n)\mbox{ or }\mbox{unequally\_directed\_lines}(m,n) ) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO251+1.p} A point in each region formed by parallel lines






$\mbox{include}(\mbox{'Axioms/GEO007+0.ax'})$

$\forall a,b,l{:}\;( \mbox{left\_apart\_point}(b,\mbox{reverse\_line}(\mbox{parallel\_through\_point}(l,a))) \ \Rightarrow \ \mbox{left\_apart\_point}(a,\mbox{parallel\_through\_point}(l,b)) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO094+1.p} Meeting point is not an endpoint of contianing curve

A meeting point of two curves is not an endpoint of any curve
 that includes both as sub-curves.




$\mbox{include}(\mbox{'Axioms/GEO004+0.ax'})$

$\forall c,c_1,c_2,p{:}\;( ( p\wedge c_1{=}c_2\mbox{ and }\mbox{part\_of}(c_1,c)\mbox{ and }\mbox{part\_of}(c_2,c) ) \ \Rightarrow \ \neg\,\mbox{end\_point}(p,c) )\qquad\mbox{fof}(\mbox{proposition\_2\_14}_3,\mbox{conjecture})$

\medskip\textbf{GEO066-2.p} Corollary 3 to collinearity






$\mbox{include}(\mbox{'Axioms/GEO002-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO002-1.ax'})$

$\mbox{between}(v,u,w)\qquad\mbox{cnf}(\mbox{u\_between\_v\_and\_w},\mbox{hypothesis})$

$\neg\,\mbox{colinear}(u,v,w)\qquad\mbox{cnf}(\mbox{prove\_uvw\_colinear},\mbox{negated\_conjecture})$

\medskip\textbf{GEO256+1.p} Property of order and betweeness






$\mbox{include}(\mbox{'Axioms/GEO007+0.ax'})$

$\forall l,a,b,c,d{:}\;( ( \mbox{distinct\_points}(a,c)\mbox{ and }\mbox{distinct\_points}(b,c)\mbox{ and }\neg\,\mbox{apart\_point\_and\_line}(c,l)\mbox{ and }\mbox{left\_apart\_point}(d,l) ) \ \Rightarrow \ ( \mbox{before\_on\_line}(l,a,b) \ \Rightarrow \ ( \mbox{before\_on\_line}(l,a,c)\mbox{ or }\mbox{before\_on\_line}(l,c,b) ) ) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO160-1.p} Oriented curves






$\mbox{include}(\mbox{'Axioms/GEO004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO004-2.ax'})$

\medskip\textbf{GEO237+3.p} Axiom of Pasch






$\mbox{include}(\mbox{'Axioms/GEO009+0.ax'})$

$\forall a,b,c,l{:}\;( \mbox{apart\_point\_and\_line}(c,l) \ \Rightarrow \ ( \mbox{divides\_points}(l,a,b) \ \Rightarrow \ ( \mbox{divides\_points}(l,a,c)\mbox{ or }\mbox{divides\_points}(l,b,c) ) ) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO239+3.p} Lemma on oriented intersection of lines with plane






$\mbox{include}(\mbox{'Axioms/GEO009+0.ax'})$

$\forall a,b,l{:}\;( ( \mbox{incident\_point\_and\_line}(a,l)\mbox{ and }\mbox{left\_apart\_point}(b,l) ) \ \Rightarrow \ \mbox{left\_convergent\_lines}(l,\mbox{line\_connecting}(a,b)) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO098-1.p} For closed curves, there are two complementary sub-curves






$\mbox{include}(\mbox{'Axioms/GEO004-0.ax'})$

$\mbox{closed}(\mbox{sk}_{14})\qquad\mbox{cnf}(\mbox{theorem\_2\_16}_{67},\mbox{negated\_conjecture})$

$\mbox{incident\_c}(\mbox{sk}_{15},\mbox{sk}_{14})\qquad\mbox{cnf}(\mbox{theorem\_2\_16}_{68},\mbox{negated\_conjecture})$

$\mbox{incident\_c}(\mbox{sk}_{16},\mbox{sk}_{14})\qquad\mbox{cnf}(\mbox{theorem\_2\_16}_{69},\mbox{negated\_conjecture})$

$\mbox{sk}_{15}\ne \mbox{sk}_{16}\qquad\mbox{cnf}(\mbox{theorem\_2\_16}_{70},\mbox{negated\_conjecture})$

$( \mbox{sk}_{15}\wedge a{=}b\mbox{ and }\mbox{sk}_{16}\wedge a{=}b ) \ \Rightarrow \ \mbox{sk}_{14}\ne a + b\qquad\mbox{cnf}(\mbox{theorem\_2\_16}_{71},\mbox{negated\_conjecture})$

\medskip\textbf{GEO204+2.p} Distinct points and equal lines

If the points X and Y are distinct, and the points Y and Z are
 equal, then X and Z are distinct, and the line connecting X
 and Y is equivalent to the line connecting X and Z.




$\mbox{include}(\mbox{'Axioms/GEO008+0.ax'})$

$\forall x,y,z{:}\;( ( \mbox{distinct\_points}(x,y)\mbox{ and }\neg\,\mbox{distinct\_points}(y,z) ) \ \Rightarrow \ ( \mbox{distinct\_points}(x,z)\mbox{ and }\neg\,\mbox{distinct\_lines}(\mbox{line\_connecting}(x,y),\mbox{line\_connecting}(x,z)) ) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO262+3.p} Lemma for parallel projection preserves or reverses order






$\mbox{include}(\mbox{'Axioms/GEO009+0.ax'})$

$\forall l,m,n,a,b,c{:}\;( ( \mbox{between\_on\_line}(l,a,b,c)\mbox{ and }\mbox{convergent\_lines}(l,m)\mbox{ and }\mbox{incident\_point\_and\_line}(b,m)\mbox{ and }\mbox{convergent\_lines}(l,n)\mbox{ and }\mbox{convergent\_lines}(m,n)\mbox{ and }\mbox{incident\_point\_and\_line}(b,n) ) \ \Rightarrow \ \mbox{between\_on\_line}(m,\mbox{intersection\_point}(m,\mbox{parallel\_through\_point}(n,a)),b,\mbox{intersection\_point}(m,\mbox{parallel\_through\_point}(n,c))) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO028-2.p} Corollary 2 to unique extension






$\mbox{include}(\mbox{'Axioms/GEO002-0.ax'})$

$\mbox{equidistant}(w,x,y,z)\qquad\mbox{cnf}(\mbox{w\_to\_x\_equals\_y\_to\_z},\mbox{hypothesis})$

$u\ne v\qquad\mbox{cnf}(\mbox{u\_not\_v},\mbox{hypothesis})$

$\mbox{extension}(u,v,w,x)\ne \mbox{extension}(u,v,y,z)\qquad\mbox{cnf}(\mbox{prove\_equal\_extensions},\mbox{negated\_conjecture})$

\medskip\textbf{GEO049-2.p} Theorem of similar situations






$\mbox{include}(\mbox{'Axioms/GEO002-0.ax'})$

$\mbox{equidistant}(u,v,u_1,v_1)\qquad\mbox{cnf}(\mbox{u\_to\_v\_equals\_u1\_to\_v}_1,\mbox{hypothesis})$

$\mbox{equidistant}(v,w,v_1,w_1)\qquad\mbox{cnf}(\mbox{v\_to\_w\_equals\_v1\_to\_w}_1,\mbox{hypothesis})$

$\mbox{equidistant}(u,w,u_1,w_1)\qquad\mbox{cnf}(\mbox{u\_to\_w\_equals\_u1\_to\_w}_1,\mbox{hypothesis})$

$\mbox{between}(u,v,w)\qquad\mbox{cnf}(\mbox{v\_between\_u\_and\_w},\mbox{hypothesis})$

$\neg\,\mbox{between}(u_1,v_1,w_1)\qquad\mbox{cnf}(\mbox{prove\_v1\_between\_u1\_and\_w}_1,\mbox{negated\_conjecture})$

\medskip\textbf{GEO199+1.p} Corollary to symmetry of incidence

If the lines X, Y, and Z are pairwise convergent, and the
 intersection point of X and Y is incident with Z, then the
 intersection point of Y and X is incident with Z.




$\mbox{include}(\mbox{'Axioms/GEO006+0.ax'})$

$\forall x,y,z{:}\;( ( \mbox{convergent\_lines}(x,y)\mbox{ and }\mbox{convergent\_lines}(z,y)\mbox{ and }\mbox{convergent\_lines}(x,z)\mbox{ and }\neg\,\mbox{apart\_point\_and\_line}(\mbox{intersection\_point}(x,y),z) ) \ \Rightarrow \ \neg\,\mbox{apart\_point\_and\_line}(\mbox{intersection\_point}(y,x),z) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO088+1.p} Endpoint of subcurve or curve

If an endpoint of a given curve lies on a sub-curve then it is
 also an endpoint of this sub-curve




$\mbox{include}(\mbox{'Axioms/GEO004+0.ax'})$

$\forall c,\mbox{cpp},p{:}\;( ( \mbox{part\_of}(\mbox{cpp},c)\mbox{ and }\mbox{end\_point}(p,c)\mbox{ and }\mbox{incident\_c}(p,\mbox{cpp}) ) \ \Rightarrow \ \mbox{end\_point}(p,\mbox{cpp}) )\qquad\mbox{fof}(\mbox{theorem\_2}_{10},\mbox{conjecture})$

\medskip\textbf{GEO180+2.p} Triangle axiom 1

If X and Y are distinct points and Z is apart from the line
 connecting X and Y, then X is apart from the line connecting
 Z and Y.




$\mbox{include}(\mbox{'Axioms/GEO008+0.ax'})$

$\forall x,y,z{:}\;( \mbox{distinct\_points}(x,y) \ \Rightarrow \ ( \mbox{apart\_point\_and\_line}(z,\mbox{line\_connecting}(x,y)) \ \Rightarrow \ \mbox{apart\_point\_and\_line}(x,\mbox{line\_connecting}(z,y)) ) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO207+3.p} Irreflexivity of line convergence






$\mbox{include}(\mbox{'Axioms/GEO006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+1.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+2.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+3.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+4.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+5.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+6.ax'})$

$\forall x{:}\;\neg\,\mbox{convergent\_lines}(x,x)\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO220+1.p} Transitivity of orthogonal

If a line L is orthogonal to line M and line N is orthogonal
 to L, then M and N are parallel.




$\mbox{include}(\mbox{'Axioms/GEO006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+4.ax'})$

$\forall l,m,n{:}\;( ( \neg\,\mbox{unorthogonal\_lines}(l,m)\mbox{ and }\neg\,\mbox{unorthogonal\_lines}(l,n) ) \ \Rightarrow \ \neg\,\mbox{convergent\_lines}(m,n) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO183+3.p} Lemma on symmetry and apartness

If X and Y are distinct points, then they are incident with
 the line that is equal to the line connecting the points.




$\mbox{include}(\mbox{'Axioms/GEO006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+1.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+2.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+3.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+4.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+5.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+6.ax'})$

$\forall x,y,u,v{:}\;( ( \mbox{distinct\_points}(x,y)\mbox{ and }\mbox{convergent\_lines}(u,v)\mbox{ and }\mbox{equal\_lines}(u,\mbox{line\_connecting}(x,y)) ) \ \Rightarrow \ ( \mbox{incident\_point\_and\_line}(x,u)\mbox{ and }\mbox{incident\_point\_and\_line}(y,u) ) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO189+3.p} Collary to symmetry of incidence

If X, Y, and Z are pairwise distinct, and Z is incident with
 the line connecting X and Y, then Y is incident with the line
 connecting X and Z.




$\mbox{include}(\mbox{'Axioms/GEO006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+1.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+2.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+3.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+4.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+5.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+6.ax'})$

$\forall x,y,z{:}\;( ( \mbox{distinct\_points}(x,y)\mbox{ and }\mbox{distinct\_points}(x,z)\mbox{ and }\mbox{distinct\_points}(y,z)\mbox{ and }\mbox{incident\_point\_and\_line}(z,\mbox{line\_connecting}(x,y)) ) \ \Rightarrow \ \mbox{incident\_point\_and\_line}(y,\mbox{line\_connecting}(x,z)) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO167+1.p} Pappus1 implies Pappus2






$\mbox{colinear}(a,b,c,l)\mbox{ and }\mbox{colinear}(d,e,f,m)\qquad\mbox{fof}(\mbox{assumption}_1,\mbox{axiom})$

$\mbox{colinear}(b,f,g,n)\mbox{ and }\mbox{colinear}(c,e,g,o)\qquad\mbox{fof}(\mbox{assumption}_2,\mbox{axiom})$

$\mbox{colinear}(b,d,h,p)\mbox{ and }\mbox{colinear}(a,e,h,q)\qquad\mbox{fof}(\mbox{assumption}_3,\mbox{axiom})$

$\mbox{colinear}(c,d,i,r)\mbox{ and }\mbox{colinear}(a,f,i,s)\qquad\mbox{fof}(\mbox{assumption}_4,\mbox{axiom})$

$\mbox{line\_equal}(n,o) \ \Rightarrow \ \mbox{goal}\qquad\mbox{fof}(\mbox{goal}_1,\mbox{axiom})$

$\mbox{line\_equal}(p,q) \ \Rightarrow \ \mbox{goal}\qquad\mbox{fof}(\mbox{goal}_2,\mbox{axiom})$

$\mbox{line\_equal}(s,r) \ \Rightarrow \ \mbox{goal}\qquad\mbox{fof}(\mbox{goal}_3,\mbox{axiom})$

$\forall a{:}\;( ( \mbox{line\_equal}(a,a)\mbox{ and }\mbox{incident}(g,a)\mbox{ and }\mbox{incident}(h,a)\mbox{ and }\mbox{incident}(i,a) ) \ \Rightarrow \ \mbox{goal} )\qquad\mbox{fof}(\mbox{goal}_4,\mbox{axiom})$

$\forall a,b,c,d{:}\;( \mbox{colinear}(a,b,c,d) \ \Rightarrow \ \mbox{incident}(a,d) )\qquad\mbox{fof}(\mbox{colinearity\_elimination}_1,\mbox{axiom})$

$\forall a,b,c,d{:}\;( \mbox{colinear}(a,b,c,d) \ \Rightarrow \ \mbox{incident}(b,d) )\qquad\mbox{fof}(\mbox{colinearity\_elimination}_2,\mbox{axiom})$

$\forall a,b,c,d{:}\;( \mbox{colinear}(a,b,c,d) \ \Rightarrow \ \mbox{incident}(c,d) )\qquad\mbox{fof}(\mbox{colinearity\_elimination}_3,\mbox{axiom})$

$\forall a,b{:}\;( \mbox{incident}(a,b) \ \Rightarrow \ \mbox{point\_equal}(a,a) )\qquad\mbox{fof}(\mbox{reflexivity\_of\_point\_equal},\mbox{axiom})$

$\forall a,b{:}\;( \mbox{point\_equal}(a,b) \ \Rightarrow \ \mbox{point\_equal}(b,a) )\qquad\mbox{fof}(\mbox{symmetry\_of\_point\_equal},\mbox{axiom})$

$\forall a,b,c{:}\;( ( \mbox{point\_equal}(a,b)\mbox{ and }\mbox{point\_equal}(b,c) ) \ \Rightarrow \ \mbox{point\_equal}(a,c) )\qquad\mbox{fof}(\mbox{transitivity\_of\_point\_equal},\mbox{axiom})$

$\forall a,b{:}\;( \mbox{incident}(a,b) \ \Rightarrow \ \mbox{line\_equal}(b,b) )\qquad\mbox{fof}(\mbox{reflexivity\_of\_line\_equal},\mbox{axiom})$

$\forall a,b{:}\;( \mbox{line\_equal}(a,b) \ \Rightarrow \ \mbox{line\_equal}(b,a) )\qquad\mbox{fof}(\mbox{symmetry\_of\_line\_equal},\mbox{axiom})$

$\forall a,b,c{:}\;( ( \mbox{line\_equal}(a,b)\mbox{ and }\mbox{line\_equal}(b,c) ) \ \Rightarrow \ \mbox{line\_equal}(a,c) )\qquad\mbox{fof}(\mbox{transitivity\_of\_line\_equal},\mbox{axiom})$

$\forall a,b,c{:}\;( ( \mbox{point\_equal}(a,b)\mbox{ and }\mbox{incident}(b,c) ) \ \Rightarrow \ \mbox{incident}(a,c) )\qquad\mbox{fof}(\mbox{pcon},\mbox{axiom})$

$\forall a,b,c{:}\;( ( \mbox{incident}(a,b)\mbox{ and }\mbox{line\_equal}(b,c) ) \ \Rightarrow \ \mbox{incident}(a,c) )\qquad\mbox{fof}(\mbox{lcon},\mbox{axiom})$

$\forall a,b,c,d,e,f,g,h,i,j,k,l,m,n,o,p,q{:}\;( ( \mbox{colinear}(a,b,c,j)\mbox{ and }\mbox{colinear}(d,e,f,k)\mbox{ and }\mbox{colinear}(b,f,g,l)\mbox{ and }\mbox{colinear}(c,e,g,m)\mbox{ and }\mbox{colinear}(b,d,h,n)\mbox{ and }\mbox{colinear}(a,e,h,o)\mbox{ and }\mbox{colinear}(c,d,i,p)\mbox{ and }\mbox{colinear}(a,f,i,q) ) \ \Rightarrow \ ( \exists r{:}\;\mbox{colinear}(g,h,i,r)\mbox{ or }\mbox{incident}(a,k)\mbox{ or }\mbox{incident}(b,k)\mbox{ or }\mbox{incident}(c,k)\mbox{ or }\mbox{incident}(d,j)\mbox{ or }\mbox{incident}(e,j)\mbox{ or }\mbox{incident}(f,j) ) )\qquad\mbox{fof}(\mbox{pappus}_1,\mbox{axiom})$

$\forall a,b,c,d{:}\;( ( \mbox{incident}(c,a)\mbox{ and }\mbox{incident}(c,b)\mbox{ and }\mbox{incident}(d,a)\mbox{ and }\mbox{incident}(d,b) ) \ \Rightarrow \ ( \mbox{point\_equal}(c,d)\mbox{ or }\mbox{line\_equal}(a,b) ) )\qquad\mbox{fof}(\mbox{unique},\mbox{axiom})$

$\forall a,b{:}\;( ( \mbox{point\_equal}(a,a)\mbox{ and }\mbox{point\_equal}(b,b) ) \ \Rightarrow \ \exists c{:}\;( \mbox{incident}(a,c)\mbox{ and }\mbox{incident}(b,c) ) )\qquad\mbox{fof}(\mbox{line},\mbox{axiom})$

$\forall a,b,c{:}\;( ( \mbox{line\_equal}(c,c)\mbox{ and }\mbox{line\_equal}(b,b) ) \ \Rightarrow \ \exists a{:}\;( \mbox{incident}(a,b)\mbox{ and }\mbox{incident}(a,c) ) )\qquad\mbox{fof}(\mbox{point},\mbox{axiom})$

$\mbox{goal}\qquad\mbox{fof}(\mbox{goal\_to\_be\_proved},\mbox{conjecture})$

\medskip\textbf{GEO225+1.p} Existence of line joining distinct points

When there are two distinct points, then a line connecting
 them can be constructed.




$\mbox{include}(\mbox{'Axioms/GEO006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+5.ax'})$

$\forall a,b{:}\;( ( \mbox{point}(a)\mbox{ and }\mbox{point}(b)\mbox{ and }\mbox{distinct\_points}(a,b) ) \ \Rightarrow \ \exists x{:}\;( \mbox{line}(x) \ \Rightarrow \ ( \neg\,\mbox{apart\_point\_and\_line}(a,x)\mbox{ and }\neg\,\mbox{apart\_point\_and\_line}(b,x) ) ) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO243+3.p} Configurations in terms of apartness






$\mbox{include}(\mbox{'Axioms/GEO009+0.ax'})$

$\forall a,b,c{:}\;( \mbox{distinct\_points}(a,b) \ \Rightarrow \ ( \mbox{left\_apart\_point}(c,\mbox{line\_connecting}(a,b)) \ \Rightarrow \ \mbox{right\_apart\_point}(c,\mbox{line\_connecting}(b,a)) ) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO115-1.p} Basic property of orderings on linear structures 5

If Q is between P and R wrt. c and Q' another point distinct
 from Q and lying on c then Q is either between P and Q' or
 between Q' and R wrt. c.




$\mbox{include}(\mbox{'Axioms/GEO004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO004-1.ax'})$

$\mbox{between\_c}(\mbox{sk}_{15},\mbox{sk}_{16},\mbox{sk}_{17},\mbox{sk}_{18})\qquad\mbox{cnf}(\mbox{theorem\_3\_8\_5}_{77},\mbox{negated\_conjecture})$

$\mbox{incident\_c}(\mbox{sk}_{19},\mbox{sk}_{15})\qquad\mbox{cnf}(\mbox{theorem\_3\_8\_5}_{78},\mbox{negated\_conjecture})$

$\mbox{sk}_{17}\ne \mbox{sk}_{19}\qquad\mbox{cnf}(\mbox{theorem\_3\_8\_5}_{79},\mbox{negated\_conjecture})$

$\neg\,\mbox{between\_c}(\mbox{sk}_{15},\mbox{sk}_{16},\mbox{sk}_{17},\mbox{sk}_{19})\qquad\mbox{cnf}(\mbox{theorem\_3\_8\_5}_{80},\mbox{negated\_conjecture})$

$\neg\,\mbox{between\_c}(\mbox{sk}_{15},\mbox{sk}_{19},\mbox{sk}_{17},\mbox{sk}_{18})\qquad\mbox{cnf}(\mbox{theorem\_3\_8\_5}_{81},\mbox{negated\_conjecture})$

\medskip\textbf{GEO030-2.p} Corollary to the outer five-segment axiom






$\mbox{include}(\mbox{'Axioms/GEO002-0.ax'})$

$\mbox{between}(u,v,w)\qquad\mbox{cnf}(\mbox{v\_between\_u\_and\_w},\mbox{hypothesis})$

$\mbox{equidistant}(u,w,u,w_1)\qquad\mbox{cnf}(\mbox{u\_to\_w\_equals\_u\_to\_w}_1,\mbox{hypothesis})$

$\mbox{equidistant}(v,w,v,w_1)\qquad\mbox{cnf}(\mbox{v\_to\_w\_equals\_v\_to\_w}_1,\mbox{hypothesis})$

$u\ne v\qquad\mbox{cnf}(\mbox{u\_not\_v},\mbox{hypothesis})$

$w\ne w_1\qquad\mbox{cnf}(\mbox{prove\_w\_is\_w}_1,\mbox{negated\_conjecture})$

\medskip\textbf{GEO179+2.p} Lemma on symmetry and apartness

If two points X and Y are distinct and a point Z is apart from
 the line connecting X and Y, then the line connecting X and Y
 is distinct from the line connecting Z and X and distinct from
 the line connecting Z and Y.




$\mbox{include}(\mbox{'Axioms/GEO008+0.ax'})$

$\forall x,y,z{:}\;( ( \mbox{distinct\_points}(x,y)\mbox{ and }\mbox{apart\_point\_and\_line}(z,\mbox{line\_connecting}(x,y)) ) \ \Rightarrow \ ( \mbox{distinct\_lines}(\mbox{line\_connecting}(x,y),\mbox{line\_connecting}(z,x))\mbox{ and }\mbox{distinct\_lines}(\mbox{line\_connecting}(x,y),\mbox{line\_connecting}(z,y)) ) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO183+1.p} Lemma on symmetry and apartness

If X and Y are distinct points, then they are incident with
 the line that is equal to the line connecting the points.




$\mbox{include}(\mbox{'Axioms/GEO006+0.ax'})$

$\forall x,y,u,v{:}\;( ( \mbox{distinct\_points}(x,y)\mbox{ and }\mbox{convergent\_lines}(u,v)\mbox{ and }\neg\,\mbox{distinct\_lines}(u,\mbox{line\_connecting}(x,y)) ) \ \Rightarrow \ ( \neg\,\mbox{apart\_point\_and\_line}(x,u)\mbox{ and }\neg\,\mbox{apart\_point\_and\_line}(y,u) ) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO025-2.p} Addition of equal segments






$\mbox{include}(\mbox{'Axioms/GEO002-0.ax'})$

$\mbox{equidistant}(u,v,u_1,v_1)\qquad\mbox{cnf}(\mbox{u\_to\_v\_equals\_u1\_to\_v}_1,\mbox{hypothesis})$

$\mbox{equidistant}(v,w,v_1,w_1)\qquad\mbox{cnf}(\mbox{v\_to\_w\_equals\_v1\_to\_w}_1,\mbox{hypothesis})$

$\mbox{between}(u,v,w)\qquad\mbox{cnf}(\mbox{v\_between\_u\_and\_w},\mbox{hypothesis})$

$\mbox{between}(u_1,v_1,w_1)\qquad\mbox{cnf}(\mbox{v1\_between\_u1\_and\_w}_1,\mbox{hypothesis})$

$\neg\,\mbox{equidistant}(u,w,u_1,w_1)\qquad\mbox{cnf}(\mbox{prove\_equal\_sums},\mbox{negated\_conjecture})$

\medskip\textbf{GEO187+3.p} Symmetry of incidence

If X and Y are distinct points, U and V are distinct points,
 X and Y are incident with the line connecting U and V, then
 U and V are incident with the line connecting X and Y.




$\mbox{include}(\mbox{'Axioms/GEO006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+1.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+2.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+3.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+4.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+5.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+6.ax'})$

$\forall x,y,u,v{:}\;( ( \mbox{distinct\_points}(x,y)\mbox{ and }\mbox{distinct\_points}(u,v)\mbox{ and }\mbox{incident\_point\_and\_line}(x,\mbox{line\_connecting}(u,v))\mbox{ and }\mbox{incident\_point\_and\_line}(y,\mbox{line\_connecting}(u,v)) ) \ \Rightarrow \ ( \mbox{incident\_point\_and\_line}(u,\mbox{line\_connecting}(x,y))\mbox{ and }\mbox{incident\_point\_and\_line}(v,\mbox{line\_connecting}(x,y)) ) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO208+1.p} Point on both parallel lines

If the point X is incident with both the lines Y and Z, and
 Y and Z are parallel, then Y and Z are equal.




$\mbox{include}(\mbox{'Axioms/GEO006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+2.ax'})$

$\forall x,y,z{:}\;( ( \neg\,\mbox{apart\_point\_and\_line}(x,y)\mbox{ and }\neg\,\mbox{apart\_point\_and\_line}(x,z)\mbox{ and }\neg\,\mbox{convergent\_lines}(y,z) ) \ \Rightarrow \ \neg\,\mbox{distinct\_lines}(y,z) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO040-3.p} Antisymmetry of betweenness in its first two arguments






$\mbox{include}(\mbox{'Axioms/GEO002-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO002-2.ax'})$

$\mbox{equidistant}(u,v,u,v)\qquad\mbox{cnf}(d_1,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(w,x,u,v)\qquad\mbox{cnf}(d_2,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(v,u,w,x)\qquad\mbox{cnf}(d_3,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(u,v,x,w)\qquad\mbox{cnf}(\mbox{d4}_1,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(v,u,x,w)\qquad\mbox{cnf}(\mbox{d4}_2,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(w,x,v,u)\qquad\mbox{cnf}(\mbox{d4}_3,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(x,w,u,v)\qquad\mbox{cnf}(\mbox{d4}_4,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(x,w,v,u)\qquad\mbox{cnf}(\mbox{d4}_5,\mbox{axiom})$

$( \mbox{equidistant}(u,v,w,x)\mbox{ and }\mbox{equidistant}(w,x,y,z) ) \ \Rightarrow \ \mbox{equidistant}(u,v,y,z)\qquad\mbox{cnf}(d_5,\mbox{axiom})$

$v = \mbox{extension}(u,v,w,w)\qquad\mbox{cnf}(e_1,\mbox{axiom})$

$y = \mbox{extension}(u,v,w,x) \ \Rightarrow \ \mbox{between}(u,v,y)\qquad\mbox{cnf}(b_0,\mbox{axiom})$

$\mbox{between}(u,v,\mbox{reflection}(u,v))\qquad\mbox{cnf}(\mbox{r2}_1,\mbox{axiom})$

$\mbox{equidistant}(v,\mbox{reflection}(u,v),u,v)\qquad\mbox{cnf}(\mbox{r2}_2,\mbox{axiom})$

$u = v \ \Rightarrow \ v = \mbox{reflection}(u,v)\qquad\mbox{cnf}(\mbox{r3}_1,\mbox{axiom})$

$u = \mbox{reflection}(u,u)\qquad\mbox{cnf}(\mbox{r3}_2,\mbox{axiom})$

$v = \mbox{reflection}(u,v) \ \Rightarrow \ u = v\qquad\mbox{cnf}(r_4,\mbox{axiom})$

$\mbox{equidistant}(u,u,v,v)\qquad\mbox{cnf}(d_7,\mbox{axiom})$

$( \mbox{equidistant}(u,v,u_1,v_1)\mbox{ and }\mbox{equidistant}(v,w,v_1,w_1)\mbox{ and }\mbox{between}(u,v,w)\mbox{ and }\mbox{between}(u_1,v_1,w_1) ) \ \Rightarrow \ \mbox{equidistant}(u,w,u_1,w_1)\qquad\mbox{cnf}(d_8,\mbox{axiom})$

$( \mbox{between}(u,v,w)\mbox{ and }\mbox{between}(u,v,x)\mbox{ and }\mbox{equidistant}(v,w,v,x) ) \ \Rightarrow \ ( u = v\mbox{ or }w = x )\qquad\mbox{cnf}(d_9,\mbox{axiom})$

$\mbox{between}(u,v,w) \ \Rightarrow \ ( u = v\mbox{ or }w = \mbox{extension}(u,v,v,w) )\qquad\mbox{cnf}(\mbox{d10}_1,\mbox{axiom})$

$\mbox{equidistant}(w,x,y,z) \ \Rightarrow \ ( \mbox{extension}(u,v,w,x) = \mbox{extension}(u,v,y,z)\mbox{ or }u = v )\qquad\mbox{cnf}(\mbox{d10}_2,\mbox{axiom})$

$\mbox{extension}(u,v,u,v) = \mbox{extension}(u,v,v,u)\mbox{ or }u = v\qquad\mbox{cnf}(\mbox{d10}_3,\mbox{axiom})$

$\mbox{equidistant}(v,u,v,\mbox{reflection}(\mbox{reflection}(u,v),v))\qquad\mbox{cnf}(r_5,\mbox{axiom})$

$u = \mbox{reflection}(\mbox{reflection}(u,v),v)\qquad\mbox{cnf}(r_6,\mbox{axiom})$

$\mbox{between}(u,v,v)\qquad\mbox{cnf}(t_3,\mbox{axiom})$

$( \mbox{between}(u,w,x)\mbox{ and }u = x ) \ \Rightarrow \ \mbox{between}(v,w,x)\qquad\mbox{cnf}(b_1,\mbox{axiom})$

$\mbox{between}(u,v,w) \ \Rightarrow \ \mbox{between}(w,v,u)\qquad\mbox{cnf}(t_1,\mbox{axiom})$

$\mbox{between}(u,u,v)\qquad\mbox{cnf}(t_2,\mbox{axiom})$

$\mbox{between}(u,v,w)\qquad\mbox{cnf}(\mbox{v\_between\_u\_and\_w},\mbox{hypothesis})$

$\mbox{between}(v,u,w)\qquad\mbox{cnf}(\mbox{u\_between\_v\_and\_w},\mbox{hypothesis})$

$u\ne v\qquad\mbox{cnf}(\mbox{prove\_u\_is\_v},\mbox{negated\_conjecture})$

\medskip\textbf{GEO022-2.p} Ordinary transitivity of equidistance

This form of transitivity is different from that expressed
 in the axioms.




$\mbox{include}(\mbox{'Axioms/GEO002-0.ax'})$

$\mbox{equidistant}(u,v,w,x)\qquad\mbox{cnf}(\mbox{u\_to\_v\_equals\_w\_to\_x},\mbox{hypothesis})$

$\mbox{equidistant}(w,x,y,z)\qquad\mbox{cnf}(\mbox{w\_to\_x\_equals\_y\_to\_z},\mbox{hypothesis})$

$\neg\,\mbox{equidistant}(u,v,y,z)\qquad\mbox{cnf}(\mbox{prove\_transitivity},\mbox{negated\_conjecture})$

\medskip\textbf{GEO213+2.p} Corollary to non-orthogonal lines and a third line

If line L is not orthogonal to line M, then a third line N
 is distinct from L or not orthogonal to M.




$\mbox{include}(\mbox{'Axioms/GEO008+0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+2.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+3.ax'})$

$\forall l,m,n{:}\;( \mbox{unorthogonal\_lines}(l,m) \ \Rightarrow \ ( \mbox{distinct\_lines}(l,n)\mbox{ or }\mbox{unorthogonal\_lines}(m,n) ) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO178+3.p} Lemma on symmetry and apartness

If two points X and Y are distinct and a point Z is apart from
 the line connecting X and Y, then Z and X are distinct, and
 Z and Y are distinct.




$\mbox{include}(\mbox{'Axioms/GEO006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+1.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+2.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+3.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+4.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+5.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+6.ax'})$

$\forall x,y,z{:}\;( ( \mbox{distinct\_points}(x,y)\mbox{ and }\mbox{apart\_point\_and\_line}(z,\mbox{line\_connecting}(x,y)) ) \ \Rightarrow \ ( \mbox{distinct\_points}(z,x)\mbox{ and }\mbox{distinct\_points}(z,y) ) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO194+2.p} Corollary to symmetry of apartness

If the lines X, Y, and Z are pairwise convergent, and the
 intersection point of X and Y is apart from a line Z, then
 the intersection point of X and Z is apart from Y.




$\mbox{include}(\mbox{'Axioms/GEO008+0.ax'})$

$\forall x,y,z{:}\;( ( \mbox{convergent\_lines}(x,y)\mbox{ and }\mbox{convergent\_lines}(z,y)\mbox{ and }\mbox{convergent\_lines}(x,z) ) \ \Rightarrow \ ( \mbox{apart\_point\_and\_line}(\mbox{intersection\_point}(x,y),z) \ \Rightarrow \ \mbox{apart\_point\_and\_line}(\mbox{intersection\_point}(x,z),y) ) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO240+1.p} Lemma on oriented intersection of lines with plane






$\mbox{include}(\mbox{'Axioms/GEO007+0.ax'})$

$\forall a,b,l{:}\;( ( \neg\,\mbox{apart\_point\_and\_line}(a,l)\mbox{ and }\mbox{left\_apart\_point}(b,\mbox{reverse\_line}(l)) ) \ \Rightarrow \ \mbox{left\_convergent\_lines}(l,\mbox{reverse\_line}(\mbox{line\_connecting}(a,b))) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO008-2.p} Betweenness for 5 points on a line (Five point theorem)

For all points x, y, z, w, and v, if y and w are between
 x and z, and $\vee$ is between y and w, then $\vee$ is between x and z.




$\mbox{include}(\mbox{'Axioms/GEO002-0.ax'})$

$\mbox{between}(a,c,e)\qquad\mbox{cnf}(\mbox{c\_between\_a\_and\_e},\mbox{hypothesis})$

$\mbox{between}(a,d,e)\qquad\mbox{cnf}(\mbox{d\_between\_a\_and\_e},\mbox{hypothesis})$

$\mbox{between}(c,b,d)\qquad\mbox{cnf}(\mbox{b\_between\_c\_and\_d},\mbox{hypothesis})$

$\neg\,\mbox{between}(a,b,e)\qquad\mbox{cnf}(\mbox{prove\_betweenness},\mbox{negated\_conjecture})$

\medskip\textbf{GEO050-2.p} First outer connectivity property of betweenness






$\mbox{include}(\mbox{'Axioms/GEO002-0.ax'})$

$\mbox{between}(u,v,w)\qquad\mbox{cnf}(\mbox{v\_between\_u\_and\_w},\mbox{hypothesis})$

$\mbox{between}(u,v,x)\qquad\mbox{cnf}(\mbox{v\_between\_u\_and\_x},\mbox{hypothesis})$

$u\ne v\qquad\mbox{cnf}(\mbox{u\_not\_v},\mbox{hypothesis})$

$\neg\,\mbox{between}(u,w,x)\qquad\mbox{cnf}(\mbox{w\_not\_between\_u\_and\_x},\mbox{hypothesis})$

$\neg\,\mbox{between}(u,x,w)\qquad\mbox{cnf}(\mbox{prove\_x\_between\_u\_and\_w},\mbox{negated\_conjecture})$

\medskip\textbf{GEO248+3.p} A point in each region formed by parallel lines






$\mbox{include}(\mbox{'Axioms/GEO009+0.ax'})$

$\forall a,b,l{:}\;( ( \mbox{apart\_point\_and\_line}(a,l)\mbox{ and }\mbox{incident\_point\_and\_line}(b,\mbox{parallel\_lines}(l,a)) ) \ \Rightarrow \ \mbox{apart\_point\_and\_line}(b,l) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO195+3.p} Corollary to symmetry of apartness

If the lines X, Y, and Z are pairwise convergent, and the
 intersection point of X and Y is apart from a line Z, then
 the intersection point of Y and X is apart from Z.




$\mbox{include}(\mbox{'Axioms/GEO006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+1.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+2.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+3.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+4.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+5.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+6.ax'})$

$\forall x,y,z{:}\;( ( \mbox{convergent\_lines}(x,y)\mbox{ and }\mbox{convergent\_lines}(z,y)\mbox{ and }\mbox{convergent\_lines}(x,z) ) \ \Rightarrow \ ( \mbox{apart\_point\_and\_line}(\mbox{intersection\_point}(x,y),z) \ \Rightarrow \ \mbox{apart\_point\_and\_line}(\mbox{intersection\_point}(y,x),z) ) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO120-1.p} Oriented curve finishing point is endpoint of underlying curve






$\mbox{include}(\mbox{'Axioms/GEO004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO004-2.ax'})$

$\mbox{finish\_point}(\mbox{sk}_{26},\mbox{sk}_{25})\qquad\mbox{cnf}(\mbox{theorem\_4\_6\_2}_{133},\mbox{negated\_conjecture})$

$\neg\,\mbox{end\_point}(\mbox{sk}_{26},\mbox{underlying\_curve}(\mbox{sk}_{25}))\qquad\mbox{cnf}(\mbox{theorem\_4\_6\_2}_{134},\mbox{negated\_conjecture})$

\medskip\textbf{GEO001\_1.p} Betweenness is symmetric in its outer arguments






$\mbox{point}{:}\;\mbox{\$tType}\qquad\mbox{tff}(\mbox{point\_type},\mbox{type})$

$\mbox{line\_point}{:}\;\mbox{\$tType}\qquad\mbox{tff}(\mbox{line\_point\_type},\mbox{type})$

$\mbox{outer\_pasch}{:}\;( \mbox{point}\times \mbox{point}\times \mbox{point}\times \mbox{point}\times \mbox{point} )\to \mbox{point}\qquad\mbox{tff}(\mbox{outer\_pasch\_type},\mbox{type})$

$\mbox{extension}{:}\;( \mbox{point}\times \mbox{point}\times \mbox{line\_point}\times \mbox{line\_point} )\to \mbox{point}\qquad\mbox{tff}(\mbox{extension\_type},\mbox{type})$

$\mbox{equidistant}{:}\;( \mbox{point}\times \mbox{point}\times \mbox{line\_point}\times \mbox{line\_point} )\to \mbox{\$o}\qquad\mbox{tff}(\mbox{equidistant\_type},\mbox{type})$

$={:}\;( \mbox{point}\times \mbox{point} )\to \mbox{\$o}\qquad\mbox{tff}(\mbox{equalish\_type},\mbox{type})$

$\mbox{between}{:}\;( \mbox{point}\times \mbox{point}\times \mbox{point} )\to \mbox{\$o}\qquad\mbox{tff}(\mbox{between\_type},\mbox{type})$

$\forall y{:}\;\mbox{point},x{:}\;\mbox{point}{:}\;( \mbox{between}(x,y,x) \ \Rightarrow \ x{=}y )\qquad\mbox{tff}(\mbox{identity\_for\_betweeness},\mbox{axiom})$

$\forall z{:}\;\mbox{line\_point},y{:}\;\mbox{point},x{:}\;\mbox{point}{:}\;( \mbox{equidistant}(x,y,z,z) \ \Rightarrow \ x{=}y )\qquad\mbox{tff}(\mbox{identity\_for\_equidistance},\mbox{axiom})$

$\forall z{:}\;\mbox{point},y{:}\;\mbox{point},v{:}\;\mbox{point},w{:}\;\mbox{point},x{:}\;\mbox{point}{:}\;( ( \mbox{between}(x,w,v)\mbox{ and }\mbox{between}(y,v,z) ) \ \Rightarrow \ \mbox{between}(x,\mbox{outer\_pasch}(w,x,y,z,v),y) )\qquad\mbox{tff}(\mbox{outer\_pasch}_1,\mbox{axiom})$

$\forall z{:}\;\mbox{point},y{:}\;\mbox{point},v{:}\;\mbox{point},w{:}\;\mbox{point},x{:}\;\mbox{point}{:}\;( ( \mbox{between}(x,w,v)\mbox{ and }\mbox{between}(y,v,z) ) \ \Rightarrow \ \mbox{between}(z,w,\mbox{outer\_pasch}(w,x,y,z,v)) )\qquad\mbox{tff}(\mbox{outer\_pasch}_2,\mbox{axiom})$

$\forall v{:}\;\mbox{line\_point},w{:}\;\mbox{line\_point},y{:}\;\mbox{point},x{:}\;\mbox{point}{:}\;\mbox{between}(x,y,\mbox{extension}(x,y,w,v))\qquad\mbox{tff}(\mbox{segment\_construction}_1,\mbox{axiom})$

$\forall v{:}\;\mbox{line\_point},w{:}\;\mbox{line\_point},x{:}\;\mbox{point},y{:}\;\mbox{point}{:}\;\mbox{equidistant}(y,\mbox{extension}(x,y,w,v),w,v)\qquad\mbox{tff}(\mbox{segment\_construction}_2,\mbox{axiom})$

$\forall z{:}\;\mbox{point},w{:}\;\mbox{point},y{:}\;\mbox{point},x{:}\;\mbox{point}{:}\;( ( x{=}y\mbox{ and }\mbox{between}(w,z,x) ) \ \Rightarrow \ \mbox{between}(w,z,y) )\qquad\mbox{tff}(\mbox{between\_substitution}_3,\mbox{axiom})$

$\forall x{:}\;\mbox{point},y{:}\;\mbox{point},z{:}\;\mbox{point}{:}\;( \mbox{between}(x,y,z) \ \Rightarrow \ \mbox{between}(z,y,x) )\qquad\mbox{tff}(\mbox{symmetric},\mbox{conjecture})$

\medskip\textbf{GEO238+3.p} Strengthened axiom of Pasch






$\mbox{include}(\mbox{'Axioms/GEO009+0.ax'})$

$\forall a,b,c,l{:}\;( ( \mbox{divides\_points}(l,a,b)\mbox{ and }\mbox{divides\_points}(l,a,c) ) \ \Rightarrow \ \neg\,\mbox{divides\_points}(l,b,c) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO177+3.p} Symmetry of apartness

If the points X and Y are distinct and U and V are distinct,
 and X or Y is apart from the line connecting U and V, then
 U or V are apart from the line connecting X and Y.




$\mbox{include}(\mbox{'Axioms/GEO006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+1.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+2.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+3.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+4.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+5.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+6.ax'})$

$\forall x,y,u,v{:}\;( ( \mbox{distinct\_points}(x,y)\mbox{ and }\mbox{distinct\_points}(u,v) ) \ \Rightarrow \ ( ( \mbox{apart\_point\_and\_line}(x,\mbox{line\_connecting}(u,v))\mbox{ or }\mbox{apart\_point\_and\_line}(y,\mbox{line\_connecting}(u,v)) ) \ \Rightarrow \ ( \mbox{apart\_point\_and\_line}(u,\mbox{line\_connecting}(x,y))\mbox{ or }\mbox{apart\_point\_and\_line}(v,\mbox{line\_connecting}(x,y)) ) ) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO106+1.p} Two common endpoints means identical or sum to whole

If two sub-curves have two common endpoints then they are
 identical or their sum is the whole curve.




$\mbox{include}(\mbox{'Axioms/GEO004+0.ax'})$

$\forall c,c_1,c_2{:}\;( ( \exists p,q{:}\;( \mbox{end\_point}(p,c_1)\mbox{ and }\mbox{end\_point}(q,c_1)\mbox{ and }\mbox{end\_point}(p,c_2)\mbox{ and }\mbox{end\_point}(q,c_2)\mbox{ and }p\ne q )\mbox{ and }\mbox{closed}(c)\mbox{ and }\mbox{part\_of}(c_1,c)\mbox{ and }\mbox{part\_of}(c_2,c) ) \ \Rightarrow \ ( c_1 = c_2\mbox{ or }c = c_1 + c_2 ) )\qquad\mbox{fof}(\mbox{theorem\_2}_{24},\mbox{conjecture})$

\medskip\textbf{GEO001-3.p} Betweenness is symmetric in its outer arguments






$\mbox{include}(\mbox{'Axioms/GEO002-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO002-2.ax'})$

$\mbox{equidistant}(u,v,u,v)\qquad\mbox{cnf}(d_1,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(w,x,u,v)\qquad\mbox{cnf}(d_2,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(v,u,w,x)\qquad\mbox{cnf}(d_3,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(u,v,x,w)\qquad\mbox{cnf}(\mbox{d4}_1,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(v,u,x,w)\qquad\mbox{cnf}(\mbox{d4}_2,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(w,x,v,u)\qquad\mbox{cnf}(\mbox{d4}_3,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(x,w,u,v)\qquad\mbox{cnf}(\mbox{d4}_4,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(x,w,v,u)\qquad\mbox{cnf}(\mbox{d4}_5,\mbox{axiom})$

$( \mbox{equidistant}(u,v,w,x)\mbox{ and }\mbox{equidistant}(w,x,y,z) ) \ \Rightarrow \ \mbox{equidistant}(u,v,y,z)\qquad\mbox{cnf}(d_5,\mbox{axiom})$

$v = \mbox{extension}(u,v,w,w)\qquad\mbox{cnf}(e_1,\mbox{axiom})$

$y = \mbox{extension}(u,v,w,x) \ \Rightarrow \ \mbox{between}(u,v,y)\qquad\mbox{cnf}(b_0,\mbox{axiom})$

$\mbox{between}(u,v,\mbox{reflection}(u,v))\qquad\mbox{cnf}(\mbox{r2}_1,\mbox{axiom})$

$\mbox{equidistant}(v,\mbox{reflection}(u,v),u,v)\qquad\mbox{cnf}(\mbox{r2}_2,\mbox{axiom})$

$u = v \ \Rightarrow \ v = \mbox{reflection}(u,v)\qquad\mbox{cnf}(\mbox{r3}_1,\mbox{axiom})$

$u = \mbox{reflection}(u,u)\qquad\mbox{cnf}(\mbox{r3}_2,\mbox{axiom})$

$v = \mbox{reflection}(u,v) \ \Rightarrow \ u = v\qquad\mbox{cnf}(r_4,\mbox{axiom})$

$\mbox{equidistant}(u,u,v,v)\qquad\mbox{cnf}(d_7,\mbox{axiom})$

$( \mbox{equidistant}(u,v,u_1,v_1)\mbox{ and }\mbox{equidistant}(v,w,v_1,w_1)\mbox{ and }\mbox{between}(u,v,w)\mbox{ and }\mbox{between}(u_1,v_1,w_1) ) \ \Rightarrow \ \mbox{equidistant}(u,w,u_1,w_1)\qquad\mbox{cnf}(d_8,\mbox{axiom})$

$( \mbox{between}(u,v,w)\mbox{ and }\mbox{between}(u,v,x)\mbox{ and }\mbox{equidistant}(v,w,v,x) ) \ \Rightarrow \ ( u = v\mbox{ or }w = x )\qquad\mbox{cnf}(d_9,\mbox{axiom})$

$\mbox{between}(u,v,w) \ \Rightarrow \ ( u = v\mbox{ or }w = \mbox{extension}(u,v,v,w) )\qquad\mbox{cnf}(\mbox{d10}_1,\mbox{axiom})$

$\mbox{equidistant}(w,x,y,z) \ \Rightarrow \ ( \mbox{extension}(u,v,w,x) = \mbox{extension}(u,v,y,z)\mbox{ or }u = v )\qquad\mbox{cnf}(\mbox{d10}_2,\mbox{axiom})$

$\mbox{extension}(u,v,u,v) = \mbox{extension}(u,v,v,u)\mbox{ or }u = v\qquad\mbox{cnf}(\mbox{d10}_3,\mbox{axiom})$

$\mbox{equidistant}(v,u,v,\mbox{reflection}(\mbox{reflection}(u,v),v))\qquad\mbox{cnf}(r_5,\mbox{axiom})$

$u = \mbox{reflection}(\mbox{reflection}(u,v),v)\qquad\mbox{cnf}(r_6,\mbox{axiom})$

$\mbox{between}(u,v,v)\qquad\mbox{cnf}(t_3,\mbox{axiom})$

$( \mbox{between}(u,w,x)\mbox{ and }u = x ) \ \Rightarrow \ \mbox{between}(v,w,x)\qquad\mbox{cnf}(b_1,\mbox{axiom})$

$\mbox{between}(a,b,c)\qquad\mbox{cnf}(\mbox{b\_between\_a\_and\_c},\mbox{hypothesis})$

$\neg\,\mbox{between}(c,b,a)\qquad\mbox{cnf}(\mbox{prove\_b\_between\_c\_and\_a},\mbox{negated\_conjecture})$

\medskip\textbf{GEO121+1.p} Endpoints are either starting or finishing points

Every endpoint of the underlying curve of an oriented curve is
 either a starting point or finishing point of the oriented curve.




$\mbox{include}(\mbox{'Axioms/GEO004+0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO004+1.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO004+2.ax'})$

$\forall o,p{:}\;( \mbox{end\_point}(p,\mbox{underlying\_curve}(o)) \ \Rightarrow \ ( \mbox{start\_point}(p,o)\mbox{ or }\mbox{finish\_point}(p,o) ) )\qquad\mbox{fof}(\mbox{theorem\_4}_7,\mbox{conjecture})$

\medskip\textbf{GEO204+1.p} Distinct points and equal lines

If the points X and Y are distinct, and the points Y and Z are
 equal, then X and Z are distinct, and the line connecting X
 and Y is equivalent to the line connecting X and Z.




$\mbox{include}(\mbox{'Axioms/GEO006+0.ax'})$

$\forall x,y,z{:}\;( ( \mbox{distinct\_points}(x,y)\mbox{ and }\neg\,\mbox{distinct\_points}(y,z) ) \ \Rightarrow \ ( \mbox{distinct\_points}(x,z)\mbox{ and }\neg\,\mbox{distinct\_lines}(\mbox{line\_connecting}(x,y),\mbox{line\_connecting}(x,z)) ) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO079-1.p} The alternate interior angles in a trapezoid are equal

The alternate interior angles formed by a diagonal of a (not
 necessarily isosceles) trapezoid are equal.




$( \mbox{right\_angle}(u,v,w)\mbox{ and }\mbox{right\_angle}(x,y,z) ) \ \Rightarrow \ \mbox{eq}(u,v,w,x,y,z)\qquad\mbox{cnf}(\mbox{right\_angles\_are\_equal},\mbox{axiom})$

$\mbox{congruent}(u,v,w,x,y,z) \ \Rightarrow \ \mbox{eq}(u,v,w,x,y,z)\qquad\mbox{cnf}(\mbox{corresponding\_angles\_are\_equal},\mbox{axiom})$

$\mbox{trapezoid}(u,v,w,x) \ \Rightarrow \ \mbox{parallel}(v,w,u,x)\qquad\mbox{cnf}(\mbox{trapezoid\_definition},\mbox{axiom})$

$\mbox{parallel}(u,v,x,y) \ \Rightarrow \ \mbox{eq}(x,v,u,v,x,y)\qquad\mbox{cnf}(\mbox{interior\_angles\_are\_equal},\mbox{axiom})$

$\mbox{trapezoid}(a,b,c,d)\qquad\mbox{cnf}(\mbox{a\_trapezoid},\mbox{hypothesis})$

$\neg\,\mbox{eq}(a,c,b,c,a,d)\qquad\mbox{cnf}(\mbox{prove\_angles\_equal},\mbox{negated\_conjecture})$

\medskip\textbf{GEO205+3.p} Convergent lines and equal points

If the lines X and Y are convergent, and Y and Z are
 equivalent, then X and Z are convergent, and the intersection
 point of X and Y, and the intersection point of X and Z are
 equal.




$\mbox{include}(\mbox{'Axioms/GEO006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+1.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+2.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+3.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+4.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+5.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+6.ax'})$

$\forall x,y,z{:}\;( ( \mbox{convergent\_lines}(x,y)\mbox{ and }\mbox{equal\_lines}(y,z) ) \ \Rightarrow \ ( \mbox{convergent\_lines}(x,z)\mbox{ and }\mbox{equal\_points}(\mbox{intersection\_point}(x,y),\mbox{intersection\_point}(x,z)) ) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO228+3.p} Obervability of equal or opposite direction






$\mbox{include}(\mbox{'Axioms/GEO009+0.ax'})$

$\forall l,m{:}\;( \mbox{parallel\_lines}(l,m)\iff ( \mbox{equally\_directed\_lines}(l,m)\mbox{ or }\mbox{equally\_directed\_opposite\_lines}(l,m) ) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO011-1.p} The axiom set points are not collinear






$\mbox{include}(\mbox{'Axioms/GEO001-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO001-1.ax'})$

$\mbox{colinear}(\mbox{lower\_dimension\_point}_1,\mbox{lower\_dimension\_point}_2,\mbox{lower\_dimension\_point}_3)\qquad\mbox{cnf}(\mbox{prove\_lower\_dimension\_points\_not\_colinear},\mbox{negated\_conjecture})$

\medskip\textbf{GEO211+1.p} A line is not orthogonal to itself






$\mbox{include}(\mbox{'Axioms/GEO006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+2.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+3.ax'})$

$\forall l{:}\;\mbox{unorthogonal\_lines}(l,l)\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO110+1.p} Betweenness for closed curves






$\mbox{include}(\mbox{'Axioms/GEO004+0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO004+1.ax'})$

$\forall c,p,q,r{:}\;( ( \mbox{closed}(c)\mbox{ and }p\ne q\mbox{ and }q\ne r\mbox{ and }p\ne r\mbox{ and }\mbox{incident\_c}(p,c)\mbox{ and }\mbox{incident\_c}(q,c)\mbox{ and }\mbox{incident\_c}(r,c) ) \ \Rightarrow \ \mbox{between\_c}(c,p,q,r) )\qquad\mbox{fof}(\mbox{theorem\_3}_7,\mbox{conjecture})$

\medskip\textbf{GEO215+2.p} Third line not orthogonal to two convergent lines

If two lines L and M are convergent, then a third line N is
 not orthogonal to L or M.




$\mbox{include}(\mbox{'Axioms/GEO008+0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+2.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+3.ax'})$

$\forall l,m,n{:}\;( \mbox{convergent\_lines}(l,m) \ \Rightarrow \ ( \mbox{unorthogonal\_lines}(l,n)\mbox{ or }\mbox{unorthogonal\_lines}(m,n) ) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO187+1.p} Symmetry of incidence

If X and Y are distinct points, U and V are distinct points,
 X and Y are incident with the line connecting U and V, then
 U and V are incident with the line connecting X and Y.




$\mbox{include}(\mbox{'Axioms/GEO006+0.ax'})$

$\forall x,y,u,v{:}\;( ( \mbox{distinct\_points}(x,y)\mbox{ and }\mbox{distinct\_points}(u,v)\mbox{ and }\neg\,\mbox{apart\_point\_and\_line}(x,\mbox{line\_connecting}(u,v))\mbox{ and }\neg\,\mbox{apart\_point\_and\_line}(y,\mbox{line\_connecting}(u,v)) ) \ \Rightarrow \ ( \neg\,\mbox{apart\_point\_and\_line}(u,\mbox{line\_connecting}(x,y))\mbox{ and }\neg\,\mbox{apart\_point\_and\_line}(v,\mbox{line\_connecting}(x,y)) ) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO201+1.p} Distinct ends means distinct lines

If the lines X and Y are convergent, then the intersection
 point of X and Y is equal to the intersection point of X and Y.




$\mbox{include}(\mbox{'Axioms/GEO006+0.ax'})$

$\forall x,y{:}\;( \mbox{convergent\_lines}(x,y) \ \Rightarrow \ \neg\,\mbox{distinct\_points}(\mbox{intersection\_point}(x,y),\mbox{intersection\_point}(y,x)) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO188+2.p} Collary to symmetry of incidence

If X, Y, and Z are pairwise distinct, and Z is incident with
 the line connecting X and Y, then X is incident with the line
 connecting Z and Y.




$\mbox{include}(\mbox{'Axioms/GEO008+0.ax'})$

$\forall x,y,z{:}\;( ( \mbox{distinct\_points}(x,y)\mbox{ and }\mbox{distinct\_points}(x,z)\mbox{ and }\mbox{distinct\_points}(y,z)\mbox{ and }\neg\,\mbox{apart\_point\_and\_line}(z,\mbox{line\_connecting}(x,y)) ) \ \Rightarrow \ \neg\,\mbox{apart\_point\_and\_line}(x,\mbox{line\_connecting}(z,y)) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO053-2.p} Unique endpoint






$\mbox{include}(\mbox{'Axioms/GEO002-0.ax'})$

$\mbox{between}(u,v,w)\qquad\mbox{cnf}(\mbox{v\_between\_u\_and\_w},\mbox{hypothesis})$

$\mbox{equidistant}(u,v,u,w)\qquad\mbox{cnf}(\mbox{u\_to\_v\_equals\_u\_to\_w},\mbox{hypothesis})$

$v\ne w\qquad\mbox{cnf}(\mbox{prove\_v\_equals\_w},\mbox{negated\_conjecture})$

\medskip\textbf{GEO080+1.p} Reflexivity of part\_of






$\mbox{include}(\mbox{'Axioms/GEO004+0.ax'})$

$\forall c{:}\;\mbox{part\_of}(c,c)\qquad\mbox{fof}(\mbox{prove\_reflexivity},\mbox{conjecture})$

\medskip\textbf{GEO117-1.p} Precedence on oriented curves is irreflexive






$\mbox{include}(\mbox{'Axioms/GEO004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO004-2.ax'})$

$\mbox{ordered\_by}(\mbox{sk}_{25},\mbox{sk}_{26},\mbox{sk}_{26})\qquad\mbox{cnf}(\mbox{theorem\_4\_4}_{133},\mbox{negated\_conjecture})$

\medskip\textbf{GEO184+2.p} Lemma on symmetry and apartness

If X and Y are distinct points, then the line that is incident
 with both points is equal to the line connecting them.




$\mbox{include}(\mbox{'Axioms/GEO008+0.ax'})$

$\forall x,y,u,v{:}\;( ( \mbox{distinct\_points}(x,y)\mbox{ and }\mbox{convergent\_lines}(u,v)\mbox{ and }\neg\,\mbox{apart\_point\_and\_line}(x,u)\mbox{ and }\neg\,\mbox{apart\_point\_and\_line}(y,u) ) \ \Rightarrow \ \neg\,\mbox{distinct\_lines}(u,\mbox{line\_connecting}(x,y)) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO118+1.p} Precedence on oriented curves is asymmetric






$\mbox{include}(\mbox{'Axioms/GEO004+0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO004+1.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO004+2.ax'})$

$\forall o,p,q{:}\;( \mbox{ordered\_by}(o,p,q) \ \Rightarrow \ \neg\,\mbox{ordered\_by}(o,q,p) )\qquad\mbox{fof}(\mbox{theorem\_4}_5,\mbox{conjecture})$

\medskip\textbf{GEO223+1.p} Corollary to uniqueness of parallels

The parallel to line L through a point A is equal to the line,
 that is orthogonal to the orthogonal to L through A, and goes
 through A as well.




$\mbox{include}(\mbox{'Axioms/GEO006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+2.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+3.ax'})$

$\forall a,l{:}\;\neg\,\mbox{distinct\_lines}(\mbox{parallel\_through\_point}(l,a),\mbox{orthogonal\_through\_point}(\mbox{orthogonal\_through\_point}(l,a),a))\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO149-1.p} Condition for meeting at two points

Objects can meet at two points only if they are ordered in the
 same way on both trajectories




$\mbox{include}(\mbox{'Axioms/GEO004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO004-2.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO004-3.ax'})$

$\mbox{connect}(\mbox{sk}_{29},\mbox{sk}_{30},\mbox{sk}_{27})\qquad\mbox{cnf}(\mbox{t15}_{156},\mbox{negated\_conjecture})$

$\mbox{connect}(\mbox{sk}_{29},\mbox{sk}_{30},\mbox{sk}_{28})\qquad\mbox{cnf}(\mbox{t15}_{157},\mbox{negated\_conjecture})$

$\mbox{ordered\_by}(\mbox{trajectory\_of}(\mbox{sk}_{29}),\mbox{sk}_{27},\mbox{sk}_{28})\qquad\mbox{cnf}(\mbox{t15}_{158},\mbox{negated\_conjecture})$

$\neg\,\mbox{ordered\_by}(\mbox{trajectory\_of}(\mbox{sk}_{30}),\mbox{sk}_{27},\mbox{sk}_{28})\qquad\mbox{cnf}(\mbox{t15}_{159},\mbox{negated\_conjecture})$

\medskip\textbf{GEO093+1.p} Sum of meeting open curves is open

If two open sub-curves of an open curve meet, then their sum
 is also open.




$\mbox{include}(\mbox{'Axioms/GEO004+0.ax'})$

$\forall c,c_1,c_2,p{:}\;( ( \mbox{open}(c)\mbox{ and }\mbox{part\_of}(c_1,c)\mbox{ and }\mbox{part\_of}(c_2,c)\mbox{ and }p\wedge c_1{=}c_2 ) \ \Rightarrow \ \mbox{open}(c_1 + c_2) )\qquad\mbox{fof}(\mbox{proposition\_2\_14}_2,\mbox{conjecture})$

\medskip\textbf{GEO087+1.p} If one curve is part of another curve then they cannot meet






$\mbox{include}(\mbox{'Axioms/GEO004+0.ax'})$

$\forall c_1,c_2{:}\;( \mbox{part\_of}(c_1,c_2) \ \Rightarrow \ \neg\,\exists p{:}\;p\wedge c_1{=}c_2 )\qquad\mbox{fof}(\mbox{corollary\_2}_9,\mbox{conjecture})$

\medskip\textbf{GEO132+1.p} Betweenness and precedence property 1






$\mbox{include}(\mbox{'Axioms/GEO004+0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO004+1.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO004+2.ax'})$

$\forall o,p,q,s{:}\;( ( \mbox{ordered\_by}(o,p,q)\mbox{ and }p\ne s\mbox{ and }\mbox{incident\_o}(s,o) ) \ \Rightarrow \ ( \mbox{ordered\_by}(o,p,s)\iff \neg\,\mbox{between}(o,s,p,q) ) )\qquad\mbox{fof}(\mbox{theorem\_4\_17}_1,\mbox{conjecture})$

\medskip\textbf{GEO186+2.p} Lemma on symmetry and apartness

If the lines U and V are convergent, then the point that is
 incident with both lines is equal to the intersection point
 of both lines.




$\mbox{include}(\mbox{'Axioms/GEO008+0.ax'})$

$\forall x,y,u,v{:}\;( ( \mbox{distinct\_points}(x,y)\mbox{ and }\mbox{convergent\_lines}(u,v)\mbox{ and }\neg\,\mbox{apart\_point\_and\_line}(x,u)\mbox{ and }\neg\,\mbox{apart\_point\_and\_line}(x,v) ) \ \Rightarrow \ \neg\,\mbox{distinct\_points}(x,\mbox{intersection\_point}(u,v)) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO041-3.p} Corollary to antisymmetry of betweenness in first 2 arguments






$\mbox{include}(\mbox{'Axioms/GEO002-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO002-2.ax'})$

$\mbox{equidistant}(u,v,u,v)\qquad\mbox{cnf}(d_1,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(w,x,u,v)\qquad\mbox{cnf}(d_2,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(v,u,w,x)\qquad\mbox{cnf}(d_3,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(u,v,x,w)\qquad\mbox{cnf}(\mbox{d4}_1,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(v,u,x,w)\qquad\mbox{cnf}(\mbox{d4}_2,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(w,x,v,u)\qquad\mbox{cnf}(\mbox{d4}_3,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(x,w,u,v)\qquad\mbox{cnf}(\mbox{d4}_4,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(x,w,v,u)\qquad\mbox{cnf}(\mbox{d4}_5,\mbox{axiom})$

$( \mbox{equidistant}(u,v,w,x)\mbox{ and }\mbox{equidistant}(w,x,y,z) ) \ \Rightarrow \ \mbox{equidistant}(u,v,y,z)\qquad\mbox{cnf}(d_5,\mbox{axiom})$

$v = \mbox{extension}(u,v,w,w)\qquad\mbox{cnf}(e_1,\mbox{axiom})$

$y = \mbox{extension}(u,v,w,x) \ \Rightarrow \ \mbox{between}(u,v,y)\qquad\mbox{cnf}(b_0,\mbox{axiom})$

$\mbox{between}(u,v,\mbox{reflection}(u,v))\qquad\mbox{cnf}(\mbox{r2}_1,\mbox{axiom})$

$\mbox{equidistant}(v,\mbox{reflection}(u,v),u,v)\qquad\mbox{cnf}(\mbox{r2}_2,\mbox{axiom})$

$u = v \ \Rightarrow \ v = \mbox{reflection}(u,v)\qquad\mbox{cnf}(\mbox{r3}_1,\mbox{axiom})$

$u = \mbox{reflection}(u,u)\qquad\mbox{cnf}(\mbox{r3}_2,\mbox{axiom})$

$v = \mbox{reflection}(u,v) \ \Rightarrow \ u = v\qquad\mbox{cnf}(r_4,\mbox{axiom})$

$\mbox{equidistant}(u,u,v,v)\qquad\mbox{cnf}(d_7,\mbox{axiom})$

$( \mbox{equidistant}(u,v,u_1,v_1)\mbox{ and }\mbox{equidistant}(v,w,v_1,w_1)\mbox{ and }\mbox{between}(u,v,w)\mbox{ and }\mbox{between}(u_1,v_1,w_1) ) \ \Rightarrow \ \mbox{equidistant}(u,w,u_1,w_1)\qquad\mbox{cnf}(d_8,\mbox{axiom})$

$( \mbox{between}(u,v,w)\mbox{ and }\mbox{between}(u,v,x)\mbox{ and }\mbox{equidistant}(v,w,v,x) ) \ \Rightarrow \ ( u = v\mbox{ or }w = x )\qquad\mbox{cnf}(d_9,\mbox{axiom})$

$\mbox{between}(u,v,w) \ \Rightarrow \ ( u = v\mbox{ or }w = \mbox{extension}(u,v,v,w) )\qquad\mbox{cnf}(\mbox{d10}_1,\mbox{axiom})$

$\mbox{equidistant}(w,x,y,z) \ \Rightarrow \ ( \mbox{extension}(u,v,w,x) = \mbox{extension}(u,v,y,z)\mbox{ or }u = v )\qquad\mbox{cnf}(\mbox{d10}_2,\mbox{axiom})$

$\mbox{extension}(u,v,u,v) = \mbox{extension}(u,v,v,u)\mbox{ or }u = v\qquad\mbox{cnf}(\mbox{d10}_3,\mbox{axiom})$

$\mbox{equidistant}(v,u,v,\mbox{reflection}(\mbox{reflection}(u,v),v))\qquad\mbox{cnf}(r_5,\mbox{axiom})$

$u = \mbox{reflection}(\mbox{reflection}(u,v),v)\qquad\mbox{cnf}(r_6,\mbox{axiom})$

$\mbox{between}(u,v,v)\qquad\mbox{cnf}(t_3,\mbox{axiom})$

$( \mbox{between}(u,w,x)\mbox{ and }u = x ) \ \Rightarrow \ \mbox{between}(v,w,x)\qquad\mbox{cnf}(b_1,\mbox{axiom})$

$\mbox{between}(u,v,w) \ \Rightarrow \ \mbox{between}(w,v,u)\qquad\mbox{cnf}(t_1,\mbox{axiom})$

$\mbox{between}(u,u,v)\qquad\mbox{cnf}(t_2,\mbox{axiom})$

$( \mbox{between}(u,v,w)\mbox{ and }\mbox{between}(v,u,w) ) \ \Rightarrow \ u = v\qquad\mbox{cnf}(b_2,\mbox{axiom})$

$\mbox{between}(u,v,w)\qquad\mbox{cnf}(\mbox{v\_between\_u\_and\_w},\mbox{hypothesis})$

$\mbox{between}(u,w,v)\qquad\mbox{cnf}(\mbox{w\_between\_u\_and\_v},\mbox{hypothesis})$

$v\ne w\qquad\mbox{cnf}(\mbox{prove\_v\_is\_w},\mbox{negated\_conjecture})$

\medskip\textbf{GEO099-1.p} Open subcurves can be complemented to form the sum

Every open sub-curve of a closed curve can be complemented by
 another curve so that their sum constitute the closed curve.




$\mbox{include}(\mbox{'Axioms/GEO004-0.ax'})$

$\mbox{closed}(\mbox{sk}_{14})\qquad\mbox{cnf}(\mbox{theorem\_2\_17}_{67},\mbox{negated\_conjecture})$

$\mbox{part\_of}(\mbox{sk}_{15},\mbox{sk}_{14})\qquad\mbox{cnf}(\mbox{theorem\_2\_17}_{68},\mbox{negated\_conjecture})$

$\mbox{end\_point}(\mbox{sk}_{16},\mbox{sk}_{15})\qquad\mbox{cnf}(\mbox{theorem\_2\_17}_{69},\mbox{negated\_conjecture})$

$\mbox{end\_point}(\mbox{sk}_{17},\mbox{sk}_{15})\qquad\mbox{cnf}(\mbox{theorem\_2\_17}_{70},\mbox{negated\_conjecture})$

$\mbox{sk}_{16}\ne \mbox{sk}_{17}\qquad\mbox{cnf}(\mbox{theorem\_2\_17}_{71},\mbox{negated\_conjecture})$

$( \mbox{sk}_{16}\wedge \mbox{sk}_{15}{=}a\mbox{ and }\mbox{sk}_{17}\wedge \mbox{sk}_{15}{=}a ) \ \Rightarrow \ \mbox{sk}_{14}\ne \mbox{sk}_{15} + a\qquad\mbox{cnf}(\mbox{theorem\_2\_17}_{72},\mbox{negated\_conjecture})$

\medskip\textbf{GEO100-1.p} Subcurves with common endpoint can be complemented

Every proper sub-curve of an open curve that has a common
 endpoint with the open curve can be complemented by another
 curve so that their sum constitute the open curve.




$\mbox{include}(\mbox{'Axioms/GEO004-0.ax'})$

$\mbox{open}(\mbox{sk}_{14})\qquad\mbox{cnf}(\mbox{theorem\_2\_18}_{67},\mbox{negated\_conjecture})$

$\mbox{part\_of}(\mbox{sk}_{15},\mbox{sk}_{14})\qquad\mbox{cnf}(\mbox{theorem\_2\_18}_{68},\mbox{negated\_conjecture})$

$\mbox{sk}_{15}\ne \mbox{sk}_{14}\qquad\mbox{cnf}(\mbox{theorem\_2\_18}_{69},\mbox{negated\_conjecture})$

$\mbox{end\_point}(\mbox{sk}_{16},\mbox{sk}_{15})\qquad\mbox{cnf}(\mbox{theorem\_2\_18}_{70},\mbox{negated\_conjecture})$

$\mbox{end\_point}(\mbox{sk}_{16},\mbox{sk}_{14})\qquad\mbox{cnf}(\mbox{theorem\_2\_18}_{71},\mbox{negated\_conjecture})$

$( a\wedge \mbox{sk}_{15}{=}b\mbox{ and }\mbox{sk}_{14} = \mbox{sk}_{15} + b ) \ \Rightarrow \ \mbox{sk}_{16} = a\qquad\mbox{cnf}(\mbox{theorem\_2\_18}_{72},\mbox{negated\_conjecture})$

\medskip\textbf{GEO045-2.p} Second outer transitivity property of betweenness






$\mbox{include}(\mbox{'Axioms/GEO002-0.ax'})$

$\mbox{between}(u,v,w)\qquad\mbox{cnf}(\mbox{v\_between\_u\_and\_w},\mbox{hypothesis})$

$\mbox{between}(v,w,x)\qquad\mbox{cnf}(\mbox{w\_between\_v\_and\_x},\mbox{hypothesis})$

$v\ne w\qquad\mbox{cnf}(\mbox{v\_not\_w},\mbox{hypothesis})$

$\neg\,\mbox{between}(u,v,x)\qquad\mbox{cnf}(\mbox{prove\_v\_between\_u\_and\_x},\mbox{negated\_conjecture})$

\medskip\textbf{GEO037-2.p} A segment can be extended






$\mbox{include}(\mbox{'Axioms/GEO002-0.ax'})$

$( \mbox{equidistant}(v,\mbox{extension}(u,v,\mbox{lower\_dimension\_point}_1,\mbox{lower\_dimension\_point}_2),x,\mbox{extension}(w,x,\mbox{lower\_dimension\_point}_1,\mbox{lower\_dimension\_point}_2))\mbox{ and }\mbox{between}(u,v,\mbox{extension}(u,v,\mbox{lower\_dimension\_point}_1,\mbox{lower\_dimension\_point}_2)) ) \ \Rightarrow \ v = \mbox{extension}(u,v,\mbox{lower\_dimension\_point}_1,\mbox{lower\_dimension\_point}_2)\qquad\mbox{cnf}(\mbox{prove\_lengthen},\mbox{negated\_conjecture})$

\medskip\textbf{GEO073-1.p} The diagonals of a non-degenerate rectancle bisect






$\mbox{include}(\mbox{'Axioms/GEO001-0.ax'})$

$\mbox{equidistant}(u,v,w,x)\qquad\mbox{cnf}(\mbox{u\_to\_v\_equals\_w\_to\_x},\mbox{hypothesis})$

$\mbox{equidistant}(v,w,x,u)\qquad\mbox{cnf}(\mbox{v\_to\_w\_equals\_x\_to\_u},\mbox{hypothesis})$

$\mbox{equidistant}(u,w,v,x)\qquad\mbox{cnf}(\mbox{u\_to\_w\_equals\_v\_to\_x},\mbox{hypothesis})$

$\mbox{between}(u,y,w)\qquad\mbox{cnf}(\mbox{y\_between\_u\_and\_w},\mbox{hypothesis})$

$\mbox{between}(v,y,x)\qquad\mbox{cnf}(\mbox{y\_between\_v\_and\_x},\mbox{hypothesis})$

$u\ne v\qquad\mbox{cnf}(\mbox{u\_not\_v},\mbox{hypothesis})$

$x\ne u\qquad\mbox{cnf}(\mbox{x\_not\_u},\mbox{hypothesis})$

$\mbox{equidistant}(u,y,w,y) \ \Rightarrow \ \neg\,\mbox{equidistant}(v,y,x,y)\qquad\mbox{cnf}(\mbox{prove\_bisection},\mbox{negated\_conjecture})$

\medskip\textbf{GEO125+1.p} Every oriented curve has at most one finishing point






$\mbox{include}(\mbox{'Axioms/GEO004+0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO004+1.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO004+2.ax'})$

$\forall o,p,q{:}\;( ( \mbox{finish\_point}(p,o)\mbox{ and }\mbox{finish\_point}(q,o) ) \ \Rightarrow \ p = q )\qquad\mbox{fof}(\mbox{corollary\_4\_10}_2,\mbox{conjecture})$

\medskip\textbf{GEO201+3.p} Distinct ends means distinct lines

If the lines X and Y are convergent, then the intersection
 point of X and Y is equal to the intersection point of X and Y.




$\mbox{include}(\mbox{'Axioms/GEO006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+1.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+2.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+3.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+4.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+5.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+6.ax'})$

$\forall x,y{:}\;( \mbox{convergent\_lines}(x,y) \ \Rightarrow \ \mbox{equal\_points}(\mbox{intersection\_point}(x,y),\mbox{intersection\_point}(y,x)) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO001-1.p} Betweenness is symmetric in its outer arguments






$\mbox{include}(\mbox{'Axioms/GEO001-0.ax'})$

$\mbox{between}(a,b,c)\qquad\mbox{cnf}(\mbox{b\_between\_a\_and\_c},\mbox{hypothesis})$

$\neg\,\mbox{between}(c,b,a)\qquad\mbox{cnf}(\mbox{prove\_b\_between\_c\_and\_a},\mbox{negated\_conjecture})$

\medskip\textbf{GEO176+2.p} Lemma on symmetry and apartness

If two lines are convergent and a point is apart from at least
 one of these lines, then this point is distinct from the
 intersection point of these lines.




$\mbox{include}(\mbox{'Axioms/GEO008+0.ax'})$

$\forall x,y,u,v{:}\;( ( \mbox{distinct\_points}(x,y)\mbox{ and }\mbox{convergent\_lines}(u,v)\mbox{ and }( \mbox{apart\_point\_and\_line}(x,u)\mbox{ or }\mbox{apart\_point\_and\_line}(x,v) ) ) \ \Rightarrow \ \mbox{distinct\_points}(x,\mbox{intersection\_point}(u,v)) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO182+1.p} Triangle axiom 3

If X and Y are distinct points and Z is apart from the line
 connecting X and Y, then Z is apart from the line connecting
 Y and X.




$\mbox{include}(\mbox{'Axioms/GEO006+0.ax'})$

$\forall x,y,z{:}\;( \mbox{distinct\_points}(x,y) \ \Rightarrow \ ( \mbox{apart\_point\_and\_line}(z,\mbox{line\_connecting}(x,y)) \ \Rightarrow \ \mbox{apart\_point\_and\_line}(z,\mbox{line\_connecting}(y,x)) ) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO122-1.p} Every curve has a finishing point






$\mbox{include}(\mbox{'Axioms/GEO004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO004-2.ax'})$

$\neg\,\mbox{finish\_point}(a,\mbox{sk}_{25})\qquad\mbox{cnf}(\mbox{corollary\_4\_8}_{133},\mbox{negated\_conjecture})$

\medskip\textbf{GEO222+2.p} Parallel to orthogonal to orthogonal

A line L is parallel to the line, that is orthogonal to the
 orthogonal to L through A, and goes through A as well.




$\mbox{include}(\mbox{'Axioms/GEO008+0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+2.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+3.ax'})$

$\forall a,l{:}\;\neg\,\mbox{convergent\_lines}(l,\mbox{orthogonal\_through\_point}(\mbox{orthogonal\_through\_point}(l,a),a))\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO065-2.p} Corollary 2 to collinearity






$\mbox{include}(\mbox{'Axioms/GEO002-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO002-1.ax'})$

$\mbox{between}(u,w,v)\qquad\mbox{cnf}(\mbox{w\_between\_u\_and\_v},\mbox{hypothesis})$

$\neg\,\mbox{colinear}(u,v,w)\qquad\mbox{cnf}(\mbox{prove\_uvw\_colinear},\mbox{negated\_conjecture})$

\medskip\textbf{GEO202+2.p} Diverging lines have equal ends

If the point X is distinct to the points Y and Z, and the lines
 connecting X and Y, and connecting X and Z are convergent, then
 the intersection point of these lines is equal to X.




$\mbox{include}(\mbox{'Axioms/GEO008+0.ax'})$

$\forall x,y,z{:}\;( ( \mbox{distinct\_points}(x,y)\mbox{ and }\mbox{distinct\_points}(x,z)\mbox{ and }\mbox{convergent\_lines}(\mbox{line\_connecting}(x,y),\mbox{line\_connecting}(x,z)) ) \ \Rightarrow \ \neg\,\mbox{distinct\_points}(\mbox{intersection\_point}(\mbox{line\_connecting}(x,y),\mbox{line\_connecting}(x,z)),x) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO191+3.p} Symmetry of apartness

If the lines X and Y are convergent, U and V are convergent,
 and the intersection point of X and Y is apart from U and V,
 then the intersection point of U and V is apart from X and Y.




$\mbox{include}(\mbox{'Axioms/GEO006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+1.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+2.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+3.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+4.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+5.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+6.ax'})$

$\forall x,y,u,v{:}\;( ( \mbox{convergent\_lines}(x,y)\mbox{ and }\mbox{convergent\_lines}(u,v)\mbox{ and }( \mbox{apart\_point\_and\_line}(\mbox{intersection\_point}(x,y),u)\mbox{ or }\mbox{apart\_point\_and\_line}(\mbox{intersection\_point}(x,y),v) ) ) \ \Rightarrow \ ( \mbox{apart\_point\_and\_line}(\mbox{intersection\_point}(u,v),x)\mbox{ or }\mbox{apart\_point\_and\_line}(\mbox{intersection\_point}(u,v),y) ) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO266+3.p} Symmetry of unequally directed opposite lines






$\mbox{include}(\mbox{'Axioms/GEO009+0.ax'})$

$\forall l,m{:}\;( \mbox{unequally\_directed\_opposite\_lines}(l,m)\iff \mbox{unequally\_directed\_opposite\_lines}(m,l) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO253+3.p} Characteristic property of parallel lines






$\mbox{include}(\mbox{'Axioms/GEO009+0.ax'})$

$\forall a,b,l{:}\;( ( \mbox{apart\_point\_and\_line}(a,l)\mbox{ and }\mbox{incident\_point\_and\_line}(b,\mbox{parallel\_through\_point}(l,a)) ) \ \Rightarrow \ \mbox{apart\_point\_and\_line}(b,l) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO248+1.p} A point in each region formed by parallel lines






$\mbox{include}(\mbox{'Axioms/GEO007+0.ax'})$

$\forall a,b,l{:}\;( ( \mbox{left\_apart\_point}(a,l)\mbox{ and }\mbox{left\_apart\_point}(b,\mbox{parallel\_through\_point}(l,a)) ) \ \Rightarrow \ \mbox{left\_apart\_point}(b,l) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO063-2.p} Insertion respects congruence in its last two arguments






$\mbox{include}(\mbox{'Axioms/GEO002-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO002-3.ax'})$

$\mbox{equidistant}(w,x,y,z)\qquad\mbox{cnf}(\mbox{w\_to\_x\_equals\_y\_to\_z},\mbox{hypothesis})$

$\mbox{insertion}(u,v,w,x)\ne \mbox{insertion}(u,v,y,z)\qquad\mbox{cnf}(\mbox{prove\_equality\_of\_insertions},\mbox{negated\_conjecture})$

\medskip\textbf{GEO085+1.p} Every open curve has at least two endpoints






$\mbox{include}(\mbox{'Axioms/GEO004+0.ax'})$

$\forall c{:}\;( \mbox{open}(c) \ \Rightarrow \ \exists p,q{:}\;( p\ne q\mbox{ and }\mbox{end\_point}(p,c)\mbox{ and }\mbox{end\_point}(q,c) ) )\qquad\mbox{fof}(\mbox{theorem\_2\_7}_1,\mbox{conjecture})$

\medskip\textbf{GEO133-1.p} Betweenness and precedence property 2






$\mbox{include}(\mbox{'Axioms/GEO004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO004-2.ax'})$

$\mbox{ordered\_by}(\mbox{sk}_{25},\mbox{sk}_{26},\mbox{sk}_{27})\qquad\mbox{cnf}(\mbox{theorem\_4\_17\_2}_{133},\mbox{negated\_conjecture})$

$\mbox{sk}_{28}\ne \mbox{sk}_{29}\qquad\mbox{cnf}(\mbox{theorem\_4\_17\_2}_{134},\mbox{negated\_conjecture})$

$\mbox{incident\_o}(\mbox{sk}_{29},\mbox{sk}_{25})\qquad\mbox{cnf}(\mbox{theorem\_4\_17\_2}_{135},\mbox{negated\_conjecture})$

$\mbox{between}(\mbox{sk}_{25},\mbox{sk}_{28},\mbox{sk}_{26},\mbox{sk}_{27})\qquad\mbox{cnf}(\mbox{theorem\_4\_17\_2}_{136},\mbox{negated\_conjecture})$

$\mbox{between}(\mbox{sk}_{25},\mbox{sk}_{29},\mbox{sk}_{28},\mbox{sk}_{27}) \ \Rightarrow \ \mbox{ordered\_by}(\mbox{sk}_{25},\mbox{sk}_{28},\mbox{sk}_{29})\qquad\mbox{cnf}(\mbox{theorem\_4\_17\_2}_{137},\mbox{negated\_conjecture})$

$\mbox{ordered\_by}(\mbox{sk}_{25},\mbox{sk}_{28},\mbox{sk}_{29}) \ \Rightarrow \ \mbox{ordered\_by}(\mbox{sk}_{25},\mbox{sk}_{28},\mbox{sk}_{29})\qquad\mbox{cnf}(\mbox{theorem\_4\_17\_2}_{138},\mbox{negated\_conjecture})$

$\mbox{between}(\mbox{sk}_{25},\mbox{sk}_{29},\mbox{sk}_{28},\mbox{sk}_{27}) \ \Rightarrow \ \mbox{between}(\mbox{sk}_{25},\mbox{sk}_{29},\mbox{sk}_{28},\mbox{sk}_{27})\qquad\mbox{cnf}(\mbox{theorem\_4\_17\_2}_{139},\mbox{negated\_conjecture})$

$\mbox{ordered\_by}(\mbox{sk}_{25},\mbox{sk}_{28},\mbox{sk}_{29}) \ \Rightarrow \ \mbox{between}(\mbox{sk}_{25},\mbox{sk}_{29},\mbox{sk}_{28},\mbox{sk}_{27})\qquad\mbox{cnf}(\mbox{theorem\_4\_17\_2}_{140},\mbox{negated\_conjecture})$

\medskip\textbf{GEO245+1.p} Configurations in terms of apartness






$\mbox{include}(\mbox{'Axioms/GEO007+0.ax'})$

$\forall a,b,c{:}\;( \mbox{distinct\_points}(a,b) \ \Rightarrow \ ( \mbox{left\_apart\_point}(c,\mbox{line\_connecting}(a,b)) \ \Rightarrow \ \mbox{left\_apart\_point}(b,\mbox{reverse\_line}(\mbox{line\_connecting}(a,c))) ) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO039-2.p} Corollary the identity axiom for betweenness






$\mbox{include}(\mbox{'Axioms/GEO002-0.ax'})$

$\mbox{between}(u,w,x)\qquad\mbox{cnf}(\mbox{w\_between\_u\_and\_x},\mbox{hypothesis})$

$u = x\qquad\mbox{cnf}(\mbox{u\_is\_x},\mbox{hypothesis})$

$\neg\,\mbox{between}(v,w,x)\qquad\mbox{cnf}(\mbox{prove\_corollary},\mbox{negated\_conjecture})$

\medskip\textbf{GEO163-1.p} Not enough axioms to prove collinearity of a finite set of points

Given a finite set of points such that for all points x, y
 there is a 3rd (different) point z collinear with x and y.
 Show that all points in the set are collinear.




$\mbox{collinear}(x,x,y)\qquad\mbox{cnf}(\mbox{two\_points\_collinear},\mbox{axiom})$

$\mbox{collinear}(x,y,z) \ \Rightarrow \ \mbox{collinear}(y,x,z)\qquad\mbox{cnf}(\mbox{rotate\_collinear},\mbox{axiom})$

$\mbox{collinear}(x,y,z) \ \Rightarrow \ \mbox{collinear}(z,x,y)\qquad\mbox{cnf}(\mbox{swap\_collinear},\mbox{axiom})$

$( \mbox{collinear}(y_1,y_2,z)\mbox{ and }\mbox{collinear}(x,y_1,y_2) ) \ \Rightarrow \ ( \mbox{collinear}(x,y_1,z)\mbox{ or }x = y_2\mbox{ or }y_1 = y_2\mbox{ or }y_2 = z )\qquad\mbox{cnf}(\mbox{transitivity\_collinear},\mbox{axiom})$

$\mbox{collinear}(x,y,\mbox{third}(x,y))\qquad\mbox{cnf}(\mbox{third\_point\_collinear},\mbox{hypothesis})$

$x\ne \mbox{third}(x,y)\qquad\mbox{cnf}(\mbox{third\_point\_different\_1a},\mbox{hypothesis})$

$y\ne \mbox{third}(x,y)\qquad\mbox{cnf}(\mbox{third\_point\_different\_1b},\mbox{hypothesis})$

$\neg\,\mbox{collinear}(p_1,p_2,p_3)\qquad\mbox{cnf}(\mbox{conjecture},\mbox{negated\_conjecture})$

\medskip\textbf{GEO205+2.p} Convergent lines and equal points

If the lines X and Y are convergent, and Y and Z are
 equivalent, then X and Z are convergent, and the intersection
 point of X and Y, and the intersection point of X and Z are
 equal.




$\mbox{include}(\mbox{'Axioms/GEO008+0.ax'})$

$\forall x,y,z{:}\;( ( \mbox{convergent\_lines}(x,y)\mbox{ and }\neg\,\mbox{distinct\_lines}(y,z) ) \ \Rightarrow \ ( \mbox{convergent\_lines}(x,z)\mbox{ and }\neg\,\mbox{distinct\_points}(\mbox{intersection\_point}(x,y),\mbox{intersection\_point}(x,z)) ) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO009-1.p} First inner connectivity property of betweenness

For all points x, y, z, and w, if y and w are between x and
 z, then either y is between x and w or w is between x and y.




$\mbox{include}(\mbox{'Axioms/GEO001-0.ax'})$

$\mbox{between}(a,c,e)\qquad\mbox{cnf}(\mbox{c\_between\_a\_and\_e},\mbox{hypothesis})$

$\mbox{between}(a,d,e)\qquad\mbox{cnf}(\mbox{d\_between\_a\_and\_e},\mbox{hypothesis})$

$\neg\,\mbox{between}(a,c,d)\qquad\mbox{cnf}(\mbox{c\_between\_a\_and\_d},\mbox{hypothesis})$

$\neg\,\mbox{between}(a,d,c)\qquad\mbox{cnf}(\mbox{prove\_d\_between\_a\_and\_c},\mbox{negated\_conjecture})$

\medskip\textbf{GEO254+3.p} Order on a line is observable






$\mbox{include}(\mbox{'Axioms/GEO009+0.ax'})$

$\forall a,b,l{:}\;( ( \mbox{distinct\_points}(a,b)\mbox{ and }\mbox{incident\_point\_and\_line}(a,l)\mbox{ and }\mbox{incident\_point\_and\_line}(b,l) ) \ \Rightarrow \ ( \mbox{before\_on\_line}(l,a,b)\mbox{ and }\mbox{before\_on\_line}(l,b,a) ) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO254+1.p} Order on a line is observable






$\mbox{include}(\mbox{'Axioms/GEO007+0.ax'})$

$\forall a,b,l{:}\;( ( \mbox{distinct\_points}(a,b)\mbox{ and }\neg\,\mbox{apart\_point\_and\_line}(a,l)\mbox{ and }\neg\,\mbox{apart\_point\_and\_line}(b,l) ) \ \Rightarrow \ ( \mbox{before\_on\_line}(l,a,b)\mbox{ and }\mbox{before\_on\_line}(l,b,a) ) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO041-2.p} Corollary to antisymmetry of betweenness in first 2 arguments






$\mbox{include}(\mbox{'Axioms/GEO002-0.ax'})$

$\mbox{between}(u,v,w)\qquad\mbox{cnf}(\mbox{v\_between\_u\_and\_w},\mbox{hypothesis})$

$\mbox{between}(u,w,v)\qquad\mbox{cnf}(\mbox{w\_between\_u\_and\_v},\mbox{hypothesis})$

$v\ne w\qquad\mbox{cnf}(\mbox{prove\_v\_is\_w},\mbox{negated\_conjecture})$

\medskip\textbf{GEO057-3.p} Corollary 2 of null extension






$\mbox{include}(\mbox{'Axioms/GEO002-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO002-2.ax'})$

$\mbox{equidistant}(u,v,u,v)\qquad\mbox{cnf}(d_1,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(w,x,u,v)\qquad\mbox{cnf}(d_2,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(v,u,w,x)\qquad\mbox{cnf}(d_3,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(u,v,x,w)\qquad\mbox{cnf}(\mbox{d4}_1,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(v,u,x,w)\qquad\mbox{cnf}(\mbox{d4}_2,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(w,x,v,u)\qquad\mbox{cnf}(\mbox{d4}_3,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(x,w,u,v)\qquad\mbox{cnf}(\mbox{d4}_4,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(x,w,v,u)\qquad\mbox{cnf}(\mbox{d4}_5,\mbox{axiom})$

$( \mbox{equidistant}(u,v,w,x)\mbox{ and }\mbox{equidistant}(w,x,y,z) ) \ \Rightarrow \ \mbox{equidistant}(u,v,y,z)\qquad\mbox{cnf}(d_5,\mbox{axiom})$

$v = \mbox{extension}(u,v,w,w)\qquad\mbox{cnf}(e_1,\mbox{axiom})$

$y = \mbox{extension}(u,v,w,x) \ \Rightarrow \ \mbox{between}(u,v,y)\qquad\mbox{cnf}(b_0,\mbox{axiom})$

$\mbox{between}(u,v,\mbox{reflection}(u,v))\qquad\mbox{cnf}(\mbox{r2}_1,\mbox{axiom})$

$\mbox{equidistant}(v,\mbox{reflection}(u,v),u,v)\qquad\mbox{cnf}(\mbox{r2}_2,\mbox{axiom})$

$u\ne \mbox{reflection}(u,u)\qquad\mbox{cnf}(\mbox{prove\_null\_extension\_by\_reflection},\mbox{negated\_conjecture})$

\medskip\textbf{GEO056-2.p} Corollary 1 to null extension






$\mbox{include}(\mbox{'Axioms/GEO002-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO002-2.ax'})$

$u = v\qquad\mbox{cnf}(\mbox{u\_equals\_v},\mbox{hypothesis})$

$v\ne \mbox{reflection}(u,v)\qquad\mbox{cnf}(\mbox{prove\_v\_equals\_reflection},\mbox{negated\_conjecture})$

\medskip\textbf{GEO203+3.p} Equal lines from points

If the lines X and Y are convergent, and X and Z are convergent,
 the intersection point of X and Y, and the intersection point
 of X and Z are distinct, then the line connecting these points
 is equal to X.




$\mbox{include}(\mbox{'Axioms/GEO006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+1.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+2.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+3.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+4.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+5.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+6.ax'})$

$\forall x,y,z{:}\;( ( \mbox{convergent\_lines}(x,y)\mbox{ and }\mbox{convergent\_lines}(x,z)\mbox{ and }\mbox{distinct\_points}(\mbox{intersection\_point}(x,y),\mbox{intersection\_point}(x,z)) ) \ \Rightarrow \ \mbox{equal\_lines}(\mbox{line\_connecting}(\mbox{intersection\_point}(x,y),\mbox{intersection\_point}(x,z)),x) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO252+3.p} Jordan-type result for half planes






$\mbox{include}(\mbox{'Axioms/GEO009+0.ax'})$

$\forall a,b,l{:}\;( ( \mbox{left\_apart\_point}(a,l)\mbox{ and }\mbox{right\_apart\_point}(b,l) ) \ \Rightarrow \ ( \mbox{distinct\_points}(a,b)\mbox{ and }\mbox{left\_convergent\_lines}(\mbox{line\_connecting}(a,b),l) ) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO158-1.p} Simple curve axioms






$\mbox{include}(\mbox{'Axioms/GEO004-0.ax'})$

\medskip\textbf{GEO168+1.p} Pappus2 implies Pappus1






$\mbox{colinear}(a,b,c,l)\mbox{ and }\mbox{colinear}(d,e,f,m)\qquad\mbox{fof}(\mbox{assumption}_1,\mbox{axiom})$

$\mbox{colinear}(b,f,g,n)\mbox{ and }\mbox{colinear}(c,e,g,o)\qquad\mbox{fof}(\mbox{assumption}_2,\mbox{axiom})$

$\mbox{colinear}(b,d,h,p)\mbox{ and }\mbox{colinear}(a,e,h,q)\qquad\mbox{fof}(\mbox{assumption}_3,\mbox{axiom})$

$\mbox{colinear}(c,d,i,r)\mbox{ and }\mbox{colinear}(a,f,i,s)\qquad\mbox{fof}(\mbox{assumption}_4,\mbox{axiom})$

$\mbox{incident}(a,m) \ \Rightarrow \ \mbox{goal}\qquad\mbox{fof}(\mbox{goalam},\mbox{axiom})$

$\mbox{incident}(b,m) \ \Rightarrow \ \mbox{goal}\qquad\mbox{fof}(\mbox{goalbm},\mbox{axiom})$

$\mbox{incident}(c,m) \ \Rightarrow \ \mbox{goal}\qquad\mbox{fof}(\mbox{goalcm},\mbox{axiom})$

$\mbox{incident}(d,l) \ \Rightarrow \ \mbox{goal}\qquad\mbox{fof}(\mbox{goaldl},\mbox{axiom})$

$\mbox{incident}(e,l) \ \Rightarrow \ \mbox{goal}\qquad\mbox{fof}(\mbox{goalel},\mbox{axiom})$

$\mbox{incident}(f,l) \ \Rightarrow \ \mbox{goal}\qquad\mbox{fof}(\mbox{goalfl},\mbox{axiom})$

$\forall a{:}\;( ( \mbox{incident}(g,a)\mbox{ and }\mbox{incident}(h,a)\mbox{ and }\mbox{incident}(i,a) ) \ \Rightarrow \ \mbox{goal} )\qquad\mbox{fof}(\mbox{goal}_4,\mbox{axiom})$

$\forall a,b,c,d{:}\;( \mbox{colinear}(a,b,c,d) \ \Rightarrow \ \mbox{incident}(a,d) )\qquad\mbox{fof}(\mbox{colinearity\_elimination}_1,\mbox{axiom})$

$\forall a,b,c,d{:}\;( \mbox{colinear}(a,b,c,d) \ \Rightarrow \ \mbox{incident}(b,d) )\qquad\mbox{fof}(\mbox{colinearity\_elimination}_2,\mbox{axiom})$

$\forall a,b,c,d{:}\;( \mbox{colinear}(a,b,c,d) \ \Rightarrow \ \mbox{incident}(c,d) )\qquad\mbox{fof}(\mbox{colinearity\_elimination}_3,\mbox{axiom})$

$\forall a,b{:}\;( \mbox{incident}(a,b) \ \Rightarrow \ \mbox{point\_equal}(a,a) )\qquad\mbox{fof}(\mbox{reflexivity\_of\_point\_equal},\mbox{axiom})$

$\forall a,b{:}\;( \mbox{point\_equal}(a,b) \ \Rightarrow \ \mbox{point\_equal}(b,a) )\qquad\mbox{fof}(\mbox{symmetry\_of\_point\_equal},\mbox{axiom})$

$\forall a,b,c{:}\;( ( \mbox{point\_equal}(a,b)\mbox{ and }\mbox{point\_equal}(b,c) ) \ \Rightarrow \ \mbox{point\_equal}(a,c) )\qquad\mbox{fof}(\mbox{transitivity\_of\_point\_equal},\mbox{axiom})$

$\forall a,b{:}\;( \mbox{incident}(a,b) \ \Rightarrow \ \mbox{line\_equal}(b,b) )\qquad\mbox{fof}(\mbox{reflexivity\_of\_line\_equal},\mbox{axiom})$

$\forall a,b{:}\;( \mbox{line\_equal}(a,b) \ \Rightarrow \ \mbox{line\_equal}(b,a) )\qquad\mbox{fof}(\mbox{symmetry\_of\_line\_equal},\mbox{axiom})$

$\forall a,b,c{:}\;( ( \mbox{line\_equal}(a,b)\mbox{ and }\mbox{line\_equal}(b,c) ) \ \Rightarrow \ \mbox{line\_equal}(a,c) )\qquad\mbox{fof}(\mbox{transitivity\_of\_line\_equal},\mbox{axiom})$

$\forall a,b,c{:}\;( ( \mbox{point\_equal}(a,b)\mbox{ and }\mbox{incident}(b,c) ) \ \Rightarrow \ \mbox{incident}(a,c) )\qquad\mbox{fof}(\mbox{pcon},\mbox{axiom})$

$\forall a,b,c{:}\;( ( \mbox{incident}(a,b)\mbox{ and }\mbox{line\_equal}(b,c) ) \ \Rightarrow \ \mbox{incident}(a,c) )\qquad\mbox{fof}(\mbox{lcon},\mbox{axiom})$

$\forall a,b,c,d,e,f,g,h,i,j,k,l,m,n,o,p,q{:}\;( ( \mbox{colinear}(a,b,c,j)\mbox{ and }\mbox{colinear}(d,e,f,k)\mbox{ and }\mbox{colinear}(b,f,g,l)\mbox{ and }\mbox{colinear}(c,e,g,m)\mbox{ and }\mbox{colinear}(b,d,h,n)\mbox{ and }\mbox{colinear}(a,e,h,o)\mbox{ and }\mbox{colinear}(c,d,i,p)\mbox{ and }\mbox{colinear}(a,f,i,q) ) \ \Rightarrow \ ( \exists r{:}\;\mbox{colinear}(g,h,i,r)\mbox{ or }\mbox{line\_equal}(l,m)\mbox{ or }\mbox{line\_equal}(n,o)\mbox{ or }\mbox{line\_equal}(p,q) ) )\qquad\mbox{fof}(\mbox{pappus}_1,\mbox{axiom})$

$\forall a,b,c,d{:}\;( ( \mbox{incident}(c,a)\mbox{ and }\mbox{incident}(c,b)\mbox{ and }\mbox{incident}(d,a)\mbox{ and }\mbox{incident}(d,b) ) \ \Rightarrow \ ( \mbox{point\_equal}(c,d)\mbox{ or }\mbox{line\_equal}(a,b) ) )\qquad\mbox{fof}(\mbox{unique},\mbox{axiom})$

$\forall a,b{:}\;( ( \mbox{point\_equal}(a,a)\mbox{ and }\mbox{point\_equal}(b,b) ) \ \Rightarrow \ \exists c{:}\;( \mbox{incident}(a,c)\mbox{ and }\mbox{incident}(b,c) ) )\qquad\mbox{fof}(\mbox{line},\mbox{axiom})$

$\forall a,b,c{:}\;( ( \mbox{line\_equal}(c,c)\mbox{ and }\mbox{line\_equal}(b,b) ) \ \Rightarrow \ \exists a{:}\;( \mbox{incident}(a,b)\mbox{ and }\mbox{incident}(a,c) ) )\qquad\mbox{fof}(\mbox{point},\mbox{axiom})$

$\mbox{goal}\qquad\mbox{fof}(\mbox{goal\_to\_be\_proved},\mbox{conjecture})$

\medskip\textbf{GEO227+3.p} Lines not directed and opposite






$\mbox{include}(\mbox{'Axioms/GEO009+0.ax'})$

$\forall l,m{:}\;\neg\,\mbox{equally\_directed\_lines}(l,m)\mbox{ and }\mbox{equally\_directed\_opposite\_lines}(l,m)\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO048-2.p} Inner points of triangle






$\mbox{include}(\mbox{'Axioms/GEO002-0.ax'})$

$\mbox{between}(u,v,w)\qquad\mbox{cnf}(\mbox{v\_between\_u\_and\_w},\mbox{hypothesis})$

$\mbox{between}(u_1,v_1,w)\qquad\mbox{cnf}(\mbox{v1\_between\_u1\_and\_w},\mbox{hypothesis})$

$\mbox{between}(u,x,u_1)\qquad\mbox{cnf}(\mbox{x\_between\_u\_and\_u}_1,\mbox{hypothesis})$

$\neg\,\mbox{between}(x,\mbox{inner\_pasch}(v_1,\mbox{inner\_pasch}(u,x,u_1,v_1,w),u,v,w),w)\qquad\mbox{cnf}(\mbox{prove\_conclusion}_1,\mbox{negated\_conjecture})$

$\neg\,\mbox{between}(v,\mbox{inner\_pasch}(v_1,\mbox{inner\_pasch}(u,x,u_1,v_1,w),u,v,w),v_1)\qquad\mbox{cnf}(\mbox{prove\_conclusion}_2,\mbox{negated\_conjecture})$

\medskip\textbf{GEO195+1.p} Corollary to symmetry of apartness

If the lines X, Y, and Z are pairwise convergent, and the
 intersection point of X and Y is apart from a line Z, then
 the intersection point of Y and X is apart from Z.




$\mbox{include}(\mbox{'Axioms/GEO006+0.ax'})$

$\forall x,y,z{:}\;( ( \mbox{convergent\_lines}(x,y)\mbox{ and }\mbox{convergent\_lines}(z,y)\mbox{ and }\mbox{convergent\_lines}(x,z) ) \ \Rightarrow \ ( \mbox{apart\_point\_and\_line}(\mbox{intersection\_point}(x,y),z) \ \Rightarrow \ \mbox{apart\_point\_and\_line}(\mbox{intersection\_point}(y,x),z) ) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO124-1.p} Every oriented curve has at most one starting point






$\mbox{include}(\mbox{'Axioms/GEO004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO004-2.ax'})$

$\mbox{start\_point}(\mbox{sk}_{26},\mbox{sk}_{25})\qquad\mbox{cnf}(\mbox{corollary\_4\_10\_1}_{133},\mbox{negated\_conjecture})$

$\mbox{start\_point}(\mbox{sk}_{27},\mbox{sk}_{25})\qquad\mbox{cnf}(\mbox{corollary\_4\_10\_1}_{134},\mbox{negated\_conjecture})$

$\mbox{sk}_{26}\ne \mbox{sk}_{27}\qquad\mbox{cnf}(\mbox{corollary\_4\_10\_1}_{135},\mbox{negated\_conjecture})$

\medskip\textbf{GEO003-1.p} For all points x and y, y is between x and y






$\mbox{include}(\mbox{'Axioms/GEO001-0.ax'})$

$\neg\,\mbox{between}(a,b,b)\qquad\mbox{cnf}(\mbox{prove\_b\_between\_a\_and\_b},\mbox{negated\_conjecture})$

\medskip\textbf{GEO256+3.p} Property of order and betweeness






$\mbox{include}(\mbox{'Axioms/GEO009+0.ax'})$

$\forall l,a,b,c,d{:}\;( ( \mbox{distinct\_points}(a,c)\mbox{ and }\mbox{distinct\_points}(b,c)\mbox{ and }\mbox{incident\_point\_and\_line}(c,l)\mbox{ and }\mbox{left\_apart\_point}(d,l) ) \ \Rightarrow \ ( \mbox{before\_on\_line}(l,a,b) \ \Rightarrow \ ( \mbox{before\_on\_line}(l,a,c)\mbox{ or }\mbox{before\_on\_line}(l,c,b) ) ) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO253+1.p} Characteristic property of parallel lines






$\mbox{include}(\mbox{'Axioms/GEO007+0.ax'})$

$\forall a,b,l{:}\;( ( \mbox{apart\_point\_and\_line}(a,l)\mbox{ and }\neg\,\mbox{apart\_point\_and\_line}(b,\mbox{parallel\_through\_point}(l,a)) ) \ \Rightarrow \ \mbox{apart\_point\_and\_line}(b,l) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO057-2.p} Corollary 2 of null extension






$\mbox{include}(\mbox{'Axioms/GEO002-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO002-2.ax'})$

$u\ne \mbox{reflection}(u,u)\qquad\mbox{cnf}(\mbox{prove\_null\_extension\_by\_reflection},\mbox{negated\_conjecture})$

\medskip\textbf{GEO247+3.p} A point in each region formed by intersecting lines






$\mbox{include}(\mbox{'Axioms/GEO009+0.ax'})$

$\forall a,b,c,d,l,m{:}\;( ( \mbox{left\_apart\_point}(a,l)\mbox{ and }\mbox{left\_apart\_point}(a,m)\mbox{ and }\mbox{right\_apart\_point}(b,l)\mbox{ and }\mbox{right\_apart\_point}(b,m)\mbox{ and }\mbox{left\_apart\_point}(c,l)\mbox{ and }\mbox{right\_apart\_point}(c,m)\mbox{ and }\mbox{right\_apart\_point}(d,l)\mbox{ and }\mbox{left\_apart\_point}(d,m) ) \ \Rightarrow \ \mbox{convergent\_lines}(l,m) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO027-2.p} Corollary 1 to unique extension






$\mbox{include}(\mbox{'Axioms/GEO002-0.ax'})$

$\mbox{between}(u,v,w)\qquad\mbox{cnf}(\mbox{v\_between\_u\_and\_w},\mbox{hypothesis})$

$u\ne v\qquad\mbox{cnf}(\mbox{u\_not\_v},\mbox{hypothesis})$

$w\ne \mbox{extension}(u,v,v,w)\qquad\mbox{cnf}(\mbox{prove\_w\_is\_an\_extension},\mbox{negated\_conjecture})$

\medskip\textbf{GEO043-3.p} Corollary to first inner transitivity property of betweenness

Corollary of first inner transitivity property of betweenness,
 using symmetry.




$\mbox{include}(\mbox{'Axioms/GEO002-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO002-2.ax'})$

$\mbox{equidistant}(u,v,u,v)\qquad\mbox{cnf}(d_1,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(w,x,u,v)\qquad\mbox{cnf}(d_2,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(v,u,w,x)\qquad\mbox{cnf}(d_3,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(u,v,x,w)\qquad\mbox{cnf}(\mbox{d4}_1,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(v,u,x,w)\qquad\mbox{cnf}(\mbox{d4}_2,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(w,x,v,u)\qquad\mbox{cnf}(\mbox{d4}_3,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(x,w,u,v)\qquad\mbox{cnf}(\mbox{d4}_4,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(x,w,v,u)\qquad\mbox{cnf}(\mbox{d4}_5,\mbox{axiom})$

$( \mbox{equidistant}(u,v,w,x)\mbox{ and }\mbox{equidistant}(w,x,y,z) ) \ \Rightarrow \ \mbox{equidistant}(u,v,y,z)\qquad\mbox{cnf}(d_5,\mbox{axiom})$

$v = \mbox{extension}(u,v,w,w)\qquad\mbox{cnf}(e_1,\mbox{axiom})$

$y = \mbox{extension}(u,v,w,x) \ \Rightarrow \ \mbox{between}(u,v,y)\qquad\mbox{cnf}(b_0,\mbox{axiom})$

$\mbox{between}(u,v,\mbox{reflection}(u,v))\qquad\mbox{cnf}(\mbox{r2}_1,\mbox{axiom})$

$\mbox{equidistant}(v,\mbox{reflection}(u,v),u,v)\qquad\mbox{cnf}(\mbox{r2}_2,\mbox{axiom})$

$u = v \ \Rightarrow \ v = \mbox{reflection}(u,v)\qquad\mbox{cnf}(\mbox{r3}_1,\mbox{axiom})$

$u = \mbox{reflection}(u,u)\qquad\mbox{cnf}(\mbox{r3}_2,\mbox{axiom})$

$v = \mbox{reflection}(u,v) \ \Rightarrow \ u = v\qquad\mbox{cnf}(r_4,\mbox{axiom})$

$\mbox{equidistant}(u,u,v,v)\qquad\mbox{cnf}(d_7,\mbox{axiom})$

$( \mbox{equidistant}(u,v,u_1,v_1)\mbox{ and }\mbox{equidistant}(v,w,v_1,w_1)\mbox{ and }\mbox{between}(u,v,w)\mbox{ and }\mbox{between}(u_1,v_1,w_1) ) \ \Rightarrow \ \mbox{equidistant}(u,w,u_1,w_1)\qquad\mbox{cnf}(d_8,\mbox{axiom})$

$( \mbox{between}(u,v,w)\mbox{ and }\mbox{between}(u,v,x)\mbox{ and }\mbox{equidistant}(v,w,v,x) ) \ \Rightarrow \ ( u = v\mbox{ or }w = x )\qquad\mbox{cnf}(d_9,\mbox{axiom})$

$\mbox{between}(u,v,w) \ \Rightarrow \ ( u = v\mbox{ or }w = \mbox{extension}(u,v,v,w) )\qquad\mbox{cnf}(\mbox{d10}_1,\mbox{axiom})$

$\mbox{equidistant}(w,x,y,z) \ \Rightarrow \ ( \mbox{extension}(u,v,w,x) = \mbox{extension}(u,v,y,z)\mbox{ or }u = v )\qquad\mbox{cnf}(\mbox{d10}_2,\mbox{axiom})$

$\mbox{extension}(u,v,u,v) = \mbox{extension}(u,v,v,u)\mbox{ or }u = v\qquad\mbox{cnf}(\mbox{d10}_3,\mbox{axiom})$

$\mbox{equidistant}(v,u,v,\mbox{reflection}(\mbox{reflection}(u,v),v))\qquad\mbox{cnf}(r_5,\mbox{axiom})$

$u = \mbox{reflection}(\mbox{reflection}(u,v),v)\qquad\mbox{cnf}(r_6,\mbox{axiom})$

$\mbox{between}(u,v,v)\qquad\mbox{cnf}(t_3,\mbox{axiom})$

$( \mbox{between}(u,w,x)\mbox{ and }u = x ) \ \Rightarrow \ \mbox{between}(v,w,x)\qquad\mbox{cnf}(b_1,\mbox{axiom})$

$\mbox{between}(u,v,w) \ \Rightarrow \ \mbox{between}(w,v,u)\qquad\mbox{cnf}(t_1,\mbox{axiom})$

$\mbox{between}(u,u,v)\qquad\mbox{cnf}(t_2,\mbox{axiom})$

$( \mbox{between}(u,v,w)\mbox{ and }\mbox{between}(v,u,w) ) \ \Rightarrow \ u = v\qquad\mbox{cnf}(b_2,\mbox{axiom})$

$( \mbox{between}(u,v,w)\mbox{ and }\mbox{between}(u,w,v) ) \ \Rightarrow \ v = w\qquad\mbox{cnf}(b_3,\mbox{axiom})$

$( \mbox{between}(u,v,w)\mbox{ and }\mbox{between}(v,u,w) ) \ \Rightarrow \ ( u = v\mbox{ or }v = w )\qquad\mbox{cnf}(\mbox{t6}_1,\mbox{axiom})$

$( \mbox{between}(u,v,w)\mbox{ and }\mbox{between}(u,w,v) ) \ \Rightarrow \ ( u = v\mbox{ or }v = w )\qquad\mbox{cnf}(\mbox{t6}_2,\mbox{axiom})$

$( \mbox{between}(u,v,w)\mbox{ and }\mbox{between}(v,w,x) ) \ \Rightarrow \ \mbox{between}(u,v,w)\qquad\mbox{cnf}(b_4,\mbox{axiom})$

$\mbox{between}(u,v,w)\qquad\mbox{cnf}(\mbox{v\_between\_u\_and\_w},\mbox{hypothesis})$

$\mbox{between}(u,w,x)\qquad\mbox{cnf}(\mbox{w\_between\_u\_and\_x},\mbox{hypothesis})$

$\neg\,\mbox{between}(v,w,x)\qquad\mbox{cnf}(\mbox{prove\_w\_between\_v\_and\_x},\mbox{negated\_conjecture})$

\medskip\textbf{GEO198+3.p} Corollary to symmetry of incidence

If the lines X, Y, and Z are pairwise convergent, and the
 intersection point of X and Y is incident with Z, then the
 intersection point of X and Z is incident with Y.




$\mbox{include}(\mbox{'Axioms/GEO006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+1.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+2.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+3.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+4.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+5.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+6.ax'})$

$\forall x,y,z{:}\;( ( \mbox{convergent\_lines}(x,y)\mbox{ and }\mbox{convergent\_lines}(z,y)\mbox{ and }\mbox{convergent\_lines}(x,z)\mbox{ and }\mbox{incident\_point\_and\_line}(\mbox{intersection\_point}(x,y),z) ) \ \Rightarrow \ \mbox{incident\_point\_and\_line}(\mbox{intersection\_point}(x,z),y) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO104+1.p} Subcurves with common endpoint meet or include

If two sub-curves of a given open curve have a common endpoint
 then the sub-curves meet or one is included in the other.




$\mbox{include}(\mbox{'Axioms/GEO004+0.ax'})$

$\forall c,c_1,c_2,p{:}\;( ( \mbox{end\_point}(p,c_1)\mbox{ and }\mbox{end\_point}(p,c_2)\mbox{ and }\mbox{part\_of}(c_1,c)\mbox{ and }\mbox{part\_of}(c_2,c)\mbox{ and }\mbox{open}(c) ) \ \Rightarrow \ ( p\wedge c_1{=}c_2\mbox{ or }\mbox{part\_of}(c_1,c_2)\mbox{ or }\mbox{part\_of}(c_2,c_1) ) )\qquad\mbox{fof}(\mbox{theorem\_2}_{22},\mbox{conjecture})$

\medskip\textbf{GEO136-1.p} Underlying curve and one pair of points sufficient for ordering

The underlying curve and one pair of points are sufficient for
 the ordering of the points on the oriented curve.




$\mbox{include}(\mbox{'Axioms/GEO004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO004-2.ax'})$

$\mbox{ordered\_by}(\mbox{sk}_{25},\mbox{sk}_{26},\mbox{sk}_{27})\qquad\mbox{cnf}(\mbox{theorem\_4\_19}_{133},\mbox{negated\_conjecture})$

$\mbox{ordered\_by}(\mbox{sk}_{25},\mbox{sk}_{28},\mbox{sk}_{29})\mbox{ or }\mbox{between}(\mbox{sk}_{25},\mbox{sk}_{28},\mbox{sk}_{26},\mbox{sk}_{27})\qquad\mbox{cnf}(\mbox{theorem\_4\_19}_{134},\mbox{negated\_conjecture})$

$\mbox{ordered\_by}(\mbox{sk}_{25},\mbox{sk}_{28},\mbox{sk}_{29})\mbox{ or }\mbox{between}(\mbox{sk}_{25},\mbox{sk}_{28},\mbox{sk}_{29},\mbox{sk}_{27})\mbox{ or }\mbox{between}(\mbox{sk}_{25},\mbox{sk}_{28},\mbox{sk}_{27},\mbox{sk}_{29})\mbox{ or }\mbox{sk}_{27} = \mbox{sk}_{29}\qquad\mbox{cnf}(\mbox{theorem\_4\_19}_{135},\mbox{negated\_conjecture})$

$\mbox{ordered\_by}(\mbox{sk}_{25},\mbox{sk}_{28},\mbox{sk}_{29}) \ \Rightarrow \ \mbox{ordered\_by}(\mbox{sk}_{25},\mbox{sk}_{28},\mbox{sk}_{29})\qquad\mbox{cnf}(\mbox{theorem\_4\_19}_{136},\mbox{negated\_conjecture})$

$( \mbox{between}(\mbox{sk}_{25},\mbox{sk}_{28},\mbox{sk}_{26},\mbox{sk}_{27})\mbox{ and }\mbox{between}(\mbox{sk}_{25},\mbox{sk}_{28},\mbox{sk}_{29},\mbox{sk}_{27}) ) \ \Rightarrow \ \mbox{between}(\mbox{sk}_{25},\mbox{sk}_{28},\mbox{sk}_{26},\mbox{sk}_{27})\qquad\mbox{cnf}(\mbox{theorem\_4\_19}_{137},\mbox{negated\_conjecture})$

$( \mbox{between}(\mbox{sk}_{25},\mbox{sk}_{28},\mbox{sk}_{26},\mbox{sk}_{27})\mbox{ and }\mbox{between}(\mbox{sk}_{25},\mbox{sk}_{28},\mbox{sk}_{29},\mbox{sk}_{27}) ) \ \Rightarrow \ ( \mbox{between}(\mbox{sk}_{25},\mbox{sk}_{28},\mbox{sk}_{29},\mbox{sk}_{27})\mbox{ or }\mbox{between}(\mbox{sk}_{25},\mbox{sk}_{28},\mbox{sk}_{27},\mbox{sk}_{29})\mbox{ or }\mbox{sk}_{27} = \mbox{sk}_{29} )\qquad\mbox{cnf}(\mbox{theorem\_4\_19}_{138},\mbox{negated\_conjecture})$

$( \mbox{between}(\mbox{sk}_{25},\mbox{sk}_{28},\mbox{sk}_{26},\mbox{sk}_{27})\mbox{ and }\mbox{between}(\mbox{sk}_{25},\mbox{sk}_{28},\mbox{sk}_{27},\mbox{sk}_{29}) ) \ \Rightarrow \ \mbox{between}(\mbox{sk}_{25},\mbox{sk}_{28},\mbox{sk}_{26},\mbox{sk}_{27})\qquad\mbox{cnf}(\mbox{theorem\_4\_19}_{139},\mbox{negated\_conjecture})$

$( \mbox{between}(\mbox{sk}_{25},\mbox{sk}_{28},\mbox{sk}_{26},\mbox{sk}_{27})\mbox{ and }\mbox{sk}_{27} = \mbox{sk}_{29} ) \ \Rightarrow \ \mbox{between}(\mbox{sk}_{25},\mbox{sk}_{28},\mbox{sk}_{26},\mbox{sk}_{27})\qquad\mbox{cnf}(\mbox{theorem\_4\_19}_{140},\mbox{negated\_conjecture})$

$( \mbox{between}(\mbox{sk}_{25},\mbox{sk}_{28},\mbox{sk}_{26},\mbox{sk}_{27})\mbox{ and }\mbox{between}(\mbox{sk}_{25},\mbox{sk}_{28},\mbox{sk}_{27},\mbox{sk}_{29}) ) \ \Rightarrow \ ( \mbox{between}(\mbox{sk}_{25},\mbox{sk}_{28},\mbox{sk}_{29},\mbox{sk}_{27})\mbox{ or }\mbox{between}(\mbox{sk}_{25},\mbox{sk}_{28},\mbox{sk}_{27},\mbox{sk}_{29})\mbox{ or }\mbox{sk}_{27} = \mbox{sk}_{29} )\qquad\mbox{cnf}(\mbox{theorem\_4\_19}_{141},\mbox{negated\_conjecture})$

$( \mbox{between}(\mbox{sk}_{25},\mbox{sk}_{28},\mbox{sk}_{26},\mbox{sk}_{27})\mbox{ and }\mbox{sk}_{27} = \mbox{sk}_{29} ) \ \Rightarrow \ ( \mbox{between}(\mbox{sk}_{25},\mbox{sk}_{28},\mbox{sk}_{29},\mbox{sk}_{27})\mbox{ or }\mbox{between}(\mbox{sk}_{25},\mbox{sk}_{28},\mbox{sk}_{27},\mbox{sk}_{29})\mbox{ or }\mbox{sk}_{27} = \mbox{sk}_{29} )\qquad\mbox{cnf}(\mbox{theorem\_4\_19}_{142},\mbox{negated\_conjecture})$

$( \mbox{between}(\mbox{sk}_{25},\mbox{sk}_{28},\mbox{sk}_{26},\mbox{sk}_{27})\mbox{ and }\mbox{between}(\mbox{sk}_{25},\mbox{sk}_{28},\mbox{sk}_{29},\mbox{sk}_{27}) ) \ \Rightarrow \ \neg\,\mbox{ordered\_by}(\mbox{sk}_{25},\mbox{sk}_{28},\mbox{sk}_{29})\qquad\mbox{cnf}(\mbox{theorem\_4\_19}_{143},\mbox{negated\_conjecture})$

$( \mbox{between}(\mbox{sk}_{25},\mbox{sk}_{28},\mbox{sk}_{26},\mbox{sk}_{27})\mbox{ and }\mbox{between}(\mbox{sk}_{25},\mbox{sk}_{28},\mbox{sk}_{27},\mbox{sk}_{29}) ) \ \Rightarrow \ \neg\,\mbox{ordered\_by}(\mbox{sk}_{25},\mbox{sk}_{28},\mbox{sk}_{29})\qquad\mbox{cnf}(\mbox{theorem\_4\_19}_{144},\mbox{negated\_conjecture})$

$( \mbox{between}(\mbox{sk}_{25},\mbox{sk}_{28},\mbox{sk}_{26},\mbox{sk}_{27})\mbox{ and }\mbox{sk}_{27} = \mbox{sk}_{29} ) \ \Rightarrow \ \neg\,\mbox{ordered\_by}(\mbox{sk}_{25},\mbox{sk}_{28},\mbox{sk}_{29})\qquad\mbox{cnf}(\mbox{theorem\_4\_19}_{145},\mbox{negated\_conjecture})$

$\mbox{between}(\mbox{sk}_{25},\mbox{sk}_{26},\mbox{sk}_{28},\mbox{sk}_{29}) \ \Rightarrow \ \neg\,\mbox{between}(\mbox{sk}_{25},\mbox{sk}_{26},\mbox{sk}_{28},\mbox{sk}_{27})\qquad\mbox{cnf}(\mbox{theorem\_4\_19}_{146},\mbox{negated\_conjecture})$

$\mbox{between}(\mbox{sk}_{25},\mbox{sk}_{26},\mbox{sk}_{28},\mbox{sk}_{29}) \ \Rightarrow \ \neg\,\mbox{between}(\mbox{sk}_{25},\mbox{sk}_{26},\mbox{sk}_{27},\mbox{sk}_{28})\qquad\mbox{cnf}(\mbox{theorem\_4\_19}_{147},\mbox{negated\_conjecture})$

$\mbox{between}(\mbox{sk}_{25},\mbox{sk}_{26},\mbox{sk}_{28},\mbox{sk}_{29}) \ \Rightarrow \ \mbox{sk}_{27}\ne \mbox{sk}_{28}\qquad\mbox{cnf}(\mbox{theorem\_4\_19}_{148},\mbox{negated\_conjecture})$

$\mbox{sk}_{26} = \mbox{sk}_{28} \ \Rightarrow \ \neg\,\mbox{between}(\mbox{sk}_{25},\mbox{sk}_{26},\mbox{sk}_{29},\mbox{sk}_{27})\qquad\mbox{cnf}(\mbox{theorem\_4\_19}_{149},\mbox{negated\_conjecture})$

$\mbox{sk}_{26} = \mbox{sk}_{28} \ \Rightarrow \ \neg\,\mbox{between}(\mbox{sk}_{25},\mbox{sk}_{26},\mbox{sk}_{27},\mbox{sk}_{29})\qquad\mbox{cnf}(\mbox{theorem\_4\_19}_{150},\mbox{negated\_conjecture})$

$\mbox{sk}_{26} = \mbox{sk}_{28} \ \Rightarrow \ \mbox{sk}_{27}\ne \mbox{sk}_{29}\qquad\mbox{cnf}(\mbox{theorem\_4\_19}_{151},\mbox{negated\_conjecture})$

\medskip\textbf{GEO090+1.p} Meeting point of curves on a subcurve

If a point P is a meeting point of two curves and lies on a
 sub-curve of one of the two curves then P is also meeting point
 of the sub-curve and the other curve.




$\mbox{include}(\mbox{'Axioms/GEO004+0.ax'})$

$\forall c_1,c_2,\mbox{cpp},p{:}\;( ( \mbox{part\_of}(c_2,c_1)\mbox{ and }\mbox{incident\_c}(p,c_2)\mbox{ and }p\wedge c_1{=}\mbox{cpp} ) \ \Rightarrow \ p\wedge c_2{=}\mbox{cpp} )\qquad\mbox{fof}(\mbox{corollary\_2}_{12},\mbox{conjecture})$

\medskip\textbf{GEO005-2.p} Isosceles triangle based on line segment

For any line segment, there exists an isoceles triangle with
 that line segment as its base.




$\mbox{include}(\mbox{'Axioms/GEO002-0.ax'})$

$\mbox{equidistant}(a,\mbox{apex}(a,b),b,\mbox{apex}(a,b)) \ \Rightarrow \ \mbox{between}(a,\mbox{apex}(a,b),b)\qquad\mbox{cnf}(\mbox{prove\_apex},\mbox{negated\_conjecture})$

\medskip\textbf{GEO135+1.p} Ordering can be determined by betweenness and incidence

The ordering of any pair of points R and S on an oriented line
 o can be determined on the basis of a given pair of points P and
 Q using betweenness and incidence only.




$\mbox{include}(\mbox{'Axioms/GEO004+0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO004+1.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO004+2.ax'})$

$\forall o,p,q{:}\;( \mbox{ordered\_by}(o,p,q) \ \Rightarrow \ \forall r,s{:}\;( \mbox{ordered\_by}(o,r,s)\iff ( ( \mbox{incident\_o}(s,o)\mbox{ and }( \mbox{between}(o,r,p,q)\mbox{ or }p = r )\mbox{ and }r\ne s\mbox{ and }\neg\,\mbox{between}(o,s,r,q) )\mbox{ or }( \mbox{between}(o,p,r,s)\mbox{ and }\neg\,\mbox{between}(o,q,p,r) ) ) ) )\qquad\mbox{fof}(\mbox{corollary\_4}_{18},\mbox{conjecture})$

\medskip\textbf{GEO223+3.p} Corollary to uniqueness of parallels

The parallel to line L through a point A is equal to the line,
 that is orthogonal to the orthogonal to L through A, and goes
 through A as well.




$\mbox{include}(\mbox{'Axioms/GEO006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+1.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+2.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+3.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+4.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+5.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+6.ax'})$

$\forall a,l{:}\;\mbox{equal\_lines}(\mbox{parallel\_through\_point}(l,a),\mbox{orthogonal\_through\_point}(\mbox{orthogonal\_through\_point}(l,a),a))\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO160+1.p} Oriented curves






$\mbox{include}(\mbox{'Axioms/GEO004+0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO004+1.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO004+2.ax'})$

\medskip\textbf{GEO073-2.p} The diagonals of a non-degenerate rectancle bisect






$\mbox{include}(\mbox{'Axioms/GEO002-0.ax'})$

$\mbox{equidistant}(u,v,w,x)\qquad\mbox{cnf}(\mbox{u\_to\_v\_equals\_w\_to\_x},\mbox{hypothesis})$

$\mbox{equidistant}(v,w,x,u)\qquad\mbox{cnf}(\mbox{v\_to\_w\_equals\_x\_to\_u},\mbox{hypothesis})$

$\mbox{equidistant}(u,w,v,x)\qquad\mbox{cnf}(\mbox{u\_to\_w\_equals\_v\_to\_x},\mbox{hypothesis})$

$\mbox{between}(u,y,w)\qquad\mbox{cnf}(\mbox{y\_between\_u\_and\_w},\mbox{hypothesis})$

$\mbox{between}(v,y,x)\qquad\mbox{cnf}(\mbox{y\_between\_v\_and\_x},\mbox{hypothesis})$

$u\ne v\qquad\mbox{cnf}(\mbox{u\_not\_v},\mbox{hypothesis})$

$x\ne u\qquad\mbox{cnf}(\mbox{x\_not\_u},\mbox{hypothesis})$

$\mbox{equidistant}(u,y,w,y) \ \Rightarrow \ \neg\,\mbox{equidistant}(v,y,x,y)\qquad\mbox{cnf}(\mbox{prove\_bisection},\mbox{negated\_conjecture})$

\medskip\textbf{GEO002-2.p} For all points x and y, x is between x and y






$\mbox{include}(\mbox{'Axioms/GEO002-0.ax'})$

$\neg\,\mbox{between}(a,a,b)\qquad\mbox{cnf}(\mbox{prove\_a\_between\_a\_and\_b},\mbox{negated\_conjecture})$

\medskip\textbf{GEO051-2.p} Second outer connectivity property of betweenness






$\mbox{include}(\mbox{'Axioms/GEO002-0.ax'})$

$\mbox{between}(u,v,w)\qquad\mbox{cnf}(\mbox{v\_between\_u\_and\_w},\mbox{hypothesis})$

$\mbox{between}(u,v,x)\qquad\mbox{cnf}(\mbox{v\_between\_u\_and\_x},\mbox{hypothesis})$

$u\ne v\qquad\mbox{cnf}(\mbox{u\_not\_v},\mbox{hypothesis})$

$\neg\,\mbox{between}(v,w,x)\qquad\mbox{cnf}(\mbox{w\_not\_between\_v\_and\_x},\mbox{hypothesis})$

$\neg\,\mbox{between}(v,x,w)\qquad\mbox{cnf}(\mbox{prove\_x\_between\_v\_and\_w},\mbox{negated\_conjecture})$

\medskip\textbf{GEO255+1.p} Property of order and betweeness






$\mbox{include}(\mbox{'Axioms/GEO007+0.ax'})$

$\forall l,a,b{:}\;( ( \mbox{line}(l)\mbox{ and }\mbox{distinct\_points}(a,b) ) \ \Rightarrow \ \neg\,\mbox{before\_on\_line}(l,a,b)\mbox{ and }\mbox{before\_on\_line}(l,b,a) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO211+3.p} A line is not orthogonal to itself






$\mbox{include}(\mbox{'Axioms/GEO006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+1.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+2.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+3.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+4.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+5.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+6.ax'})$

$\forall l{:}\;\mbox{not\_orthogonal\_lines}(l,l)\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO033-2.p} First inner five-segment theorem






$\mbox{include}(\mbox{'Axioms/GEO002-0.ax'})$

$\mbox{equidistant}(u,v,u_1,v_1)\qquad\mbox{cnf}(\mbox{u\_to\_v\_equals\_u1\_to\_v}_1,\mbox{hypothesis})$

$\mbox{equidistant}(v,w,v_1,w_1)\qquad\mbox{cnf}(\mbox{v\_to\_w\_equals\_v1\_to\_w}_1,\mbox{hypothesis})$

$\mbox{equidistant}(u,x,u_1,x_1)\qquad\mbox{cnf}(\mbox{u\_to\_x\_equals\_u1\_to\_x}_1,\mbox{hypothesis})$

$\mbox{equidistant}(w,x,w_1,x_1)\qquad\mbox{cnf}(\mbox{w\_to\_x\_equals\_w1\_to\_x}_1,\mbox{hypothesis})$

$\mbox{between}(u,v,w)\qquad\mbox{cnf}(\mbox{v\_between\_u\_and\_w},\mbox{hypothesis})$

$\mbox{between}(u_1,v_1,w_1)\qquad\mbox{cnf}(\mbox{v1\_between\_u1\_and\_w}_1,\mbox{hypothesis})$

$\neg\,\mbox{equidistant}(v,x,v_1,x_1)\qquad\mbox{cnf}(\mbox{prove\_v\_to\_x\_equals\_v1\_to\_x}_1,\mbox{negated\_conjecture})$

\medskip\textbf{GEO229+3.p} Uniqueness of reversed lines






$\mbox{include}(\mbox{'Axioms/GEO009+0.ax'})$

$\forall l,m{:}\;( \mbox{equally\_directed\_opposite\_lines}(l,\mbox{reverse\_line}(m)) \ \Rightarrow \ \mbox{equally\_directed\_lines}(l,\mbox{reverse\_line}(m)) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO185+1.p} Lemma on symmetry and apartness

If the lines U and V are convergent, then the point that is
 equal to the intersection point is incident to both lines.




$\mbox{include}(\mbox{'Axioms/GEO006+0.ax'})$

$\forall x,y,u,v{:}\;( ( \mbox{distinct\_points}(x,y)\mbox{ and }\mbox{convergent\_lines}(u,v)\mbox{ and }\neg\,\mbox{distinct\_points}(x,\mbox{intersection\_point}(u,v)) ) \ \Rightarrow \ ( \neg\,\mbox{apart\_point\_and\_line}(x,u)\mbox{ and }\neg\,\mbox{apart\_point\_and\_line}(x,v) ) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO190+1.p} Collary to symmetry of incidence

If X, Y, and Z are pairwise distinct, and Z is incident with
 the line connecting X and Y, then Z is incident with the line
 connecting Y and X.




$\mbox{include}(\mbox{'Axioms/GEO006+0.ax'})$

$\forall x,y,z{:}\;( ( \mbox{distinct\_points}(x,y)\mbox{ and }\mbox{distinct\_points}(x,z)\mbox{ and }\mbox{distinct\_points}(y,z)\mbox{ and }\neg\,\mbox{apart\_point\_and\_line}(z,\mbox{line\_connecting}(x,y)) ) \ \Rightarrow \ \neg\,\mbox{apart\_point\_and\_line}(z,\mbox{line\_connecting}(y,x)) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO252+1.p} Jordan-type result for half planes






$\mbox{include}(\mbox{'Axioms/GEO007+0.ax'})$

$\forall a,b,l{:}\;( ( \mbox{left\_apart\_point}(a,l)\mbox{ and }\mbox{left\_apart\_point}(b,\mbox{reverse\_line}(l)) ) \ \Rightarrow \ ( \mbox{distinct\_points}(a,b)\mbox{ and }\mbox{left\_convergent\_lines}(\mbox{line\_connecting}(a,b),l) ) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO020-3.p} Corollary 4 to symmetries of equidistance

Show that if the distance from A to B equals the distance
 from C to D, then the distance from D to C equals the
 distance from A to B.




$\mbox{include}(\mbox{'Axioms/GEO002-0.ax'})$

$\mbox{equidistant}(u,v,u,v)\qquad\mbox{cnf}(d_1,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(w,x,u,v)\qquad\mbox{cnf}(d_2,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(v,u,w,x)\qquad\mbox{cnf}(d_3,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x)\qquad\mbox{cnf}(\mbox{u\_to\_v\_equals\_w\_to\_x},\mbox{hypothesis})$

$\neg\,\mbox{equidistant}(x,w,u,v)\qquad\mbox{cnf}(\mbox{prove\_symmetry},\mbox{negated\_conjecture})$

\medskip\textbf{GEO145+1.p} Starting point and precedence

If R is the starting point of o, then P precedes Q wrt. o, iff
 P is identical with R and Q is on o but different from R or P
 is between R and Q on o.




$\mbox{include}(\mbox{'Axioms/GEO004+0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO004+1.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO004+2.ax'})$

$\forall o,r{:}\;( \mbox{start\_point}(r,o) \ \Rightarrow \ \forall p,q{:}\;( \mbox{ordered\_by}(o,p,q)\iff ( ( p = r\mbox{ and }q\ne r\mbox{ and }\mbox{incident\_o}(q,o) )\mbox{ or }\mbox{between}(o,r,p,q) ) ) )\qquad\mbox{fof}(\mbox{theorem\_4}_{24},\mbox{conjecture})$

\medskip\textbf{GEO013-2.p} Collinearity for 5 points

If z1, z2, and z3 are each collinear with distinct points
 x and y, then z1, z2, and z3 are collinear.




$\mbox{include}(\mbox{'Axioms/GEO002-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO002-1.ax'})$

$a\ne b\qquad\mbox{cnf}(\mbox{a\_not\_b},\mbox{hypothesis})$

$\mbox{colinear}(a,b,d_1)\qquad\mbox{cnf}(\mbox{and1\_colinear},\mbox{hypothesis})$

$\mbox{colinear}(a,b,d_2)\qquad\mbox{cnf}(\mbox{abd2\_colinear},\mbox{hypothesis})$

$\mbox{colinear}(a,b,d_3)\qquad\mbox{cnf}(\mbox{abd3\_colinear},\mbox{hypothesis})$

$\neg\,\mbox{colinear}(d_1,d_2,d_3)\qquad\mbox{cnf}(\mbox{prove\_d1d2d3\_colinear},\mbox{negated\_conjecture})$

\medskip\textbf{GEO102-1.p} Common endpoint of subcurves means inclusion

If two sub-curves of one curve have a common endpoint and
 include a sub-curve starting at this endpoint, then one of the
 two sub-curves is included in the other.




$\mbox{include}(\mbox{'Axioms/GEO004-0.ax'})$

$\mbox{part\_of}(\mbox{sk}_{16},\mbox{sk}_{14})\qquad\mbox{cnf}(\mbox{theorem\_2\_20}_{67},\mbox{negated\_conjecture})$

$\mbox{part\_of}(\mbox{sk}_{16},\mbox{sk}_{15})\qquad\mbox{cnf}(\mbox{theorem\_2\_20}_{68},\mbox{negated\_conjecture})$

$\mbox{end\_point}(\mbox{sk}_{17},\mbox{sk}_{14})\qquad\mbox{cnf}(\mbox{theorem\_2\_20}_{69},\mbox{negated\_conjecture})$

$\mbox{end\_point}(\mbox{sk}_{17},\mbox{sk}_{15})\qquad\mbox{cnf}(\mbox{theorem\_2\_20}_{70},\mbox{negated\_conjecture})$

$\mbox{end\_point}(\mbox{sk}_{17},\mbox{sk}_{16})\qquad\mbox{cnf}(\mbox{theorem\_2\_20}_{71},\mbox{negated\_conjecture})$

$\mbox{part\_of}(\mbox{sk}_{14},\mbox{sk}_{18})\qquad\mbox{cnf}(\mbox{theorem\_2\_20}_{72},\mbox{negated\_conjecture})$

$\mbox{part\_of}(\mbox{sk}_{15},\mbox{sk}_{18})\qquad\mbox{cnf}(\mbox{theorem\_2\_20}_{73},\mbox{negated\_conjecture})$

$\neg\,\mbox{part\_of}(\mbox{sk}_{14},\mbox{sk}_{15})\qquad\mbox{cnf}(\mbox{theorem\_2\_20}_{74},\mbox{negated\_conjecture})$

$\neg\,\mbox{part\_of}(\mbox{sk}_{15},\mbox{sk}_{14})\qquad\mbox{cnf}(\mbox{theorem\_2\_20}_{75},\mbox{negated\_conjecture})$

\medskip\textbf{GEO216+3.p} A line is not orthogonal to itself






$\mbox{include}(\mbox{'Axioms/GEO006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+1.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+2.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+3.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+4.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+5.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+6.ax'})$

$\forall l{:}\;\neg\,\mbox{orthogonal\_lines}(l,l)\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO257+1.p} Transitivity of order on a line






$\mbox{include}(\mbox{'Axioms/GEO007+0.ax'})$

$\forall l,a,b,c,d{:}\;( ( \mbox{distinct\_points}(a,c)\mbox{ and }\mbox{distinct\_points}(b,c)\mbox{ and }\neg\,\mbox{apart\_point\_and\_line}(c,l)\mbox{ and }\mbox{left\_apart\_point}(d,l) ) \ \Rightarrow \ ( ( \mbox{before\_on\_line}(l,a,b)\mbox{ and }\mbox{before\_on\_line}(l,b,c) ) \ \Rightarrow \ \mbox{before\_on\_line}(l,a,c) ) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO172+2.p} Uniqueness of constructed points

If two lines are convergent and there is a point that is
 incident with both lines, then this point is equivalent to the
 intersection point of these lines.




$\mbox{include}(\mbox{'Axioms/GEO008+0.ax'})$

$\forall x,y,z{:}\;( ( \mbox{convergent\_lines}(x,y)\mbox{ and }\neg\,\mbox{apart\_point\_and\_line}(z,x)\mbox{ and }\neg\,\mbox{apart\_point\_and\_line}(z,y) ) \ \Rightarrow \ \neg\,\mbox{distinct\_points}(z,\mbox{intersection\_point}(x,y)) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO143+1.p} Unique oppositely oriented curve 4

For every oriented line there is exactly one uniquely determined
 oriented line with the same underlying curve that orders the
 points in the opposite way.




$\mbox{include}(\mbox{'Axioms/GEO004+0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO004+1.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO004+2.ax'})$

$\forall o,p,q,r{:}\;( \mbox{ordered\_by}(o,p,q) \ \Rightarrow \ ( \mbox{ordered\_by}(o,q,r)\iff \mbox{between}(o,p,q,r) ) )\qquad\mbox{fof}(\mbox{theorem\_4\_23}_4,\mbox{conjecture})$

\medskip\textbf{GEO151+1.p} Object stays still while one moves

If an object is in a position before and after another object
 moves, then it stays in this position while the other one moves




$\mbox{include}(\mbox{'Axioms/GEO004+0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO004+1.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO004+2.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO004+3.ax'})$

$\forall p,q_1,q_2,q_3,x,y{:}\;( ( \mbox{once}(\mbox{at\_the\_same\_time}(\mbox{at}(x,p),\mbox{at}(y,q_1)))\mbox{ and }\mbox{once}(\mbox{at\_the\_same\_time}(\mbox{at}(x,p),\mbox{at}(y,q_3)))\mbox{ and }\mbox{between\_o}(\mbox{trajectory\_of}(x),q_1,q_2,q_3) ) \ \Rightarrow \ \mbox{once}(\mbox{at\_the\_same\_time}(\mbox{at}(x,p),\mbox{at}(x,q_2))) )\qquad\mbox{fof}(t_{17},\mbox{conjecture})$

\medskip\textbf{GEO148-1.p} No meeting if someone has already passed

A point can only be a meeting point of two moving objects if
 it is not the case that one object already passed through it
 when the other object was still moving towards it




$\mbox{include}(\mbox{'Axioms/GEO004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO004-2.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO004-3.ax'})$

$\mbox{connect}(\mbox{sk}_{28},\mbox{sk}_{29},\mbox{sk}_{27})\qquad\mbox{cnf}(\mbox{t14}_{156},\mbox{negated\_conjecture})$

$\mbox{ordered\_by}(\mbox{trajectory\_of}(\mbox{sk}_{29}),\mbox{sk}_{31},\mbox{sk}_{27})\qquad\mbox{cnf}(\mbox{t14}_{157},\mbox{negated\_conjecture})$

$\mbox{ordered\_by}(\mbox{trajectory\_of}(\mbox{sk}_{28}),\mbox{sk}_{27},\mbox{sk}_{30})\qquad\mbox{cnf}(\mbox{t14}_{158},\mbox{negated\_conjecture})$

$\mbox{once}(\mbox{at\_the\_same\_time}(\mbox{at}(\mbox{sk}_{28},\mbox{sk}_{30}),\mbox{at}(\mbox{sk}_{29},\mbox{sk}_{31})))\qquad\mbox{cnf}(\mbox{t14}_{159},\mbox{negated\_conjecture})$

\medskip\textbf{GEO355+1.p} Ordered affine geometry with definitions






$\mbox{include}(\mbox{'Axioms/GEO009+0.ax'})$

\medskip\textbf{GEO263+3.p} Parallel projection preserves or reverses order






$\mbox{include}(\mbox{'Axioms/GEO009+0.ax'})$

$\forall l,m,n,a,b,c{:}\;( ( \mbox{between\_on\_line}(l,a,b,c)\mbox{ and }\mbox{convergent\_lines}(l,m)\mbox{ and }\mbox{convergent\_lines}(l,n)\mbox{ and }\mbox{convergent\_lines}(m,n) ) \ \Rightarrow \ \mbox{between\_on\_line}(m,\mbox{intersection\_point}(m,\mbox{parallel\_through\_point}(n,a)),\mbox{intersection\_point}(m,\mbox{parallel\_through\_point}(n,b)),\mbox{intersection\_point}(m,\mbox{parallel\_through\_point}(n,c))) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO006-3.p} Betweenness for 3 points on a line

For any three distinct points x, y, and z, if y is between
 x and z, then both x is not between y and z and z is not
 between x and y.




$\mbox{include}(\mbox{'Axioms/GEO002-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO002-2.ax'})$

$\mbox{equidistant}(u,v,u,v)\qquad\mbox{cnf}(d_1,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(w,x,u,v)\qquad\mbox{cnf}(d_2,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(v,u,w,x)\qquad\mbox{cnf}(d_3,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(u,v,x,w)\qquad\mbox{cnf}(\mbox{d4}_1,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(v,u,x,w)\qquad\mbox{cnf}(\mbox{d4}_2,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(w,x,v,u)\qquad\mbox{cnf}(\mbox{d4}_3,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(x,w,u,v)\qquad\mbox{cnf}(\mbox{d4}_4,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(x,w,v,u)\qquad\mbox{cnf}(\mbox{d4}_5,\mbox{axiom})$

$( \mbox{equidistant}(u,v,w,x)\mbox{ and }\mbox{equidistant}(w,x,y,z) ) \ \Rightarrow \ \mbox{equidistant}(u,v,y,z)\qquad\mbox{cnf}(d_5,\mbox{axiom})$

$v = \mbox{extension}(u,v,w,w)\qquad\mbox{cnf}(e_1,\mbox{axiom})$

$y = \mbox{extension}(u,v,w,x) \ \Rightarrow \ \mbox{between}(u,v,y)\qquad\mbox{cnf}(b_0,\mbox{axiom})$

$\mbox{between}(u,v,\mbox{reflection}(u,v))\qquad\mbox{cnf}(\mbox{r2}_1,\mbox{axiom})$

$\mbox{equidistant}(v,\mbox{reflection}(u,v),u,v)\qquad\mbox{cnf}(\mbox{r2}_2,\mbox{axiom})$

$u = v \ \Rightarrow \ v = \mbox{reflection}(u,v)\qquad\mbox{cnf}(\mbox{r3}_1,\mbox{axiom})$

$u = \mbox{reflection}(u,u)\qquad\mbox{cnf}(\mbox{r3}_2,\mbox{axiom})$

$v = \mbox{reflection}(u,v) \ \Rightarrow \ u = v\qquad\mbox{cnf}(r_4,\mbox{axiom})$

$\mbox{equidistant}(u,u,v,v)\qquad\mbox{cnf}(d_7,\mbox{axiom})$

$( \mbox{equidistant}(u,v,u_1,v_1)\mbox{ and }\mbox{equidistant}(v,w,v_1,w_1)\mbox{ and }\mbox{between}(u,v,w)\mbox{ and }\mbox{between}(u_1,v_1,w_1) ) \ \Rightarrow \ \mbox{equidistant}(u,w,u_1,w_1)\qquad\mbox{cnf}(d_8,\mbox{axiom})$

$( \mbox{between}(u,v,w)\mbox{ and }\mbox{between}(u,v,x)\mbox{ and }\mbox{equidistant}(v,w,v,x) ) \ \Rightarrow \ ( u = v\mbox{ or }w = x )\qquad\mbox{cnf}(d_9,\mbox{axiom})$

$\mbox{between}(u,v,w) \ \Rightarrow \ ( u = v\mbox{ or }w = \mbox{extension}(u,v,v,w) )\qquad\mbox{cnf}(\mbox{d10}_1,\mbox{axiom})$

$\mbox{equidistant}(w,x,y,z) \ \Rightarrow \ ( \mbox{extension}(u,v,w,x) = \mbox{extension}(u,v,y,z)\mbox{ or }u = v )\qquad\mbox{cnf}(\mbox{d10}_2,\mbox{axiom})$

$\mbox{extension}(u,v,u,v) = \mbox{extension}(u,v,v,u)\mbox{ or }u = v\qquad\mbox{cnf}(\mbox{d10}_3,\mbox{axiom})$

$\mbox{equidistant}(v,u,v,\mbox{reflection}(\mbox{reflection}(u,v),v))\qquad\mbox{cnf}(r_5,\mbox{axiom})$

$u = \mbox{reflection}(\mbox{reflection}(u,v),v)\qquad\mbox{cnf}(r_6,\mbox{axiom})$

$\mbox{between}(u,v,v)\qquad\mbox{cnf}(t_3,\mbox{axiom})$

$( \mbox{between}(u,w,x)\mbox{ and }u = x ) \ \Rightarrow \ \mbox{between}(v,w,x)\qquad\mbox{cnf}(b_1,\mbox{axiom})$

$\mbox{between}(u,v,w) \ \Rightarrow \ \mbox{between}(w,v,u)\qquad\mbox{cnf}(t_1,\mbox{axiom})$

$\mbox{between}(u,u,v)\qquad\mbox{cnf}(t_2,\mbox{axiom})$

$( \mbox{between}(u,v,w)\mbox{ and }\mbox{between}(v,u,w) ) \ \Rightarrow \ u = v\qquad\mbox{cnf}(b_2,\mbox{axiom})$

$( \mbox{between}(u,v,w)\mbox{ and }\mbox{between}(u,w,v) ) \ \Rightarrow \ v = w\qquad\mbox{cnf}(b_3,\mbox{axiom})$

$a\ne c\qquad\mbox{cnf}(\mbox{a\_not\_c},\mbox{hypothesis})$

$a\ne d\qquad\mbox{cnf}(\mbox{a\_not\_d},\mbox{hypothesis})$

$c\ne d\qquad\mbox{cnf}(\mbox{c\_not\_d},\mbox{hypothesis})$

$\mbox{between}(a,c,d)\qquad\mbox{cnf}(\mbox{c\_between\_a\_and\_d},\mbox{hypothesis})$

$\mbox{between}(c,a,d)\mbox{ or }\mbox{between}(a,d,c)\qquad\mbox{cnf}(\mbox{prove\_not\_between\_others},\mbox{negated\_conjecture})$

\medskip\textbf{GEO024-3.p} All null segments are congruent






$\mbox{include}(\mbox{'Axioms/GEO002-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO002-2.ax'})$

$\mbox{equidistant}(u,v,u,v)\qquad\mbox{cnf}(d_1,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(w,x,u,v)\qquad\mbox{cnf}(d_2,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(v,u,w,x)\qquad\mbox{cnf}(d_3,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(u,v,x,w)\qquad\mbox{cnf}(\mbox{d4}_1,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(v,u,x,w)\qquad\mbox{cnf}(\mbox{d4}_2,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(w,x,v,u)\qquad\mbox{cnf}(\mbox{d4}_3,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(x,w,u,v)\qquad\mbox{cnf}(\mbox{d4}_4,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(x,w,v,u)\qquad\mbox{cnf}(\mbox{d4}_5,\mbox{axiom})$

$( \mbox{equidistant}(u,v,w,x)\mbox{ and }\mbox{equidistant}(w,x,y,z) ) \ \Rightarrow \ \mbox{equidistant}(u,v,y,z)\qquad\mbox{cnf}(d_5,\mbox{axiom})$

$v = \mbox{extension}(u,v,w,w)\qquad\mbox{cnf}(e_1,\mbox{axiom})$

$y = \mbox{extension}(u,v,w,x) \ \Rightarrow \ \mbox{between}(u,v,y)\qquad\mbox{cnf}(b_0,\mbox{axiom})$

$\mbox{between}(u,v,\mbox{reflection}(u,v))\qquad\mbox{cnf}(\mbox{r2}_1,\mbox{axiom})$

$\mbox{equidistant}(v,\mbox{reflection}(u,v),u,v)\qquad\mbox{cnf}(\mbox{r2}_2,\mbox{axiom})$

$u = v \ \Rightarrow \ v = \mbox{reflection}(u,v)\qquad\mbox{cnf}(\mbox{r3}_1,\mbox{axiom})$

$u = \mbox{reflection}(u,u)\qquad\mbox{cnf}(\mbox{r3}_2,\mbox{axiom})$

$v = \mbox{reflection}(u,v) \ \Rightarrow \ u = v\qquad\mbox{cnf}(r_4,\mbox{axiom})$

$\neg\,\mbox{equidistant}(u,u,v,v)\qquad\mbox{cnf}(\mbox{prove\_congruence},\mbox{negated\_conjecture})$

\medskip\textbf{GEO082+1.p} Antisymmetry of part\_of






$\mbox{include}(\mbox{'Axioms/GEO004+0.ax'})$

$\forall c_1,c_2{:}\;( ( \mbox{part\_of}(c_1,c_2)\mbox{ and }\mbox{part\_of}(c_2,c_1) ) \ \Rightarrow \ c_1 = c_2 )\qquad\mbox{fof}(\mbox{part\_of\_antisymmetry},\mbox{conjecture})$

\medskip\textbf{GEO038-3.p} Corollary 1 to the segment contruction axiom






$\mbox{include}(\mbox{'Axioms/GEO002-0.ax'})$

$\mbox{equidistant}(u,v,u,v)\qquad\mbox{cnf}(d_1,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(w,x,u,v)\qquad\mbox{cnf}(d_2,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(v,u,w,x)\qquad\mbox{cnf}(d_3,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(u,v,x,w)\qquad\mbox{cnf}(\mbox{d4}_1,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(v,u,x,w)\qquad\mbox{cnf}(\mbox{d4}_2,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(w,x,v,u)\qquad\mbox{cnf}(\mbox{d4}_3,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(x,w,u,v)\qquad\mbox{cnf}(\mbox{d4}_4,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(x,w,v,u)\qquad\mbox{cnf}(\mbox{d4}_5,\mbox{axiom})$

$( \mbox{equidistant}(u,v,w,x)\mbox{ and }\mbox{equidistant}(w,x,y,z) ) \ \Rightarrow \ \mbox{equidistant}(u,v,y,z)\qquad\mbox{cnf}(d_5,\mbox{axiom})$

$v = \mbox{extension}(u,v,w,w)\qquad\mbox{cnf}(e_1,\mbox{axiom})$

$y = \mbox{extension}(u,v,w,x)\qquad\mbox{cnf}(\mbox{y\_is\_extension},\mbox{hypothesis})$

$\neg\,\mbox{between}(u,v,y)\qquad\mbox{cnf}(\mbox{prove\_corollary},\mbox{negated\_conjecture})$

\medskip\textbf{GEO268+3.p} Equivalence of unequally directed opposite and reversed lines






$\mbox{include}(\mbox{'Axioms/GEO009+0.ax'})$

$\forall l,m{:}\;( \mbox{unequally\_directed\_lines}(l,m)\iff \mbox{unequally\_directed\_opposite\_lines}(l,\mbox{reverse\_line}(m)) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO090-1.p} Meeting point of curves on a subcurve

If a point P is a meeting point of two curves and lies on a
 sub-curve of one of the two curves then P is also meeting point
 of the sub-curve and the other curve.




$\mbox{include}(\mbox{'Axioms/GEO004-0.ax'})$

$\mbox{part\_of}(\mbox{sk}_{15},\mbox{sk}_{14})\qquad\mbox{cnf}(\mbox{corollary\_2\_12}_{67},\mbox{negated\_conjecture})$

$\mbox{incident\_c}(\mbox{sk}_{17},\mbox{sk}_{15})\qquad\mbox{cnf}(\mbox{corollary\_2\_12}_{68},\mbox{negated\_conjecture})$

$\mbox{sk}_{17}\wedge \mbox{sk}_{14}{=}\mbox{sk}_{16}\qquad\mbox{cnf}(\mbox{corollary\_2\_12}_{69},\mbox{negated\_conjecture})$

$\neg\,\mbox{sk}_{17}\wedge \mbox{sk}_{15}{=}\mbox{sk}_{16}\qquad\mbox{cnf}(\mbox{corollary\_2\_12}_{70},\mbox{negated\_conjecture})$

\medskip\textbf{GEO247+1.p} A point in each region formed by intersecting lines






$\mbox{include}(\mbox{'Axioms/GEO007+0.ax'})$

$\forall a,b,c,d,l,m{:}\;( ( \mbox{left\_apart\_point}(a,l)\mbox{ and }\mbox{left\_apart\_point}(a,m)\mbox{ and }\mbox{left\_apart\_point}(b,\mbox{reverse\_line}(l))\mbox{ and }\mbox{left\_apart\_point}(b,\mbox{reverse\_line}(m))\mbox{ and }\mbox{left\_apart\_point}(c,l)\mbox{ and }\mbox{left\_apart\_point}(c,\mbox{reverse\_line}(m))\mbox{ and }\mbox{left\_apart\_point}(d,\mbox{reverse\_line}(l))\mbox{ and }\mbox{left\_apart\_point}(d,m) ) \ \Rightarrow \ \mbox{convergent\_lines}(l,m) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO017-2.p} Corollary 1 to symmetries of equidistance

Show that if the distance from A to B equals the distance
 from C to D, then the distance from A to B equals the
 distance from D to C.




$\mbox{include}(\mbox{'Axioms/GEO002-0.ax'})$

$\mbox{equidistant}(u,v,w,x)\qquad\mbox{cnf}(\mbox{u\_to\_v\_equals\_w\_to\_x},\mbox{hypothesis})$

$\neg\,\mbox{equidistant}(u,v,x,w)\qquad\mbox{cnf}(\mbox{prove\_symmetry},\mbox{negated\_conjecture})$

\medskip\textbf{GEO224+2.p} Find point incident to line

Assume orthogonal geometry. Given a point and a line, to find
 a point incident with the line.




$\mbox{include}(\mbox{'Axioms/GEO008+0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+2.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+3.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+5.ax'})$

$\forall x,y{:}\;( ( \mbox{point}(x)\mbox{ and }\mbox{line}(y) ) \ \Rightarrow \ \exists z{:}\;( \mbox{point}(z)\mbox{ and }\neg\,\mbox{apt}(z,y) ) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO249+3.p} A point in each region formed by parallel lines






$\mbox{include}(\mbox{'Axioms/GEO009+0.ax'})$

$\forall a,b,l{:}\;( ( \mbox{right\_apart\_point}(a,l)\mbox{ and }\mbox{right\_apart\_point}(b,\mbox{parallel\_through\_point}(l,a)) ) \ \Rightarrow \ \mbox{right\_apart\_point}(b,l) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO142+1.p} Unique oppositely oriented curve 3

For every oriented line there is exactly one uniquely determined
 oriented line with the same underlying curve that orders the
 points in the opposite way.




$\mbox{include}(\mbox{'Axioms/GEO004+0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO004+1.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO004+2.ax'})$

$\forall o,p,q,r{:}\;( \mbox{ordered\_by}(o,p,q) \ \Rightarrow \ ( \mbox{ordered\_by}(o,p,r)\iff ( \mbox{between}(o,p,r,q)\mbox{ or }\mbox{between}(o,p,q,r)\mbox{ or }q = r ) ) )\qquad\mbox{fof}(\mbox{theorem\_4\_23}_3,\mbox{conjecture})$

\medskip\textbf{GEO229+1.p} Uniqueness of reversed lines






$\mbox{include}(\mbox{'Axioms/GEO007+0.ax'})$

$\forall l,m{:}\;( \neg\,\mbox{unequally\_directed\_lines}(l,\mbox{reverse\_line}(m)) \ \Rightarrow \ \neg\,\mbox{unequally\_directed\_lines}(l,\mbox{reverse\_line}(m)) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO064-2.p} Corollary 1 to collinearity






$\mbox{include}(\mbox{'Axioms/GEO002-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO002-1.ax'})$

$\mbox{between}(w,v,u)\qquad\mbox{cnf}(\mbox{v\_between\_w\_and\_u},\mbox{hypothesis})$

$\neg\,\mbox{colinear}(u,v,w)\qquad\mbox{cnf}(\mbox{prove\_uvw\_colinear},\mbox{negated\_conjecture})$

\medskip\textbf{GEO233+3.p} Reverse is idempotent for direction






$\mbox{include}(\mbox{'Axioms/GEO009+0.ax'})$

$\forall l{:}\;\mbox{equally\_directed\_lines}(\mbox{reverse\_line}(\mbox{reverse\_line}(l)),l)\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO139+1.p} Oppositely oriented curve exists

For every oriented curve there is an oppositely oriented curve
 with the same underlying curve.




$\mbox{include}(\mbox{'Axioms/GEO004+0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO004+1.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO004+2.ax'})$

$\forall o{:}\;\exists \mbox{opp}{:}\;( \mbox{underlying\_curve}(o) = \mbox{underlying\_curve}(\mbox{opp})\mbox{ and }\forall p,q{:}\;( \mbox{ordered\_by}(o,p,q) \ \Rightarrow \ \mbox{ordered\_by}(\mbox{opp},q,p) ) )\qquad\mbox{fof}(\mbox{theorem\_4}_{22},\mbox{conjecture})$

\medskip\textbf{GEO185+3.p} Lemma on symmetry and apartness

If the lines U and V are convergent, then the point that is
 equal to the intersection point is incident to both lines.




$\mbox{include}(\mbox{'Axioms/GEO006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+1.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+2.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+3.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+4.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+5.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+6.ax'})$

$\forall x,y,u,v{:}\;( ( \mbox{distinct\_points}(x,y)\mbox{ and }\mbox{convergent\_lines}(u,v)\mbox{ and }\mbox{equal\_points}(x,\mbox{intersection\_point}(u,v)) ) \ \Rightarrow \ ( \mbox{incident\_point\_and\_line}(x,u)\mbox{ and }\mbox{incident\_point\_and\_line}(x,v) ) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO354+1.p} Ordered affine geometry






$\mbox{include}(\mbox{'Axioms/GEO007+0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO007+1.ax'})$

\medskip\textbf{GEO175+2.p} Lemma on symmetry and apartness

If two lines are convergent and a point is distinct from the
 intersection point then this point is apart from at least one of
 these lines.




$\mbox{include}(\mbox{'Axioms/GEO008+0.ax'})$

$\forall x,y,u,v{:}\;( ( \mbox{distinct\_points}(x,y)\mbox{ and }\mbox{convergent\_lines}(u,v)\mbox{ and }\mbox{distinct\_points}(x,\mbox{intersection\_point}(u,v)) ) \ \Rightarrow \ ( \mbox{apart\_point\_and\_line}(x,u)\mbox{ or }\mbox{apart\_point\_and\_line}(x,v) ) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO208+2.p} Point on both parallel lines

If the point X is incident with both the lines Y and Z, and
 Y and Z are parallel, then Y and Z are equal.




$\mbox{include}(\mbox{'Axioms/GEO008+0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+2.ax'})$

$\forall x,y,z{:}\;( ( \neg\,\mbox{apart\_point\_and\_line}(x,y)\mbox{ and }\neg\,\mbox{apart\_point\_and\_line}(x,z)\mbox{ and }\neg\,\mbox{convergent\_lines}(y,z) ) \ \Rightarrow \ \neg\,\mbox{distinct\_lines}(y,z) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO121-1.p} Endpoints are either starting or finishing points

Every endpoint of the underlying curve of an oriented curve is
 either a starting point or finishing point of the oriented curve.




$\mbox{include}(\mbox{'Axioms/GEO004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO004-2.ax'})$

$\mbox{end\_point}(\mbox{sk}_{26},\mbox{underlying\_curve}(\mbox{sk}_{25}))\qquad\mbox{cnf}(\mbox{theorem\_4\_7}_{133},\mbox{negated\_conjecture})$

$\neg\,\mbox{start\_point}(\mbox{sk}_{26},\mbox{sk}_{25})\qquad\mbox{cnf}(\mbox{theorem\_4\_7}_{134},\mbox{negated\_conjecture})$

$\neg\,\mbox{finish\_point}(\mbox{sk}_{26},\mbox{sk}_{25})\qquad\mbox{cnf}(\mbox{theorem\_4\_7}_{135},\mbox{negated\_conjecture})$

\medskip\textbf{GEO012-2.p} Collinearity for 4 points

If any three distinct points x, y, and z are collinear and
 a fourth point w is collinear with x and y, then
 w is collinear with x and z and also with x and y.




$\mbox{include}(\mbox{'Axioms/GEO002-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO002-1.ax'})$

$a\ne b\qquad\mbox{cnf}(\mbox{a\_not\_b},\mbox{hypothesis})$

$\mbox{colinear}(a,b,c)\qquad\mbox{cnf}(\mbox{abc\_colinear},\mbox{hypothesis})$

$\mbox{colinear}(a,b,d)\qquad\mbox{cnf}(\mbox{abd\_colinear},\mbox{hypothesis})$

$\mbox{colinear}(a,c,d) \ \Rightarrow \ \neg\,\mbox{colinear}(b,c,d)\qquad\mbox{cnf}(\mbox{prove\_colinearity},\mbox{negated\_conjecture})$

\medskip\textbf{GEO008-1.p} Betweenness for 5 points on a line (Five point theorem)

For all points x, y, z, w, and v, if y and w are between
 x and z, and $\vee$ is between y and w, then $\vee$ is between x and z.




$\mbox{include}(\mbox{'Axioms/GEO001-0.ax'})$

$\mbox{between}(a,c,e)\qquad\mbox{cnf}(\mbox{c\_between\_a\_and\_e},\mbox{hypothesis})$

$\mbox{between}(a,d,e)\qquad\mbox{cnf}(\mbox{d\_between\_a\_and\_e},\mbox{hypothesis})$

$\mbox{between}(c,b,d)\qquad\mbox{cnf}(\mbox{b\_between\_c\_and\_d},\mbox{hypothesis})$

$\neg\,\mbox{between}(a,b,e)\qquad\mbox{cnf}(\mbox{prove\_betweenness},\mbox{negated\_conjecture})$

\medskip\textbf{GEO230+1.p} Reversed lines are equal and conversely directed






$\mbox{include}(\mbox{'Axioms/GEO007+0.ax'})$

$\forall a,b{:}\;( \mbox{distinct\_points}(a,b) \ \Rightarrow \ ( \neg\,\mbox{distinct\_lines}(\mbox{line\_connecting}(a,b),\mbox{line\_connecting}(b,a))\mbox{ and }\neg\,\mbox{unequally\_directed\_lines}(\mbox{line\_connecting}(b,a),\mbox{line\_connecting}(a,b)) ) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO212+1.p} Non-orthogonal lines and a third line

If a line L is not orthogonal to M, then a third line N is
 convergent to L or not orthogonal to M.




$\mbox{include}(\mbox{'Axioms/GEO006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+2.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+3.ax'})$

$\forall l,m,n{:}\;( \mbox{unorthogonal\_lines}(l,m) \ \Rightarrow \ ( \mbox{convergent\_lines}(l,n)\mbox{ or }\mbox{unorthogonal\_lines}(m,n) ) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO179+3.p} Lemma on symmetry and apartness

If two points X and Y are distinct and a point Z is apart from
 the line connecting X and Y, then the line connecting X and Y
 is distinct from the line connecting Z and X and distinct from
 the line connecting Z and Y.




$\mbox{include}(\mbox{'Axioms/GEO006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+1.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+2.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+3.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+4.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+5.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+6.ax'})$

$\forall x,y,z{:}\;( ( \mbox{distinct\_points}(x,y)\mbox{ and }\mbox{apart\_point\_and\_line}(z,\mbox{line\_connecting}(x,y)) ) \ \Rightarrow \ ( \mbox{distinct\_lines}(\mbox{line\_connecting}(x,y),\mbox{line\_connecting}(z,x))\mbox{ and }\mbox{distinct\_lines}(\mbox{line\_connecting}(x,y),\mbox{line\_connecting}(z,y)) ) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO040-2.p} Antisymmetry of betweenness in its first two arguments






$\mbox{include}(\mbox{'Axioms/GEO002-0.ax'})$

$\mbox{between}(u,v,w)\qquad\mbox{cnf}(\mbox{v\_between\_u\_and\_w},\mbox{hypothesis})$

$\mbox{between}(v,u,w)\qquad\mbox{cnf}(\mbox{u\_between\_v\_and\_w},\mbox{hypothesis})$

$u\ne v\qquad\mbox{cnf}(\mbox{prove\_u\_is\_v},\mbox{negated\_conjecture})$

\medskip\textbf{GEO219+2.p} Transitivity of orthogonal and parallel

If line L is orthogonal to M and a line N is parallel to L,
 then M is orthogonal to N.




$\mbox{include}(\mbox{'Axioms/GEO008+0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+4.ax'})$

$\forall l,m,n{:}\;( ( \neg\,\mbox{unorthogonal\_lines}(l,m)\mbox{ and }\neg\,\mbox{convergent\_lines}(l,n) ) \ \Rightarrow \ \neg\,\mbox{unorthogonal\_lines}(m,n) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO239+1.p} Lemma on oriented intersection of lines with plane






$\mbox{include}(\mbox{'Axioms/GEO007+0.ax'})$

$\forall a,b,l{:}\;( ( \neg\,\mbox{apart\_point\_and\_line}(a,l)\mbox{ and }\mbox{left\_apart\_point}(b,l) ) \ \Rightarrow \ \mbox{left\_convergent\_lines}(l,\mbox{line\_connecting}(a,b)) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO240+3.p} Lemma on oriented intersection of lines with plane






$\mbox{include}(\mbox{'Axioms/GEO009+0.ax'})$

$\forall a,b,l{:}\;( ( \mbox{incident\_point\_and\_line}(a,l)\mbox{ and }\mbox{right\_apart\_point}(b,l) ) \ \Rightarrow \ \mbox{right\_convergent\_lines}(l,\mbox{line\_connecting}(a,b)) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO044-2.p} First outer transitivity property for betweenness






$\mbox{include}(\mbox{'Axioms/GEO002-0.ax'})$

$\mbox{between}(u,v,w)\qquad\mbox{cnf}(\mbox{v\_between\_u\_and\_w},\mbox{hypothesis})$

$\mbox{between}(v,w,x)\qquad\mbox{cnf}(\mbox{w\_between\_v\_and\_x},\mbox{hypothesis})$

$v\ne w\qquad\mbox{cnf}(\mbox{v\_not\_w},\mbox{hypothesis})$

$\neg\,\mbox{between}(u,w,x)\qquad\mbox{cnf}(\mbox{prove\_w\_between\_u\_and\_x},\mbox{negated\_conjecture})$

\medskip\textbf{GEO003-2.p} For all points x and y, y is between x and y






$\mbox{include}(\mbox{'Axioms/GEO002-0.ax'})$

$\neg\,\mbox{between}(a,b,b)\qquad\mbox{cnf}(\mbox{prove\_b\_between\_a\_and\_b},\mbox{negated\_conjecture})$

\medskip\textbf{GEO119-1.p} Oriented curve starting point is endpoint of underlying curve






$\mbox{include}(\mbox{'Axioms/GEO004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO004-2.ax'})$

$\mbox{start\_point}(\mbox{sk}_{26},\mbox{sk}_{25})\qquad\mbox{cnf}(\mbox{theorem\_4\_6\_1}_{133},\mbox{negated\_conjecture})$

$\neg\,\mbox{end\_point}(\mbox{sk}_{26},\mbox{underlying\_curve}(\mbox{sk}_{25}))\qquad\mbox{cnf}(\mbox{theorem\_4\_6\_1}_{134},\mbox{negated\_conjecture})$

\medskip\textbf{GEO217+2.p} Transitivity of parallel

If a line L is parallel to the lines M and N, then M and N
 are parallel.




$\mbox{include}(\mbox{'Axioms/GEO008+0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+4.ax'})$

$\forall l,m,n{:}\;( ( \neg\,\mbox{convergent\_lines}(l,m)\mbox{ and }\neg\,\mbox{convergent\_lines}(l,n) ) \ \Rightarrow \ \neg\,\mbox{convergent\_lines}(m,n) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO231+3.p} Oppositely and equally directed lines






$\mbox{include}(\mbox{'Axioms/GEO009+0.ax'})$

$\forall l,m,n{:}\;( ( \mbox{equally\_directed\_opposite\_lines}(l,m)\mbox{ and }\mbox{equally\_directed\_lines}(l,n) ) \ \Rightarrow \ \mbox{equally\_directed\_opposite\_lines}(m,n) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO124+1.p} Every oriented curve has at most one starting point






$\mbox{include}(\mbox{'Axioms/GEO004+0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO004+1.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO004+2.ax'})$

$\forall o,p,q{:}\;( ( \mbox{start\_point}(p,o)\mbox{ and }\mbox{start\_point}(q,o) ) \ \Rightarrow \ p = q )\qquad\mbox{fof}(\mbox{corollary\_4\_10}_1,\mbox{conjecture})$

\medskip\textbf{GEO221+2.p} Lemma on orthogonality

If a point B is incident with the orthogonal to a line L
 through point A, then this orthogonal is equal to the
 orthogonal to L through B.




$\mbox{include}(\mbox{'Axioms/GEO008+0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+2.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+3.ax'})$

$\forall a,b,l{:}\;( \neg\,\mbox{apart\_point\_and\_line}(b,\mbox{orthogonal\_through\_point}(l,a)) \ \Rightarrow \ \neg\,\mbox{distinct\_lines}(\mbox{orthogonal\_through\_point}(l,a),\mbox{orthogonal\_through\_point}(l,b)) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO129+1.p} Precedence on an oriented curve is a transitive relation






$\mbox{include}(\mbox{'Axioms/GEO004+0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO004+1.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO004+2.ax'})$

$\forall o,p,q,r{:}\;( ( \mbox{ordered\_by}(o,p,q)\mbox{ and }\mbox{ordered\_by}(o,q,r) ) \ \Rightarrow \ \mbox{ordered\_by}(o,p,r) )\qquad\mbox{fof}(\mbox{theorem\_4}_{14},\mbox{conjecture})$

\medskip\textbf{GEO147+1.p} Meeting is possible only if there is a common position






$\mbox{include}(\mbox{'Axioms/GEO004+0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO004+1.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO004+2.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO004+3.ax'})$

$\forall p,x,y{:}\;( \mbox{connect}(x,y,p) \ \Rightarrow \ ( \mbox{incident\_o}(p,\mbox{trajectory\_of}(x))\mbox{ and }\mbox{incident\_o}(p,\mbox{trajectory\_of}(y)) ) )\qquad\mbox{fof}(t_{13},\mbox{conjecture})$

\medskip\textbf{GEO255+3.p} Property of order and betweeness






$\mbox{include}(\mbox{'Axioms/GEO009+0.ax'})$

$\forall l,a,b{:}\;\neg\,\mbox{before\_on\_line}(l,a,b)\mbox{ and }\mbox{before\_on\_line}(l,b,a)\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO055-2.p} Corollary 3 to the segment construction axiom






$\mbox{include}(\mbox{'Axioms/GEO002-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO002-2.ax'})$

$\neg\,\mbox{equidistant}(v,\mbox{reflection}(u,v),u,v)\qquad\mbox{cnf}(\mbox{prove\_equidistance},\mbox{negated\_conjecture})$

\medskip\textbf{GEO015-2.p} Equidistance is symmetric between its argument pairs

Show that if the distance from A to B equals the distance
 from C to D, then the distance from C to D equals the
 distance from A to B.




$\mbox{include}(\mbox{'Axioms/GEO002-0.ax'})$

$\mbox{equidistant}(u,v,w,x)\qquad\mbox{cnf}(\mbox{u\_to\_v\_equals\_w\_to\_x},\mbox{hypothesis})$

$\neg\,\mbox{equidistant}(w,x,u,v)\qquad\mbox{cnf}(\mbox{prove\_symmetry},\mbox{negated\_conjecture})$

\medskip\textbf{GEO120+1.p} Oriented curve finishing point is endpoint of underlying curve






$\mbox{include}(\mbox{'Axioms/GEO004+0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO004+1.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO004+2.ax'})$

$\forall o,p{:}\;( \mbox{finish\_point}(p,o) \ \Rightarrow \ \mbox{end\_point}(p,\mbox{underlying\_curve}(o)) )\qquad\mbox{fof}(\mbox{theorem\_4\_6}_2,\mbox{conjecture})$

\medskip\textbf{GEO220+3.p} Transitivity of orthogonal

If a line L is orthogonal to line M and line N is orthogonal
 to L, then M and N are parallel.




$\mbox{include}(\mbox{'Axioms/GEO006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+1.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+2.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+3.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+4.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+5.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+6.ax'})$

$\forall l,m,n{:}\;( ( \mbox{orthogonal\_lines}(l,m)\mbox{ and }\mbox{orthogonal\_lines}(l,n) ) \ \Rightarrow \ \mbox{parallel\_lines}(m,n) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO154-2.p} Colinearity axioms for the GEO002 geometry axioms






$\mbox{include}(\mbox{'Axioms/GEO002-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO002-1.ax'})$

\medskip\textbf{GEO059-3.p} Congruence for double reflection






$\mbox{include}(\mbox{'Axioms/GEO002-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO002-2.ax'})$

$\mbox{equidistant}(u,v,u,v)\qquad\mbox{cnf}(d_1,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(w,x,u,v)\qquad\mbox{cnf}(d_2,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(v,u,w,x)\qquad\mbox{cnf}(d_3,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(u,v,x,w)\qquad\mbox{cnf}(\mbox{d4}_1,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(v,u,x,w)\qquad\mbox{cnf}(\mbox{d4}_2,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(w,x,v,u)\qquad\mbox{cnf}(\mbox{d4}_3,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(x,w,u,v)\qquad\mbox{cnf}(\mbox{d4}_4,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(x,w,v,u)\qquad\mbox{cnf}(\mbox{d4}_5,\mbox{axiom})$

$( \mbox{equidistant}(u,v,w,x)\mbox{ and }\mbox{equidistant}(w,x,y,z) ) \ \Rightarrow \ \mbox{equidistant}(u,v,y,z)\qquad\mbox{cnf}(d_5,\mbox{axiom})$

$v = \mbox{extension}(u,v,w,w)\qquad\mbox{cnf}(e_1,\mbox{axiom})$

$y = \mbox{extension}(u,v,w,x) \ \Rightarrow \ \mbox{between}(u,v,y)\qquad\mbox{cnf}(b_0,\mbox{axiom})$

$\mbox{between}(u,v,\mbox{reflection}(u,v))\qquad\mbox{cnf}(\mbox{r2}_1,\mbox{axiom})$

$\mbox{equidistant}(v,\mbox{reflection}(u,v),u,v)\qquad\mbox{cnf}(\mbox{r2}_2,\mbox{axiom})$

$u = v \ \Rightarrow \ v = \mbox{reflection}(u,v)\qquad\mbox{cnf}(\mbox{r3}_1,\mbox{axiom})$

$u = \mbox{reflection}(u,u)\qquad\mbox{cnf}(\mbox{r3}_2,\mbox{axiom})$

$v = \mbox{reflection}(u,v) \ \Rightarrow \ u = v\qquad\mbox{cnf}(r_4,\mbox{axiom})$

$\mbox{equidistant}(u,u,v,v)\qquad\mbox{cnf}(d_7,\mbox{axiom})$

$( \mbox{equidistant}(u,v,u_1,v_1)\mbox{ and }\mbox{equidistant}(v,w,v_1,w_1)\mbox{ and }\mbox{between}(u,v,w)\mbox{ and }\mbox{between}(u_1,v_1,w_1) ) \ \Rightarrow \ \mbox{equidistant}(u,w,u_1,w_1)\qquad\mbox{cnf}(d_8,\mbox{axiom})$

$( \mbox{between}(u,v,w)\mbox{ and }\mbox{between}(u,v,x)\mbox{ and }\mbox{equidistant}(v,w,v,x) ) \ \Rightarrow \ ( u = v\mbox{ or }w = x )\qquad\mbox{cnf}(d_9,\mbox{axiom})$

$\mbox{between}(u,v,w) \ \Rightarrow \ ( u = v\mbox{ or }w = \mbox{extension}(u,v,v,w) )\qquad\mbox{cnf}(\mbox{d10}_1,\mbox{axiom})$

$\mbox{equidistant}(w,x,y,z) \ \Rightarrow \ ( \mbox{extension}(u,v,w,x) = \mbox{extension}(u,v,y,z)\mbox{ or }u = v )\qquad\mbox{cnf}(\mbox{d10}_2,\mbox{axiom})$

$\mbox{extension}(u,v,u,v) = \mbox{extension}(u,v,v,u)\mbox{ or }u = v\qquad\mbox{cnf}(\mbox{d10}_3,\mbox{axiom})$

$\neg\,\mbox{equidistant}(v,u,v,\mbox{reflection}(\mbox{reflection}(u,v),v))\qquad\mbox{cnf}(\mbox{prove\_congruence},\mbox{negated\_conjecture})$

\medskip\textbf{GEO108+1.p} Equivalence of betweenness definitions 1 and 3






$\mbox{include}(\mbox{'Axioms/GEO004+0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO004+1.ax'})$

$\forall c,p,q,r{:}\;( \mbox{between\_c}_2(c,p,q,r)\iff ( p\ne q\mbox{ and }p\ne r\mbox{ and }q\ne r\mbox{ and }\exists c_1,c_2{:}\;( q\wedge c_1{=}c_2\mbox{ and }\mbox{part\_of}(c_1,c)\mbox{ and }\mbox{part\_of}(c_2,c)\mbox{ and }\mbox{end\_point}(p,c_1)\mbox{ and }\mbox{end\_point}(r,c_2) ) ) )\qquad\mbox{fof}(\mbox{between\_c2\_defn},\mbox{hypothesis})$

$\forall c,p,q,r{:}\;( \mbox{between\_c}_3(c,p,q,r)\iff ( p\ne q\mbox{ and }p\ne r\mbox{ and }q\ne r\mbox{ and }\exists c_1,c_2{:}\;( q\wedge c_1{=}c_2\mbox{ and }c_1 + c_2 = c\mbox{ and }\mbox{incident\_c}(p,c_1)\mbox{ and }\mbox{incident\_c}(r,c_2) ) ) )\qquad\mbox{fof}(\mbox{between\_c3\_defn},\mbox{hypothesis})$

$\forall c,p,q,r{:}\;( \mbox{between\_c}_2(c,p,q,r)\iff \mbox{between\_c}_3(c,p,q,r) )\qquad\mbox{fof}(\mbox{theorem\_3}_5,\mbox{conjecture})$

\medskip\textbf{GEO081+1.p} Transitivity of part\_of






$\mbox{include}(\mbox{'Axioms/GEO004+0.ax'})$

$\forall c_1,c_2,c_3{:}\;( ( \mbox{part\_of}(c_1,c_2)\mbox{ and }\mbox{part\_of}(c_2,c_3) ) \ \Rightarrow \ \mbox{part\_of}(c_1,c_3) )\qquad\mbox{fof}(\mbox{part\_of\_transitivity},\mbox{conjecture})$

\medskip\textbf{GEO094-1.p} Meeting point is not an endpoint of contianing curve

A meeting point of two curves is not an endpoint of any curve
 that includes both as sub-curves.




$\mbox{include}(\mbox{'Axioms/GEO004-0.ax'})$

$\mbox{sk}_{17}\wedge \mbox{sk}_{15}{=}\mbox{sk}_{16}\qquad\mbox{cnf}(\mbox{proposition\_2\_14\_3}_{67},\mbox{negated\_conjecture})$

$\mbox{part\_of}(\mbox{sk}_{15},\mbox{sk}_{14})\qquad\mbox{cnf}(\mbox{proposition\_2\_14\_3}_{68},\mbox{negated\_conjecture})$

$\mbox{part\_of}(\mbox{sk}_{16},\mbox{sk}_{14})\qquad\mbox{cnf}(\mbox{proposition\_2\_14\_3}_{69},\mbox{negated\_conjecture})$

$\mbox{end\_point}(\mbox{sk}_{17},\mbox{sk}_{14})\qquad\mbox{cnf}(\mbox{proposition\_2\_14\_3}_{70},\mbox{negated\_conjecture})$

\medskip\textbf{GEO127+1.p} Incidence on oriented curves can be defined using precedence






$\mbox{include}(\mbox{'Axioms/GEO004+0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO004+1.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO004+2.ax'})$

$\forall o,p{:}\;( \mbox{incident\_o}(p,o)\iff \exists q{:}\;( \mbox{ordered\_by}(o,p,q)\mbox{ or }\mbox{ordered\_by}(o,q,p) ) )\qquad\mbox{fof}(\mbox{theorem\_4}_{12},\mbox{conjecture})$

\medskip\textbf{GEO177+2.p} Symmetry of apartness

If the points X and Y are distinct and U and V are distinct,
 and X or Y is apart from the line connecting U and V, then
 U or V are apart from the line connecting X and Y.




$\mbox{include}(\mbox{'Axioms/GEO008+0.ax'})$

$\forall x,y,u,v{:}\;( ( \mbox{distinct\_points}(x,y)\mbox{ and }\mbox{distinct\_points}(u,v) ) \ \Rightarrow \ ( ( \mbox{apart\_point\_and\_line}(x,\mbox{line\_connecting}(u,v))\mbox{ or }\mbox{apart\_point\_and\_line}(y,\mbox{line\_connecting}(u,v)) ) \ \Rightarrow \ ( \mbox{apart\_point\_and\_line}(u,\mbox{line\_connecting}(x,y))\mbox{ or }\mbox{apart\_point\_and\_line}(v,\mbox{line\_connecting}(x,y)) ) ) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO118-1.p} Precedence on oriented curves is asymmetric






$\mbox{include}(\mbox{'Axioms/GEO004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO004-2.ax'})$

$\mbox{ordered\_by}(\mbox{sk}_{25},\mbox{sk}_{26},\mbox{sk}_{27})\qquad\mbox{cnf}(\mbox{theorem\_4\_5}_{133},\mbox{negated\_conjecture})$

$\mbox{ordered\_by}(\mbox{sk}_{25},\mbox{sk}_{27},\mbox{sk}_{26})\qquad\mbox{cnf}(\mbox{theorem\_4\_5}_{134},\mbox{negated\_conjecture})$

\medskip\textbf{GEO111+1.p} Basic property of orderings on linear structures 1

If Q is between P and R wrt. c, then P, Q and R are incident
 with c and are pairwise distinct




$\mbox{include}(\mbox{'Axioms/GEO004+0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO004+1.ax'})$

$\forall c,p,q,r{:}\;( \mbox{between\_c}(c,p,q,r) \ \Rightarrow \ ( \mbox{incident\_c}(p,c)\mbox{ and }\mbox{incident\_c}(q,c)\mbox{ and }\mbox{incident\_c}(r,c)\mbox{ and }p\ne q\mbox{ and }q\ne r\mbox{ and }p\ne r ) )\qquad\mbox{fof}(\mbox{theorem\_3\_8}_1,\mbox{conjecture})$

\medskip\textbf{GEO208+3.p} Point on both parallel lines

If the point X is incident with both the lines Y and Z, and
 Y and Z are parallel, then Y and Z are equal.




$\mbox{include}(\mbox{'Axioms/GEO006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+1.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+2.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+3.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+4.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+5.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+6.ax'})$

$\forall x,y,z{:}\;( ( \mbox{incident\_point\_and\_line}(x,y)\mbox{ and }\mbox{incident\_point\_and\_line}(x,z)\mbox{ and }\mbox{parallel\_lines}(y,z) ) \ \Rightarrow \ \mbox{equal\_lines}(y,z) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO162-1.p} Hilbert geometry axioms, adapted to respect multi-sortedness






$\mbox{include}(\mbox{'Axioms/GEO005-0.ax'})$

\medskip\textbf{GEO245+3.p} Configurations in terms of apartness






$\mbox{include}(\mbox{'Axioms/GEO009+0.ax'})$

$\forall a,b,c{:}\;( \mbox{distinct\_points}(a,b) \ \Rightarrow \ ( \mbox{left\_apart\_point}(c,\mbox{line\_connecting}(a,b)) \ \Rightarrow \ \mbox{right\_apart\_point}(b,\mbox{line\_connecting}(a,c)) ) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO019-2.p} Corollary 3 to symmetries of equidistance

Show that if the distance from A to B equals the distance
 from C to D, then the distance from C to D equals the
 distance from B to A.




$\mbox{include}(\mbox{'Axioms/GEO002-0.ax'})$

$\mbox{equidistant}(u,v,w,x)\qquad\mbox{cnf}(\mbox{u\_to\_v\_equals\_w\_to\_x},\mbox{hypothesis})$

$\neg\,\mbox{equidistant}(w,x,v,u)\qquad\mbox{cnf}(\mbox{prove\_symmetry},\mbox{negated\_conjecture})$

\medskip\textbf{GEO215+1.p} Third line not orthogonal to two convergent lines

If two lines L and M are convergent, then a third line N is
 not orthogonal to L or M.




$\mbox{include}(\mbox{'Axioms/GEO006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+2.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+3.ax'})$

$\forall l,m,n{:}\;( \mbox{convergent\_lines}(l,m) \ \Rightarrow \ ( \mbox{unorthogonal\_lines}(l,n)\mbox{ or }\mbox{unorthogonal\_lines}(m,n) ) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO243+1.p} Configurations in terms of apartness






$\mbox{include}(\mbox{'Axioms/GEO007+0.ax'})$

$\forall a,b,c{:}\;( \mbox{distinct\_points}(a,b) \ \Rightarrow \ ( \mbox{left\_apart\_point}(c,\mbox{line\_connecting}(a,b)) \ \Rightarrow \ \mbox{left\_apart\_point}(c,\mbox{reverse\_line}(\mbox{line\_connecting}(b,a))) ) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO224+3.p} Find point incident to line

Assume orthogonal geometry. Given a point and a line, to find
 a point incident with the line.




$\mbox{include}(\mbox{'Axioms/GEO006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+1.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+2.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+3.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+4.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+5.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+6.ax'})$

$\forall x,y{:}\;( ( \mbox{point}(x)\mbox{ and }\mbox{line}(y) ) \ \Rightarrow \ \exists z{:}\;( \mbox{point}(z)\mbox{ and }\mbox{incident\_point\_and\_line}(z,y) ) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO013-1.p} Collinearity for 5 points

If z1, z2, and z3 are each collinear with distinct points
 x and y, then z1, z2, and z3 are collinear.




$\mbox{include}(\mbox{'Axioms/GEO001-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO001-1.ax'})$

$a\ne b\qquad\mbox{cnf}(\mbox{a\_not\_b},\mbox{hypothesis})$

$\mbox{colinear}(a,b,d_1)\qquad\mbox{cnf}(\mbox{and1\_colinear},\mbox{hypothesis})$

$\mbox{colinear}(a,b,d_2)\qquad\mbox{cnf}(\mbox{abd2\_colinear},\mbox{hypothesis})$

$\mbox{colinear}(a,b,d_3)\qquad\mbox{cnf}(\mbox{abd3\_colinear},\mbox{hypothesis})$

$\neg\,\mbox{colinear}(d_1,d_2,d_3)\qquad\mbox{cnf}(\mbox{prove\_d1d2d3\_colinear},\mbox{negated\_conjecture})$

\medskip\textbf{GEO224+1.p} Find point incident to line

Assume orthogonal geometry. Given a point and a line, to find
 a point incident with the line.




$\mbox{include}(\mbox{'Axioms/GEO006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+2.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+3.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+5.ax'})$

$\forall x,y{:}\;( ( \mbox{point}(x)\mbox{ and }\mbox{line}(y) ) \ \Rightarrow \ \exists z{:}\;( \mbox{point}(z)\mbox{ and }\neg\,\mbox{apt}(z,y) ) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO036-3.p} The 3 axiom set points are distinct






$\mbox{include}(\mbox{'Axioms/GEO002-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO002-2.ax'})$

$\mbox{equidistant}(u,v,u,v)\qquad\mbox{cnf}(d_1,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(w,x,u,v)\qquad\mbox{cnf}(d_2,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(v,u,w,x)\qquad\mbox{cnf}(d_3,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(u,v,x,w)\qquad\mbox{cnf}(\mbox{d4}_1,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(v,u,x,w)\qquad\mbox{cnf}(\mbox{d4}_2,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(w,x,v,u)\qquad\mbox{cnf}(\mbox{d4}_3,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(x,w,u,v)\qquad\mbox{cnf}(\mbox{d4}_4,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(x,w,v,u)\qquad\mbox{cnf}(\mbox{d4}_5,\mbox{axiom})$

$( \mbox{equidistant}(u,v,w,x)\mbox{ and }\mbox{equidistant}(w,x,y,z) ) \ \Rightarrow \ \mbox{equidistant}(u,v,y,z)\qquad\mbox{cnf}(d_5,\mbox{axiom})$

$v = \mbox{extension}(u,v,w,w)\qquad\mbox{cnf}(e_1,\mbox{axiom})$

$y = \mbox{extension}(u,v,w,x) \ \Rightarrow \ \mbox{between}(u,v,y)\qquad\mbox{cnf}(b_0,\mbox{axiom})$

$\mbox{between}(u,v,\mbox{reflection}(u,v))\qquad\mbox{cnf}(\mbox{r2}_1,\mbox{axiom})$

$\mbox{equidistant}(v,\mbox{reflection}(u,v),u,v)\qquad\mbox{cnf}(\mbox{r2}_2,\mbox{axiom})$

$u = v \ \Rightarrow \ v = \mbox{reflection}(u,v)\qquad\mbox{cnf}(\mbox{r3}_1,\mbox{axiom})$

$u = \mbox{reflection}(u,u)\qquad\mbox{cnf}(\mbox{r3}_2,\mbox{axiom})$

$v = \mbox{reflection}(u,v) \ \Rightarrow \ u = v\qquad\mbox{cnf}(r_4,\mbox{axiom})$

$\mbox{equidistant}(u,u,v,v)\qquad\mbox{cnf}(d_7,\mbox{axiom})$

$( \mbox{equidistant}(u,v,u_1,v_1)\mbox{ and }\mbox{equidistant}(v,w,v_1,w_1)\mbox{ and }\mbox{between}(u,v,w)\mbox{ and }\mbox{between}(u_1,v_1,w_1) ) \ \Rightarrow \ \mbox{equidistant}(u,w,u_1,w_1)\qquad\mbox{cnf}(d_8,\mbox{axiom})$

$( \mbox{between}(u,v,w)\mbox{ and }\mbox{between}(u,v,x)\mbox{ and }\mbox{equidistant}(v,w,v,x) ) \ \Rightarrow \ ( u = v\mbox{ or }w = x )\qquad\mbox{cnf}(d_9,\mbox{axiom})$

$\mbox{between}(u,v,w) \ \Rightarrow \ ( u = v\mbox{ or }w = \mbox{extension}(u,v,v,w) )\qquad\mbox{cnf}(\mbox{d10}_1,\mbox{axiom})$

$\mbox{equidistant}(w,x,y,z) \ \Rightarrow \ ( \mbox{extension}(u,v,w,x) = \mbox{extension}(u,v,y,z)\mbox{ or }u = v )\qquad\mbox{cnf}(\mbox{d10}_2,\mbox{axiom})$

$\mbox{extension}(u,v,u,v) = \mbox{extension}(u,v,v,u)\mbox{ or }u = v\qquad\mbox{cnf}(\mbox{d10}_3,\mbox{axiom})$

$\mbox{equidistant}(v,u,v,\mbox{reflection}(\mbox{reflection}(u,v),v))\qquad\mbox{cnf}(r_5,\mbox{axiom})$

$u = \mbox{reflection}(\mbox{reflection}(u,v),v)\qquad\mbox{cnf}(r_6,\mbox{axiom})$

$\mbox{between}(u,v,v)\qquad\mbox{cnf}(t_3,\mbox{axiom})$

$( \mbox{between}(u,w,x)\mbox{ and }u = x ) \ \Rightarrow \ \mbox{between}(v,w,x)\qquad\mbox{cnf}(b_1,\mbox{axiom})$

$\mbox{between}(u,v,w) \ \Rightarrow \ \mbox{between}(w,v,u)\qquad\mbox{cnf}(t_1,\mbox{axiom})$

$\mbox{between}(u,u,v)\qquad\mbox{cnf}(t_2,\mbox{axiom})$

$( \mbox{between}(u,v,w)\mbox{ and }\mbox{between}(v,u,w) ) \ \Rightarrow \ u = v\qquad\mbox{cnf}(b_2,\mbox{axiom})$

$( \mbox{between}(u,v,w)\mbox{ and }\mbox{between}(u,w,v) ) \ \Rightarrow \ v = w\qquad\mbox{cnf}(b_3,\mbox{axiom})$

$( \mbox{between}(u,v,w)\mbox{ and }\mbox{between}(v,u,w) ) \ \Rightarrow \ ( u = v\mbox{ or }v = w )\qquad\mbox{cnf}(\mbox{t6}_1,\mbox{axiom})$

$( \mbox{between}(u,v,w)\mbox{ and }\mbox{between}(u,w,v) ) \ \Rightarrow \ ( u = v\mbox{ or }v = w )\qquad\mbox{cnf}(\mbox{t6}_2,\mbox{axiom})$

$( \mbox{between}(u,v,w)\mbox{ and }\mbox{between}(v,w,x) ) \ \Rightarrow \ \mbox{between}(u,v,w)\qquad\mbox{cnf}(b_4,\mbox{axiom})$

$( \mbox{between}(u,v,w)\mbox{ and }\mbox{between}(u,w,x) ) \ \Rightarrow \ \mbox{between}(v,w,x)\qquad\mbox{cnf}(b_5,\mbox{axiom})$

$( \mbox{between}(u,v,w)\mbox{ and }\mbox{between}(v,w,x) ) \ \Rightarrow \ ( \mbox{between}(u,w,x)\mbox{ or }v = w )\qquad\mbox{cnf}(b_6,\mbox{axiom})$

$( \mbox{between}(u,v,w)\mbox{ and }\mbox{between}(v,w,x) ) \ \Rightarrow \ ( \mbox{between}(u,v,x)\mbox{ or }v = w )\qquad\mbox{cnf}(b_7,\mbox{axiom})$

$( \mbox{between}(u,v,x)\mbox{ and }\mbox{between}(v,w,x) ) \ \Rightarrow \ \mbox{between}(u,w,x)\qquad\mbox{cnf}(b_8,\mbox{axiom})$

$( \mbox{between}(u,v,w)\mbox{ and }\mbox{between}(u,w,x) ) \ \Rightarrow \ \mbox{between}(u,v,x)\qquad\mbox{cnf}(b_9,\mbox{axiom})$

$\mbox{lower\_dimension\_point}_1 = \mbox{lower\_dimension\_point}_2\mbox{ or }\mbox{lower\_dimension\_point}_2 = \mbox{lower\_dimension\_point}_3\mbox{ or }\mbox{lower\_dimension\_point}_1 = \mbox{lower\_dimension\_point}_3\qquad\mbox{cnf}(\mbox{prove\_axioms\_points\_are\_distinct},\mbox{negated\_conjecture})$

\medskip\textbf{GEO132-1.p} Betweenness and precedence property 1






$\mbox{include}(\mbox{'Axioms/GEO004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO004-2.ax'})$

$\mbox{ordered\_by}(\mbox{sk}_{25},\mbox{sk}_{26},\mbox{sk}_{27})\qquad\mbox{cnf}(\mbox{theorem\_4\_17\_1}_{133},\mbox{negated\_conjecture})$

$\mbox{sk}_{26}\ne \mbox{sk}_{28}\qquad\mbox{cnf}(\mbox{theorem\_4\_17\_1}_{134},\mbox{negated\_conjecture})$

$\mbox{incident\_o}(\mbox{sk}_{28},\mbox{sk}_{25})\qquad\mbox{cnf}(\mbox{theorem\_4\_17\_1}_{135},\mbox{negated\_conjecture})$

$\mbox{between}(\mbox{sk}_{25},\mbox{sk}_{28},\mbox{sk}_{26},\mbox{sk}_{27}) \ \Rightarrow \ \mbox{ordered\_by}(\mbox{sk}_{25},\mbox{sk}_{26},\mbox{sk}_{28})\qquad\mbox{cnf}(\mbox{theorem\_4\_17\_1}_{136},\mbox{negated\_conjecture})$

$\mbox{ordered\_by}(\mbox{sk}_{25},\mbox{sk}_{26},\mbox{sk}_{28}) \ \Rightarrow \ \mbox{ordered\_by}(\mbox{sk}_{25},\mbox{sk}_{26},\mbox{sk}_{28})\qquad\mbox{cnf}(\mbox{theorem\_4\_17\_1}_{137},\mbox{negated\_conjecture})$

$\mbox{between}(\mbox{sk}_{25},\mbox{sk}_{28},\mbox{sk}_{26},\mbox{sk}_{27}) \ \Rightarrow \ \mbox{between}(\mbox{sk}_{25},\mbox{sk}_{28},\mbox{sk}_{26},\mbox{sk}_{27})\qquad\mbox{cnf}(\mbox{theorem\_4\_17\_1}_{138},\mbox{negated\_conjecture})$

$\mbox{ordered\_by}(\mbox{sk}_{25},\mbox{sk}_{26},\mbox{sk}_{28}) \ \Rightarrow \ \mbox{between}(\mbox{sk}_{25},\mbox{sk}_{28},\mbox{sk}_{26},\mbox{sk}_{27})\qquad\mbox{cnf}(\mbox{theorem\_4\_17\_1}_{139},\mbox{negated\_conjecture})$

\medskip\textbf{GEO021-2.p} Corollary 5 to symmetries of equidistance

Show that if the distance from A to B equals the distance
 from C to D, then the distance from D to C equals the
 distance from B to A.




$\mbox{include}(\mbox{'Axioms/GEO002-0.ax'})$

$\mbox{equidistant}(u,v,w,x)\qquad\mbox{cnf}(\mbox{u\_to\_v\_equals\_w\_to\_x},\mbox{hypothesis})$

$\neg\,\mbox{equidistant}(x,w,v,u)\qquad\mbox{cnf}(\mbox{prove\_symmetry},\mbox{negated\_conjecture})$

\medskip\textbf{GEO202+1.p} Diverging lines have equal ends

If the point X is distinct to the points Y and Z, and the lines
 connecting X and Y, and connecting X and Z are convergent, then
 the intersection point of these lines is equal to X.




$\mbox{include}(\mbox{'Axioms/GEO006+0.ax'})$

$\forall x,y,z{:}\;( ( \mbox{distinct\_points}(x,y)\mbox{ and }\mbox{distinct\_points}(x,z)\mbox{ and }\mbox{convergent\_lines}(\mbox{line\_connecting}(x,y),\mbox{line\_connecting}(x,z)) ) \ \Rightarrow \ \neg\,\mbox{distinct\_points}(\mbox{intersection\_point}(\mbox{line\_connecting}(x,y),\mbox{line\_connecting}(x,z)),x) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO141+1.p} Unique oppositely oriented curve 2

For every oriented line there is exactly one uniquely determined
 oriented line with the same underlying curve that orders the
 points in the opposite way.




$\mbox{include}(\mbox{'Axioms/GEO004+0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO004+1.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO004+2.ax'})$

$\forall o,p,q,r{:}\;( \mbox{ordered\_by}(o,p,q) \ \Rightarrow \ ( \mbox{ordered\_by}(o,r,q)\iff ( \mbox{between}(o,r,p,q)\mbox{ or }\mbox{between}(o,p,r,q)\mbox{ or }p = r ) ) )\qquad\mbox{fof}(\mbox{theorem\_4\_23}_2,\mbox{conjecture})$

\medskip\textbf{GEO138-1.p} Curve and ordered points determine oriented curve

A curve and a ordered pair of points uniquely determine an
 oriented curve.




$\mbox{include}(\mbox{'Axioms/GEO004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO004-2.ax'})$

$\mbox{underlying\_curve}(\mbox{sk}_{25}) = \mbox{underlying\_curve}(\mbox{sk}_{26})\qquad\mbox{cnf}(\mbox{corollary\_4\_21}_{133},\mbox{negated\_conjecture})$

$\mbox{ordered\_by}(\mbox{sk}_{25},\mbox{sk}_{27},\mbox{sk}_{28})\qquad\mbox{cnf}(\mbox{corollary\_4\_21}_{134},\mbox{negated\_conjecture})$

$\mbox{ordered\_by}(\mbox{sk}_{26},\mbox{sk}_{27},\mbox{sk}_{28})\qquad\mbox{cnf}(\mbox{corollary\_4\_21}_{135},\mbox{negated\_conjecture})$

$\mbox{sk}_{25}\ne \mbox{sk}_{26}\qquad\mbox{cnf}(\mbox{corollary\_4\_21}_{136},\mbox{negated\_conjecture})$

\medskip\textbf{GEO014-2.p} Ordinary reflexivity of equidistance

This shows that the distance from A to B is the same as the
 distance from A to B. This is different from the axiom which
 states that the distance from A to B is the same as the
 distance from B to A.




$\mbox{include}(\mbox{'Axioms/GEO002-0.ax'})$

$\neg\,\mbox{equidistant}(u,v,u,v)\qquad\mbox{cnf}(\mbox{prove\_reflexivity},\mbox{negated\_conjecture})$

\medskip\textbf{GEO236+3.p} Left and right apart leads to distinctness of lines






$\mbox{include}(\mbox{'Axioms/GEO009+0.ax'})$

$\forall a,l,m{:}\;( ( \mbox{left\_apart\_point}(a,l)\mbox{ and }\mbox{right\_apart\_point}(a,m) ) \ \Rightarrow \ ( \mbox{unequally\_directed\_lines}(l,m)\mbox{ or }\mbox{distinct\_lines}(l,m) ) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO137+1.p} Identical oriented lines

Oriented lines consisting of the same points and ordering one
 pair of points in the same way, are identical.




$\mbox{include}(\mbox{'Axioms/GEO004+0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO004+1.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO004+2.ax'})$

$\forall o_1,o_2{:}\;( ( \forall p{:}\;( \mbox{incident\_o}(p,o_1)\iff \mbox{incident\_o}(p,o_2) )\mbox{ and }\exists p,q{:}\;( \mbox{ordered\_by}(o_1,p,q)\mbox{ and }\mbox{ordered\_by}(o_2,p,q) ) ) \ \Rightarrow \ o_1 = o_2 )\qquad\mbox{fof}(\mbox{theorem\_4}_{20},\mbox{conjecture})$

\medskip\textbf{GEO193+1.p} Corollary to symmetry of apartness

If the lines X, Y, and Z are pairwise convergent, and the
 intersection point of X and Y is apart from a line Z, then
 the intersection point of Y and Z is apart from X.




$\mbox{include}(\mbox{'Axioms/GEO006+0.ax'})$

$\forall x,y,z{:}\;( ( \mbox{convergent\_lines}(x,y)\mbox{ and }\mbox{convergent\_lines}(z,y)\mbox{ and }\mbox{convergent\_lines}(x,z) ) \ \Rightarrow \ ( \mbox{apart\_point\_and\_line}(\mbox{intersection\_point}(x,y),z) \ \Rightarrow \ \mbox{apart\_point\_and\_line}(\mbox{intersection\_point}(z,y),x) ) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO142-1.p} Unique oppositely oriented curve 3

For every oriented line there is exactly one uniquely determined
 oriented line with the same underlying curve that orders the
 points in the opposite way.




$\mbox{include}(\mbox{'Axioms/GEO004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO004-2.ax'})$

$\mbox{ordered\_by}(\mbox{sk}_{25},\mbox{sk}_{26},\mbox{sk}_{27})\qquad\mbox{cnf}(\mbox{theorem\_4\_23\_3}_{133},\mbox{negated\_conjecture})$

$\mbox{ordered\_by}(\mbox{sk}_{25},\mbox{sk}_{26},\mbox{sk}_{28})\mbox{ or }\mbox{between}(\mbox{sk}_{25},\mbox{sk}_{26},\mbox{sk}_{28},\mbox{sk}_{27})\mbox{ or }\mbox{between}(\mbox{sk}_{25},\mbox{sk}_{26},\mbox{sk}_{27},\mbox{sk}_{28})\mbox{ or }\mbox{sk}_{27} = \mbox{sk}_{28}\qquad\mbox{cnf}(\mbox{theorem\_4\_23\_3}_{134},\mbox{negated\_conjecture})$

$\mbox{ordered\_by}(\mbox{sk}_{25},\mbox{sk}_{26},\mbox{sk}_{28}) \ \Rightarrow \ \mbox{ordered\_by}(\mbox{sk}_{25},\mbox{sk}_{26},\mbox{sk}_{28})\qquad\mbox{cnf}(\mbox{theorem\_4\_23\_3}_{135},\mbox{negated\_conjecture})$

$\mbox{between}(\mbox{sk}_{25},\mbox{sk}_{26},\mbox{sk}_{28},\mbox{sk}_{27}) \ \Rightarrow \ ( \mbox{between}(\mbox{sk}_{25},\mbox{sk}_{26},\mbox{sk}_{28},\mbox{sk}_{27})\mbox{ or }\mbox{between}(\mbox{sk}_{25},\mbox{sk}_{26},\mbox{sk}_{27},\mbox{sk}_{28})\mbox{ or }\mbox{sk}_{27} = \mbox{sk}_{28} )\qquad\mbox{cnf}(\mbox{theorem\_4\_23\_3}_{136},\mbox{negated\_conjecture})$

$\mbox{between}(\mbox{sk}_{25},\mbox{sk}_{26},\mbox{sk}_{27},\mbox{sk}_{28}) \ \Rightarrow \ ( \mbox{between}(\mbox{sk}_{25},\mbox{sk}_{26},\mbox{sk}_{28},\mbox{sk}_{27})\mbox{ or }\mbox{between}(\mbox{sk}_{25},\mbox{sk}_{26},\mbox{sk}_{27},\mbox{sk}_{28})\mbox{ or }\mbox{sk}_{27} = \mbox{sk}_{28} )\qquad\mbox{cnf}(\mbox{theorem\_4\_23\_3}_{137},\mbox{negated\_conjecture})$

$\mbox{sk}_{27} = \mbox{sk}_{28} \ \Rightarrow \ ( \mbox{between}(\mbox{sk}_{25},\mbox{sk}_{26},\mbox{sk}_{28},\mbox{sk}_{27})\mbox{ or }\mbox{between}(\mbox{sk}_{25},\mbox{sk}_{26},\mbox{sk}_{27},\mbox{sk}_{28})\mbox{ or }\mbox{sk}_{27} = \mbox{sk}_{28} )\qquad\mbox{cnf}(\mbox{theorem\_4\_23\_3}_{138},\mbox{negated\_conjecture})$

$\mbox{between}(\mbox{sk}_{25},\mbox{sk}_{26},\mbox{sk}_{28},\mbox{sk}_{27}) \ \Rightarrow \ \neg\,\mbox{ordered\_by}(\mbox{sk}_{25},\mbox{sk}_{26},\mbox{sk}_{28})\qquad\mbox{cnf}(\mbox{theorem\_4\_23\_3}_{139},\mbox{negated\_conjecture})$

$\mbox{between}(\mbox{sk}_{25},\mbox{sk}_{26},\mbox{sk}_{27},\mbox{sk}_{28}) \ \Rightarrow \ \neg\,\mbox{ordered\_by}(\mbox{sk}_{25},\mbox{sk}_{26},\mbox{sk}_{28})\qquad\mbox{cnf}(\mbox{theorem\_4\_23\_3}_{140},\mbox{negated\_conjecture})$

$\mbox{sk}_{27} = \mbox{sk}_{28} \ \Rightarrow \ \neg\,\mbox{ordered\_by}(\mbox{sk}_{25},\mbox{sk}_{26},\mbox{sk}_{28})\qquad\mbox{cnf}(\mbox{theorem\_4\_23\_3}_{141},\mbox{negated\_conjecture})$

\medskip\textbf{GEO096+1.p} Endpoints of curves are endpoints of sum

If two curves meet, than the endpoints of the sum are exactly
 those endpoints of the two curves that are not meeting-points
 of them.




$\mbox{include}(\mbox{'Axioms/GEO004+0.ax'})$

$\forall c_1,c_2{:}\;( \exists p{:}\;p\wedge c_1{=}c_2 \ \Rightarrow \ \forall q{:}\;( \mbox{end\_point}(q,c_1 + c_2)\iff ( \neg\,q\wedge c_1{=}c_2\mbox{ and }( \mbox{end\_point}(q,c_1)\mbox{ or }\mbox{end\_point}(q,c_2) ) ) ) )\qquad\mbox{fof}(\mbox{proposition\_2\_14}_5,\mbox{conjecture})$

\medskip\textbf{GEO175+3.p} Lemma on symmetry and apartness

If two lines are convergent and a point is distinct from the
 intersection point then this point is apart from at least one of
 these lines.




$\mbox{include}(\mbox{'Axioms/GEO006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+1.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+2.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+3.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+4.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+5.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+6.ax'})$

$\forall x,y,u,v{:}\;( ( \mbox{distinct\_points}(x,y)\mbox{ and }\mbox{convergent\_lines}(u,v)\mbox{ and }\mbox{distinct\_points}(x,\mbox{intersection\_point}(u,v)) ) \ \Rightarrow \ ( \mbox{apart\_point\_and\_line}(x,u)\mbox{ or }\mbox{apart\_point\_and\_line}(x,v) ) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO119+1.p} Oriented curve starting point is endpoint of underlying curve






$\mbox{include}(\mbox{'Axioms/GEO004+0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO004+1.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO004+2.ax'})$

$\forall o,p{:}\;( \mbox{start\_point}(p,o) \ \Rightarrow \ \mbox{end\_point}(p,\mbox{underlying\_curve}(o)) )\qquad\mbox{fof}(\mbox{theorem\_4\_6}_1,\mbox{conjecture})$

\medskip\textbf{GEO074-2.p} Prove the Outer Pasch Axiom






$\mbox{include}(\mbox{'Axioms/GEO002-0.ax'})$

$\mbox{between}(u,w,x)\qquad\mbox{cnf}(\mbox{w\_between\_u\_and\_x},\mbox{hypothesis})$

$\mbox{between}(v,x,y)\qquad\mbox{cnf}(\mbox{x\_between\_v\_and\_y},\mbox{hypothesis})$

$\mbox{between}(u,\mbox{outer\_pasch}(u,v,x,y,w),v) \ \Rightarrow \ \neg\,\mbox{between}(y,w,\mbox{outer\_pasch}(u,v,x,y,w))\qquad\mbox{cnf}(\mbox{prove\_outer\_pasch},\mbox{negated\_conjecture})$

\medskip\textbf{GEO170+2.p} Uniqueness of constructed lines

If two distinct points are incident with a line, then this line
 is equivalent with the connecting line of these points.




$\mbox{include}(\mbox{'Axioms/GEO008+0.ax'})$

$\forall x,y,z{:}\;( ( \mbox{distinct\_points}(x,y)\mbox{ and }\neg\,\mbox{apart\_point\_and\_line}(x,z)\mbox{ and }\neg\,\mbox{apart\_point\_and\_line}(y,z) ) \ \Rightarrow \ \neg\,\mbox{distinct\_lines}(z,\mbox{line\_connecting}(x,y)) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO159+1.p} Betweenness for simple curves






$\mbox{include}(\mbox{'Axioms/GEO004+0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO004+1.ax'})$

\medskip\textbf{GEO209+1.p} Pont and three parallel lines

If the point A is apart from the line L, but incident with the
 lines M and N, and L is parallel to M and N, then M and N are
 equal.




$\mbox{include}(\mbox{'Axioms/GEO006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+2.ax'})$

$\forall a,l,m,n{:}\;( ( \mbox{apart\_point\_and\_line}(a,l)\mbox{ and }\neg\,\mbox{apart\_point\_and\_line}(a,m)\mbox{ and }\neg\,\mbox{apart\_point\_and\_line}(a,n)\mbox{ and }\neg\,\mbox{convergent\_lines}(m,l)\mbox{ and }\neg\,\mbox{convergent\_lines}(n,l) ) \ \Rightarrow \ \neg\,\mbox{distinct\_lines}(m,n) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO173+2.p} Lemma on symmetry and apartness

If two points are distinct and a line U is distinct from the
 line connecting the points, then U is apart from at least one
 of these points.




$\mbox{include}(\mbox{'Axioms/GEO008+0.ax'})$

$\forall x,y,u,v{:}\;( ( \mbox{distinct\_points}(x,y)\mbox{ and }\mbox{convergent\_lines}(u,v)\mbox{ and }\mbox{distinct\_lines}(u,\mbox{line\_connecting}(x,y)) ) \ \Rightarrow \ ( \mbox{apart\_point\_and\_line}(x,u)\mbox{ or }\mbox{apart\_point\_and\_line}(y,u) ) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO235+1.p} Left and right apart leads to distinctness of points






$\mbox{include}(\mbox{'Axioms/GEO007+0.ax'})$

$\forall a,b,l{:}\;( ( \mbox{left\_apart\_point}(a,l)\mbox{ and }\mbox{left\_apart\_point}(b,\mbox{reverse\_line}(l)) ) \ \Rightarrow \ \mbox{distinct\_points}(a,b) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO259+3.p} Betweeness






$\mbox{include}(\mbox{'Axioms/GEO009+0.ax'})$

$\forall l,a,b,c{:}\;\neg\,\mbox{between\_on\_line}(l,a,b,c)\mbox{ and }\mbox{between\_on\_line}(l,b,a,c)\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO199+3.p} Corollary to symmetry of incidence

If the lines X, Y, and Z are pairwise convergent, and the
 intersection point of X and Y is incident with Z, then the
 intersection point of Y and X is incident with Z.




$\mbox{include}(\mbox{'Axioms/GEO006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+1.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+2.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+3.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+4.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+5.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+6.ax'})$

$\forall x,y,z{:}\;( ( \mbox{convergent\_lines}(x,y)\mbox{ and }\mbox{convergent\_lines}(z,y)\mbox{ and }\mbox{convergent\_lines}(x,z)\mbox{ and }\mbox{incident\_point\_and\_line}(\mbox{intersection\_point}(x,y),z) ) \ \Rightarrow \ \mbox{incident\_point\_and\_line}(\mbox{intersection\_point}(y,x),z) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO078-7.p} Every plane contains 3 noncollinear points






$\mbox{include}(\mbox{'Axioms/GEO005-0.ax'})$

$( \mbox{point}(x_1)\mbox{ and }\mbox{point}(x_2)\mbox{ and }\mbox{point}(x_3)\mbox{ and }\mbox{point\_on\_line}(x_1,y_1)\mbox{ and }\mbox{point\_on\_line}(x_2,y_1)\mbox{ and }\mbox{line}(y_1)\mbox{ and }\mbox{collinear}(x_1,x_2,x_3) ) \ \Rightarrow \ ( x_1 = x_2\mbox{ or }x_1 = x_3\mbox{ or }x_2 = x_3\mbox{ or }\mbox{point\_on\_line}(x_3,y_1) )\qquad\mbox{cnf}(\mbox{points\_not\_collinear},\mbox{axiom})$

$\mbox{plane}(\mbox{a\_plane})\qquad\mbox{cnf}(\mbox{there\_is\_a\_plane},\mbox{hypothesis})$

$( \mbox{point}(x_1)\mbox{ and }\mbox{point}(x_2)\mbox{ and }\mbox{point}(x_3)\mbox{ and }\mbox{point\_on\_plane}(x_1,\mbox{a\_plane})\mbox{ and }\mbox{point\_on\_plane}(x_2,\mbox{a\_plane})\mbox{ and }\mbox{point\_on\_plane}(x_3,\mbox{a\_plane}) ) \ \Rightarrow \ ( \mbox{collinear}(x_1,x_2,x_3)\mbox{ or }x_1 = x_2\mbox{ or }x_1 = x_3\mbox{ or }x_2 = x_3 )\qquad\mbox{cnf}(\mbox{prove\_every\_plane\_contains\_3\_noncollinear\_points},\mbox{negated\_conjecture})$

\medskip\textbf{GEO226+1.p} Existence of point incident to line

Assume orthogonal geometry. Given a point and a line, to find
 a point incident with the line.




$\mbox{include}(\mbox{'Axioms/GEO006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+5.ax'})$

$\forall l,m{:}\;( ( \mbox{line}(l)\mbox{ and }\mbox{line}(m)\mbox{ and }\mbox{convergent\_lines}(l,m) ) \ \Rightarrow \ \exists x{:}\;( \mbox{point}(x) \ \Rightarrow \ ( \neg\,\mbox{apart\_point\_and\_line}(x,l)\mbox{ and }\neg\,\mbox{apart\_point\_and\_line}(x,m) ) ) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO171+3.p} Two convergent lines are distinct






$\mbox{include}(\mbox{'Axioms/GEO006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+1.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+2.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+3.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+4.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+5.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+6.ax'})$

$\forall x,y{:}\;( \mbox{convergent\_lines}(x,y) \ \Rightarrow \ \mbox{distinct\_lines}(x,y) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO107-1.p} Equivalence of betweenness definitions 1 and 2






$\mbox{include}(\mbox{'Axioms/GEO004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO004-1.ax'})$

$\mbox{between\_c}_2(a,b,c,d) \ \Rightarrow \ b\ne c\qquad\mbox{cnf}(\mbox{between\_c2\_defn}_{81},\mbox{hypothesis})$

$\mbox{between\_c}_2(a,b,c,d) \ \Rightarrow \ b\ne d\qquad\mbox{cnf}(\mbox{between\_c2\_defn}_{82},\mbox{hypothesis})$

$\mbox{between\_c}_2(a,b,c,d) \ \Rightarrow \ c\ne d\qquad\mbox{cnf}(\mbox{between\_c2\_defn}_{83},\mbox{hypothesis})$

$\mbox{between\_c}_2(a,b,c,d) \ \Rightarrow \ c\wedge \mbox{sk}_{15}(d,c,b,a){=}\mbox{sk}_{16}(d,c,b,a)\qquad\mbox{cnf}(\mbox{between\_c2\_defn}_{84},\mbox{hypothesis})$

$\mbox{between\_c}_2(a,b,c,d) \ \Rightarrow \ \mbox{part\_of}(\mbox{sk}_{15}(d,c,b,a),a)\qquad\mbox{cnf}(\mbox{between\_c2\_defn}_{85},\mbox{hypothesis})$

$\mbox{between\_c}_2(a,b,c,d) \ \Rightarrow \ \mbox{part\_of}(\mbox{sk}_{16}(d,c,b,a),a)\qquad\mbox{cnf}(\mbox{between\_c2\_defn}_{86},\mbox{hypothesis})$

$\mbox{between\_c}_2(a,b,c,d) \ \Rightarrow \ \mbox{end\_point}(b,\mbox{sk}_{15}(d,c,b,a))\qquad\mbox{cnf}(\mbox{between\_c2\_defn}_{87},\mbox{hypothesis})$

$\mbox{between\_c}_2(a,b,c,d) \ \Rightarrow \ \mbox{end\_point}(d,\mbox{sk}_{16}(d,c,b,a))\qquad\mbox{cnf}(\mbox{between\_c2\_defn}_{88},\mbox{hypothesis})$

$( b\wedge d{=}e\mbox{ and }\mbox{part\_of}(d,f)\mbox{ and }\mbox{part\_of}(e,f)\mbox{ and }\mbox{end\_point}(a,d)\mbox{ and }\mbox{end\_point}(c,e) ) \ \Rightarrow \ ( a = b\mbox{ or }a = c\mbox{ or }b = c\mbox{ or }\mbox{between\_c}_2(f,a,b,c) )\qquad\mbox{cnf}(\mbox{between\_c2\_defn}_{89},\mbox{hypothesis})$

$\mbox{between\_c}(\mbox{sk}_{17},\mbox{sk}_{18},\mbox{sk}_{19},\mbox{sk}_{20})\mbox{ or }\mbox{between\_c}_2(\mbox{sk}_{17},\mbox{sk}_{18},\mbox{sk}_{19},\mbox{sk}_{20})\qquad\mbox{cnf}(\mbox{theorem\_3\_3}_{90},\mbox{negated\_conjecture})$

$\mbox{between\_c}(\mbox{sk}_{17},\mbox{sk}_{18},\mbox{sk}_{19},\mbox{sk}_{20}) \ \Rightarrow \ \mbox{between\_c}(\mbox{sk}_{17},\mbox{sk}_{18},\mbox{sk}_{19},\mbox{sk}_{20})\qquad\mbox{cnf}(\mbox{theorem\_3\_3}_{91},\mbox{negated\_conjecture})$

$\mbox{between\_c}_2(\mbox{sk}_{17},\mbox{sk}_{18},\mbox{sk}_{19},\mbox{sk}_{20}) \ \Rightarrow \ \mbox{between\_c}_2(\mbox{sk}_{17},\mbox{sk}_{18},\mbox{sk}_{19},\mbox{sk}_{20})\qquad\mbox{cnf}(\mbox{theorem\_3\_3}_{92},\mbox{negated\_conjecture})$

$\mbox{between\_c}_2(\mbox{sk}_{17},\mbox{sk}_{18},\mbox{sk}_{19},\mbox{sk}_{20}) \ \Rightarrow \ \neg\,\mbox{between\_c}(\mbox{sk}_{17},\mbox{sk}_{18},\mbox{sk}_{19},\mbox{sk}_{20})\qquad\mbox{cnf}(\mbox{theorem\_3\_3}_{93},\mbox{negated\_conjecture})$

\medskip\textbf{GEO161+1.p} Trajectories






$\mbox{include}(\mbox{'Axioms/GEO004+0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO004+1.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO004+2.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO004+3.ax'})$

\medskip\textbf{GEO108-1.p} Equivalence of betweenness definitions 1 and 3






$\mbox{include}(\mbox{'Axioms/GEO004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO004-1.ax'})$

$\mbox{between\_c}_2(a,b,c,d) \ \Rightarrow \ b\ne c\qquad\mbox{cnf}(\mbox{between\_c2\_defn}_{85},\mbox{hypothesis})$

$\mbox{between\_c}_2(a,b,c,d) \ \Rightarrow \ b\ne d\qquad\mbox{cnf}(\mbox{between\_c2\_defn}_{86},\mbox{hypothesis})$

$\mbox{between\_c}_2(a,b,c,d) \ \Rightarrow \ c\ne d\qquad\mbox{cnf}(\mbox{between\_c2\_defn}_{87},\mbox{hypothesis})$

$\mbox{between\_c}_2(a,b,c,d) \ \Rightarrow \ c\wedge \mbox{sk}_{15}(d,c,b,a){=}\mbox{sk}_{16}(d,c,b,a)\qquad\mbox{cnf}(\mbox{between\_c2\_defn}_{88},\mbox{hypothesis})$

$\mbox{between\_c}_2(a,b,c,d) \ \Rightarrow \ \mbox{part\_of}(\mbox{sk}_{15}(d,c,b,a),a)\qquad\mbox{cnf}(\mbox{between\_c2\_defn}_{89},\mbox{hypothesis})$

$\mbox{between\_c}_2(a,b,c,d) \ \Rightarrow \ \mbox{part\_of}(\mbox{sk}_{16}(d,c,b,a),a)\qquad\mbox{cnf}(\mbox{between\_c2\_defn}_{90},\mbox{hypothesis})$

$\mbox{between\_c}_2(a,b,c,d) \ \Rightarrow \ \mbox{end\_point}(b,\mbox{sk}_{15}(d,c,b,a))\qquad\mbox{cnf}(\mbox{between\_c2\_defn}_{91},\mbox{hypothesis})$

$\mbox{between\_c}_2(a,b,c,d) \ \Rightarrow \ \mbox{end\_point}(d,\mbox{sk}_{16}(d,c,b,a))\qquad\mbox{cnf}(\mbox{between\_c2\_defn}_{92},\mbox{hypothesis})$

$( b\wedge d{=}e\mbox{ and }\mbox{part\_of}(d,f)\mbox{ and }\mbox{part\_of}(e,f)\mbox{ and }\mbox{end\_point}(a,d)\mbox{ and }\mbox{end\_point}(c,e) ) \ \Rightarrow \ ( a = b\mbox{ or }a = c\mbox{ or }b = c\mbox{ or }\mbox{between\_c}_2(f,a,b,c) )\qquad\mbox{cnf}(\mbox{between\_c2\_defn}_{93},\mbox{hypothesis})$

$\mbox{between\_c}_3(a,b,c,d) \ \Rightarrow \ b\ne c\qquad\mbox{cnf}(\mbox{between\_c3\_defn}_{94},\mbox{hypothesis})$

$\mbox{between\_c}_3(a,b,c,d) \ \Rightarrow \ b\ne d\qquad\mbox{cnf}(\mbox{between\_c3\_defn}_{95},\mbox{hypothesis})$

$\mbox{between\_c}_3(a,b,c,d) \ \Rightarrow \ c\ne d\qquad\mbox{cnf}(\mbox{between\_c3\_defn}_{96},\mbox{hypothesis})$

$\mbox{between\_c}_3(a,b,c,d) \ \Rightarrow \ c\wedge \mbox{sk}_{17}(d,c,b,a){=}\mbox{sk}_{18}(d,c,b,a)\qquad\mbox{cnf}(\mbox{between\_c3\_defn}_{97},\mbox{hypothesis})$

$\mbox{between\_c}_3(a,b,c,d) \ \Rightarrow \ \mbox{sk}_{17}(d,c,b,a) + \mbox{sk}_{18}(d,c,b,a) = a\qquad\mbox{cnf}(\mbox{between\_c3\_defn}_{98},\mbox{hypothesis})$

$\mbox{between\_c}_3(a,b,c,d) \ \Rightarrow \ \mbox{incident\_c}(b,\mbox{sk}_{17}(d,c,b,a))\qquad\mbox{cnf}(\mbox{between\_c3\_defn}_{99},\mbox{hypothesis})$

$\mbox{between\_c}_3(a,b,c,d) \ \Rightarrow \ \mbox{incident\_c}(d,\mbox{sk}_{18}(d,c,b,a))\qquad\mbox{cnf}(\mbox{between\_c3\_defn}_{100},\mbox{hypothesis})$

$( b\wedge d{=}e\mbox{ and }d + e = f\mbox{ and }\mbox{incident\_c}(a,d)\mbox{ and }\mbox{incident\_c}(c,e) ) \ \Rightarrow \ ( a = b\mbox{ or }a = c\mbox{ or }b = c\mbox{ or }\mbox{between\_c}_3(f,a,b,c) )\qquad\mbox{cnf}(\mbox{between\_c3\_defn}_{101},\mbox{hypothesis})$

$\mbox{between\_c}_2(\mbox{sk}_{19},\mbox{sk}_{20},\mbox{sk}_{21},\mbox{sk}_{22})\mbox{ or }\mbox{between\_c}_3(\mbox{sk}_{19},\mbox{sk}_{20},\mbox{sk}_{21},\mbox{sk}_{22})\qquad\mbox{cnf}(\mbox{theorem\_3\_5}_{102},\mbox{negated\_conjecture})$

$\mbox{between\_c}_2(\mbox{sk}_{19},\mbox{sk}_{20},\mbox{sk}_{21},\mbox{sk}_{22}) \ \Rightarrow \ \mbox{between\_c}_2(\mbox{sk}_{19},\mbox{sk}_{20},\mbox{sk}_{21},\mbox{sk}_{22})\qquad\mbox{cnf}(\mbox{theorem\_3\_5}_{103},\mbox{negated\_conjecture})$

$\mbox{between\_c}_3(\mbox{sk}_{19},\mbox{sk}_{20},\mbox{sk}_{21},\mbox{sk}_{22}) \ \Rightarrow \ \mbox{between\_c}_3(\mbox{sk}_{19},\mbox{sk}_{20},\mbox{sk}_{21},\mbox{sk}_{22})\qquad\mbox{cnf}(\mbox{theorem\_3\_5}_{104},\mbox{negated\_conjecture})$

$\mbox{between\_c}_3(\mbox{sk}_{19},\mbox{sk}_{20},\mbox{sk}_{21},\mbox{sk}_{22}) \ \Rightarrow \ \neg\,\mbox{between\_c}_2(\mbox{sk}_{19},\mbox{sk}_{20},\mbox{sk}_{21},\mbox{sk}_{22})\qquad\mbox{cnf}(\mbox{theorem\_3\_5}_{105},\mbox{negated\_conjecture})$

\medskip\textbf{GEO184+1.p} Lemma on symmetry and apartness

If X and Y are distinct points, then the line that is incident
 with both points is equal to the line connecting them.




$\mbox{include}(\mbox{'Axioms/GEO006+0.ax'})$

$\forall x,y,u,v{:}\;( ( \mbox{distinct\_points}(x,y)\mbox{ and }\mbox{convergent\_lines}(u,v)\mbox{ and }\neg\,\mbox{apart\_point\_and\_line}(x,u)\mbox{ and }\neg\,\mbox{apart\_point\_and\_line}(y,u) ) \ \Rightarrow \ \neg\,\mbox{distinct\_lines}(u,\mbox{line\_connecting}(x,y)) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO186+1.p} Lemma on symmetry and apartness

If the lines U and V are convergent, then the point that is
 incident with both lines is equal to the intersection point
 of both lines.




$\mbox{include}(\mbox{'Axioms/GEO006+0.ax'})$

$\forall x,y,u,v{:}\;( ( \mbox{distinct\_points}(x,y)\mbox{ and }\mbox{convergent\_lines}(u,v)\mbox{ and }\neg\,\mbox{apart\_point\_and\_line}(x,u)\mbox{ and }\neg\,\mbox{apart\_point\_and\_line}(x,v) ) \ \Rightarrow \ \neg\,\mbox{distinct\_points}(x,\mbox{intersection\_point}(u,v)) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO105+1.p} If subcurves meet at an endpoint then there's a meeting

If two sub-curves of an open curve meet at a point and this
 point is an endpoint for another sub-curve then this sub-curve
 meets one of the former sub-curves at this point.




$\mbox{include}(\mbox{'Axioms/GEO004+0.ax'})$

$\forall c,c_1,c_2,c_3,p{:}\;( ( \mbox{part\_of}(c_1,c)\mbox{ and }\mbox{part\_of}(c_2,c)\mbox{ and }\mbox{part\_of}(c_3,c)\mbox{ and }p\wedge c_1{=}c_2\mbox{ and }\mbox{end\_point}(p,c_3)\mbox{ and }\mbox{open}(c) ) \ \Rightarrow \ ( p\wedge c_1{=}c_3\mbox{ or }p\wedge c_2{=}c_3 ) )\qquad\mbox{fof}(\mbox{proposition\_2}_{23},\mbox{conjecture})$

\medskip\textbf{GEO203+2.p} Equal lines from points

If the lines X and Y are convergent, and X and Z are convergent,
 the intersection point of X and Y, and the intersection point
 of X and Z are distinct, then the line connecting these points
 is equal to X.




$\mbox{include}(\mbox{'Axioms/GEO008+0.ax'})$

$\forall x,y,z{:}\;( ( \mbox{convergent\_lines}(x,y)\mbox{ and }\mbox{convergent\_lines}(x,z)\mbox{ and }\mbox{distinct\_points}(\mbox{intersection\_point}(x,y),\mbox{intersection\_point}(x,z)) ) \ \Rightarrow \ \neg\,\mbox{distinct\_lines}(\mbox{line\_connecting}(\mbox{intersection\_point}(x,y),\mbox{intersection\_point}(x,z)),x) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO155-1.p} Reflection axioms for the GEO002 geometry axioms






$\mbox{include}(\mbox{'Axioms/GEO002-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO002-2.ax'})$

\medskip\textbf{GEO192+3.p} Lemma on symmetry and apartness

If the lines X and Y are convergent, and the intersection
 point of X and Y is apart from a line Z, then Z is distinct
 from X and Y.




$\mbox{include}(\mbox{'Axioms/GEO006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+1.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+2.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+3.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+4.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+5.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+6.ax'})$

$\forall x,y,z{:}\;( \mbox{convergent\_lines}(x,y) \ \Rightarrow \ ( \mbox{apart\_point\_and\_line}(\mbox{intersection\_point}(x,y),z) \ \Rightarrow \ ( \mbox{distinct\_lines}(x,z)\mbox{ and }\mbox{distinct\_lines}(y,z) ) ) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO211+2.p} A line is not orthogonal to itself

A Line is not orthogonal to itself.




$\mbox{include}(\mbox{'Axioms/GEO008+0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+2.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+3.ax'})$

$\forall l{:}\;\mbox{unorthogonal\_lines}(l,l)\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO260+1.p} Betweeness






$\mbox{include}(\mbox{'Axioms/GEO007+0.ax'})$

$\forall l,a,b,c{:}\;\forall l,a,b,c{:}\;( ( \mbox{line}(l)\mbox{ and }\mbox{distinct\_points}(a,c)\mbox{ and }\mbox{distinct\_points}(b,c)\mbox{ and }\mbox{distinct\_points}(a,b) ) \ \Rightarrow \ \neg\,\mbox{between\_on\_line}(l,a,b,c)\mbox{ and }\mbox{between\_on\_line}(l,a,c,b) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO137-1.p} Identical oriented lines

Oriented lines consisting of the same points and ordering one
 pair of points in the same way, are identical.




$\mbox{include}(\mbox{'Axioms/GEO004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO004-2.ax'})$

$\mbox{incident\_o}(a,\mbox{sk}_{25}) \ \Rightarrow \ \mbox{incident\_o}(a,\mbox{sk}_{26})\qquad\mbox{cnf}(\mbox{theorem\_4\_20}_{133},\mbox{negated\_conjecture})$

$\mbox{incident\_o}(a,\mbox{sk}_{26}) \ \Rightarrow \ \mbox{incident\_o}(a,\mbox{sk}_{25})\qquad\mbox{cnf}(\mbox{theorem\_4\_20}_{134},\mbox{negated\_conjecture})$

$\mbox{ordered\_by}(\mbox{sk}_{25},\mbox{sk}_{27},\mbox{sk}_{28})\qquad\mbox{cnf}(\mbox{theorem\_4\_20}_{135},\mbox{negated\_conjecture})$

$\mbox{ordered\_by}(\mbox{sk}_{26},\mbox{sk}_{27},\mbox{sk}_{28})\qquad\mbox{cnf}(\mbox{theorem\_4\_20}_{136},\mbox{negated\_conjecture})$

$\mbox{sk}_{25}\ne \mbox{sk}_{26}\qquad\mbox{cnf}(\mbox{theorem\_4\_20}_{137},\mbox{negated\_conjecture})$

\medskip\textbf{GEO024-2.p} All null segments are congruent






$\mbox{include}(\mbox{'Axioms/GEO002-0.ax'})$

$\neg\,\mbox{equidistant}(u,u,v,v)\qquad\mbox{cnf}(\mbox{prove\_congruence},\mbox{negated\_conjecture})$

\medskip\textbf{GEO189+1.p} Collary to symmetry of incidence

If X, Y, and Z are pairwise distinct, and Z is incident with
 the line connecting X and Y, then Y is incident with the line
 connecting X and Z.




$\mbox{include}(\mbox{'Axioms/GEO006+0.ax'})$

$\forall x,y,z{:}\;( ( \mbox{distinct\_points}(x,y)\mbox{ and }\mbox{distinct\_points}(x,z)\mbox{ and }\mbox{distinct\_points}(y,z)\mbox{ and }\neg\,\mbox{apart\_point\_and\_line}(z,\mbox{line\_connecting}(x,y)) ) \ \Rightarrow \ \neg\,\mbox{apart\_point\_and\_line}(y,\mbox{line\_connecting}(x,z)) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO047-2.p} Corollary to second inner inner transitivity of betweenness

Corollary of second inner transitivity property of betweenness,
 using symmetry.




$\mbox{include}(\mbox{'Axioms/GEO002-0.ax'})$

$\mbox{between}(u,v,w)\qquad\mbox{cnf}(\mbox{v\_between\_u\_and\_w},\mbox{hypothesis})$

$\mbox{between}(u,w,x)\qquad\mbox{cnf}(\mbox{w\_between\_u\_and\_x},\mbox{hypothesis})$

$\neg\,\mbox{between}(u,v,x)\qquad\mbox{cnf}(\mbox{prove\_v\_between\_u\_and\_x},\mbox{negated\_conjecture})$

\medskip\textbf{GEO076-4.p} There is no point on every line






$\mbox{include}(\mbox{'Axioms/GEO003-0.ax'})$

$\mbox{point}(\mbox{a\_point})\qquad\mbox{cnf}(\mbox{there\_is\_a\_point},\mbox{hypothesis})$

$\mbox{line}(\mbox{line}) \ \Rightarrow \ \mbox{on}(\mbox{a\_point},\mbox{line})\qquad\mbox{cnf}(\mbox{prove\_point\_is\_not\_on\_every\_line},\mbox{negated\_conjecture})$

\medskip\textbf{GEO177+1.p} Symmetry of apartness

If the points X and Y are distinct and U and V are distinct,
 and X or Y is apart from the line connecting U and V, then
 U or V are apart from the line connecting X and Y.




$\mbox{include}(\mbox{'Axioms/GEO006+0.ax'})$

$\forall x,y,u,v{:}\;( ( \mbox{distinct\_points}(x,y)\mbox{ and }\mbox{distinct\_points}(u,v) ) \ \Rightarrow \ ( ( \mbox{apart\_point\_and\_line}(x,\mbox{line\_connecting}(u,v))\mbox{ or }\mbox{apart\_point\_and\_line}(y,\mbox{line\_connecting}(u,v)) ) \ \Rightarrow \ ( \mbox{apart\_point\_and\_line}(u,\mbox{line\_connecting}(x,y))\mbox{ or }\mbox{apart\_point\_and\_line}(v,\mbox{line\_connecting}(x,y)) ) ) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO181+3.p} Triangle axiom 2

If X and Y are distinct points and Z is apart from the line
 connecting X and Y, then Y is apart from the line connecting
 X and Z.




$\mbox{include}(\mbox{'Axioms/GEO006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+1.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+2.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+3.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+4.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+5.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+6.ax'})$

$\forall x,y,z{:}\;( \mbox{distinct\_points}(x,y) \ \Rightarrow \ ( \mbox{apart\_point\_and\_line}(z,\mbox{line\_connecting}(x,y)) \ \Rightarrow \ \mbox{apart\_point\_and\_line}(y,\mbox{line\_connecting}(x,z)) ) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO072-2.p} Corollary 2 to non-collinear points theorem






$\mbox{include}(\mbox{'Axioms/GEO002-0.ax'})$

$\mbox{equidistant}(u,v,w,x)\qquad\mbox{cnf}(\mbox{u\_to\_v\_equals\_w\_to\_x},\mbox{hypothesis})$

$\mbox{equidistant}(v,w,x,u)\qquad\mbox{cnf}(\mbox{v\_to\_w\_equals\_x\_to\_u},\mbox{hypothesis})$

$\mbox{equidistant}(u,w,v,x)\qquad\mbox{cnf}(\mbox{u\_to\_w\_equals\_v\_to\_x},\mbox{hypothesis})$

$\mbox{between}(u,y,w)\qquad\mbox{cnf}(\mbox{y\_between\_u\_and\_w},\mbox{hypothesis})$

$\mbox{between}(v,y,x)\qquad\mbox{cnf}(\mbox{y\_between\_v\_and\_x},\mbox{hypothesis})$

$u\ne v\qquad\mbox{cnf}(\mbox{u\_not\_v},\mbox{hypothesis})$

$x\ne u\qquad\mbox{cnf}(\mbox{x\_not\_u},\mbox{hypothesis})$

$v\ne x\qquad\mbox{cnf}(\mbox{prove\_v\_equals\_x},\mbox{negated\_conjecture})$

\medskip\textbf{GEO080-1.p} Reflexivity of part\_of






$\mbox{include}(\mbox{'Axioms/GEO004-0.ax'})$

$\neg\,\mbox{part\_of}(\mbox{sk}_{14},\mbox{sk}_{14})\qquad\mbox{cnf}(\mbox{prove\_reflexivity},\mbox{negated\_conjecture})$

\medskip\textbf{GEO195+2.p} Corollary to symmetry of apartness

If the lines X, Y, and Z are pairwise convergent, and the
 intersection point of X and Y is apart from a line Z, then
 the intersection point of Y and X is apart from Z.




$\mbox{include}(\mbox{'Axioms/GEO008+0.ax'})$

$\forall x,y,z{:}\;( ( \mbox{convergent\_lines}(x,y)\mbox{ and }\mbox{convergent\_lines}(z,y)\mbox{ and }\mbox{convergent\_lines}(x,z) ) \ \Rightarrow \ ( \mbox{apart\_point\_and\_line}(\mbox{intersection\_point}(x,y),z) \ \Rightarrow \ \mbox{apart\_point\_and\_line}(\mbox{intersection\_point}(y,x),z) ) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO173+3.p} Lemma on symmetry and apartness

If two points are distinct and a line U is distinct from the
 line connecting the points, then U is apart from at least one
 of these points.




$\mbox{include}(\mbox{'Axioms/GEO006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+1.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+2.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+3.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+4.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+5.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+6.ax'})$

$\forall x,y,u,v{:}\;( ( \mbox{distinct\_points}(x,y)\mbox{ and }\mbox{convergent\_lines}(u,v)\mbox{ and }\mbox{distinct\_lines}(u,\mbox{line\_connecting}(x,y)) ) \ \Rightarrow \ ( \mbox{apart\_point\_and\_line}(x,u)\mbox{ or }\mbox{apart\_point\_and\_line}(y,u) ) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO201+2.p} Distinct ends means distinct lines

If the lines X and Y are convergent, then the intersection
 point of X and Y is equal to the intersection point of X and Y.




$\mbox{include}(\mbox{'Axioms/GEO008+0.ax'})$

$\forall x,y{:}\;( \mbox{convergent\_lines}(x,y) \ \Rightarrow \ \neg\,\mbox{distinct\_points}(\mbox{intersection\_point}(x,y),\mbox{intersection\_point}(y,x)) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO029-3.p} Corollary 3 to unique extension






$\mbox{include}(\mbox{'Axioms/GEO002-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO002-2.ax'})$

$\mbox{equidistant}(u,v,u,v)\qquad\mbox{cnf}(d_1,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(w,x,u,v)\qquad\mbox{cnf}(d_2,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(v,u,w,x)\qquad\mbox{cnf}(d_3,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(u,v,x,w)\qquad\mbox{cnf}(\mbox{d4}_1,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(v,u,x,w)\qquad\mbox{cnf}(\mbox{d4}_2,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(w,x,v,u)\qquad\mbox{cnf}(\mbox{d4}_3,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(x,w,u,v)\qquad\mbox{cnf}(\mbox{d4}_4,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(x,w,v,u)\qquad\mbox{cnf}(\mbox{d4}_5,\mbox{axiom})$

$( \mbox{equidistant}(u,v,w,x)\mbox{ and }\mbox{equidistant}(w,x,y,z) ) \ \Rightarrow \ \mbox{equidistant}(u,v,y,z)\qquad\mbox{cnf}(d_5,\mbox{axiom})$

$v = \mbox{extension}(u,v,w,w)\qquad\mbox{cnf}(e_1,\mbox{axiom})$

$y = \mbox{extension}(u,v,w,x) \ \Rightarrow \ \mbox{between}(u,v,y)\qquad\mbox{cnf}(b_0,\mbox{axiom})$

$\mbox{between}(u,v,\mbox{reflection}(u,v))\qquad\mbox{cnf}(\mbox{r2}_1,\mbox{axiom})$

$\mbox{equidistant}(v,\mbox{reflection}(u,v),u,v)\qquad\mbox{cnf}(\mbox{r2}_2,\mbox{axiom})$

$u = v \ \Rightarrow \ v = \mbox{reflection}(u,v)\qquad\mbox{cnf}(\mbox{r3}_1,\mbox{axiom})$

$u = \mbox{reflection}(u,u)\qquad\mbox{cnf}(\mbox{r3}_2,\mbox{axiom})$

$v = \mbox{reflection}(u,v) \ \Rightarrow \ u = v\qquad\mbox{cnf}(r_4,\mbox{axiom})$

$\mbox{equidistant}(u,u,v,v)\qquad\mbox{cnf}(d_7,\mbox{axiom})$

$( \mbox{equidistant}(u,v,u_1,v_1)\mbox{ and }\mbox{equidistant}(v,w,v_1,w_1)\mbox{ and }\mbox{between}(u,v,w)\mbox{ and }\mbox{between}(u_1,v_1,w_1) ) \ \Rightarrow \ \mbox{equidistant}(u,w,u_1,w_1)\qquad\mbox{cnf}(d_8,\mbox{axiom})$

$( \mbox{between}(u,v,w)\mbox{ and }\mbox{between}(u,v,x)\mbox{ and }\mbox{equidistant}(v,w,v,x) ) \ \Rightarrow \ ( u = v\mbox{ or }w = x )\qquad\mbox{cnf}(d_9,\mbox{axiom})$

$u\ne v\qquad\mbox{cnf}(\mbox{u\_not\_v},\mbox{hypothesis})$

$\mbox{extension}(u,v,u,v)\ne \mbox{extension}(u,v,v,u)\qquad\mbox{cnf}(\mbox{prove\_equal\_extensions},\mbox{negated\_conjecture})$

\medskip\textbf{GEO170+3.p} Uniqueness of constructed lines

If two distinct points are incident with a line, then this line
 is equivalent with the connecting line of these points.




$\mbox{include}(\mbox{'Axioms/GEO006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+1.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+2.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+3.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+4.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+5.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+6.ax'})$

$\forall x,y,z{:}\;( ( \mbox{distinct\_points}(x,y)\mbox{ and }\mbox{incident\_point\_and\_line}(x,z)\mbox{ and }\mbox{incident\_point\_and\_line}(y,z) ) \ \Rightarrow \ \mbox{equal\_lines}(z,\mbox{line\_connecting}(x,y)) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO260+3.p} Betweeness






$\mbox{include}(\mbox{'Axioms/GEO009+0.ax'})$

$\forall l,a,b,c{:}\;\neg\,\mbox{between\_on\_line}(l,a,b,c)\mbox{ and }\mbox{between\_on\_line}(l,a,c,b)\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO188+3.p} Collary to symmetry of incidence

If X, Y, and Z are pairwise distinct, and Z is incident with
 the line connecting X and Y, then X is incident with the line
 connecting Z and Y.




$\mbox{include}(\mbox{'Axioms/GEO006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+1.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+2.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+3.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+4.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+5.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+6.ax'})$

$\forall x,y,z{:}\;( ( \mbox{distinct\_points}(x,y)\mbox{ and }\mbox{distinct\_points}(x,z)\mbox{ and }\mbox{distinct\_points}(y,z)\mbox{ and }\mbox{incident\_point\_and\_line}(z,\mbox{line\_connecting}(x,y)) ) \ \Rightarrow \ \mbox{incident\_point\_and\_line}(x,\mbox{line\_connecting}(z,y)) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO130-1.p} Betweenness and precedence for three points






$\mbox{include}(\mbox{'Axioms/GEO004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO004-2.ax'})$

$\mbox{between}(\mbox{sk}_{25},\mbox{sk}_{26},\mbox{sk}_{27},\mbox{sk}_{28})\qquad\mbox{cnf}(\mbox{theorem\_4\_15}_{133},\mbox{negated\_conjecture})$

$\mbox{ordered\_by}(\mbox{sk}_{25},\mbox{sk}_{26},\mbox{sk}_{27})\mbox{ or }\mbox{ordered\_by}(\mbox{sk}_{25},\mbox{sk}_{27},\mbox{sk}_{28})\qquad\mbox{cnf}(\mbox{theorem\_4\_15}_{134},\mbox{negated\_conjecture})$

$\mbox{ordered\_by}(\mbox{sk}_{25},\mbox{sk}_{26},\mbox{sk}_{27}) \ \Rightarrow \ \mbox{ordered\_by}(\mbox{sk}_{25},\mbox{sk}_{26},\mbox{sk}_{27})\qquad\mbox{cnf}(\mbox{theorem\_4\_15}_{135},\mbox{negated\_conjecture})$

$\mbox{ordered\_by}(\mbox{sk}_{25},\mbox{sk}_{27},\mbox{sk}_{28}) \ \Rightarrow \ \mbox{ordered\_by}(\mbox{sk}_{25},\mbox{sk}_{27},\mbox{sk}_{28})\qquad\mbox{cnf}(\mbox{theorem\_4\_15}_{136},\mbox{negated\_conjecture})$

$\mbox{ordered\_by}(\mbox{sk}_{25},\mbox{sk}_{27},\mbox{sk}_{28}) \ \Rightarrow \ \neg\,\mbox{ordered\_by}(\mbox{sk}_{25},\mbox{sk}_{26},\mbox{sk}_{27})\qquad\mbox{cnf}(\mbox{theorem\_4\_15}_{137},\mbox{negated\_conjecture})$

\medskip\textbf{GEO172+1.p} Uniqueness of constructed points

If two lines are convergent and there is a point that is
 incident with both lines, then this point is equivalent to the
 intersection point of these lines.




$\mbox{include}(\mbox{'Axioms/GEO006+0.ax'})$

$\forall x,y,z{:}\;( ( \mbox{convergent\_lines}(x,y)\mbox{ and }\neg\,\mbox{apart\_point\_and\_line}(z,x)\mbox{ and }\neg\,\mbox{apart\_point\_and\_line}(z,y) ) \ \Rightarrow \ \neg\,\mbox{distinct\_points}(z,\mbox{intersection\_point}(x,y)) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO144+1.p} Unique oppositely oriented curve 5

For every oriented line there is exactly one uniquely determined
 oriented line with the same underlying curve that orders the
 points in the opposite way.




$\mbox{include}(\mbox{'Axioms/GEO004+0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO004+1.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO004+2.ax'})$

$\forall o,p,q,r,s{:}\;( ( \mbox{ordered\_by}(o,p,q)\mbox{ and }p\ne r\mbox{ and }p\ne s\mbox{ and }q\ne s\mbox{ and }q\ne r ) \ \Rightarrow \ ( \mbox{ordered\_by}(o,r,s)\iff ( ( \mbox{between}(o,r,p,q)\mbox{ and }( \mbox{between}(o,r,s,q)\mbox{ or }\mbox{between}(o,r,q,s) ) )\mbox{ or }( \mbox{between}(o,p,r,s)\mbox{ and }( \mbox{between}(o,p,r,q)\mbox{ or }\mbox{between}(o,p,q,r) ) ) ) ) )\qquad\mbox{fof}(\mbox{theorem\_4\_23}_5,\mbox{conjecture})$

\medskip\textbf{GEO192+2.p} Lemma on symmetry and apartness

If the lines X and Y are convergent, and the intersection
 point of X and Y is apart from a line Z, then Z is distinct
 from X and Y.




$\mbox{include}(\mbox{'Axioms/GEO008+0.ax'})$

$\forall x,y,z{:}\;( \mbox{convergent\_lines}(x,y) \ \Rightarrow \ ( \mbox{apart\_point\_and\_line}(\mbox{intersection\_point}(x,y),z) \ \Rightarrow \ ( \mbox{distinct\_lines}(x,z)\mbox{ and }\mbox{distinct\_lines}(y,z) ) ) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO159-1.p} Betweenness for simple curves






$\mbox{include}(\mbox{'Axioms/GEO004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO004-1.ax'})$

\medskip\textbf{GEO212+2.p} Non-orthogonal lines and a third line

If a line L is not orthogonal to M, then a third line N is
 convergent to L or not orthogonal to M.




$\mbox{include}(\mbox{'Axioms/GEO008+0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+2.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+3.ax'})$

$\forall l,m,n{:}\;( \mbox{unorthogonal\_lines}(l,m) \ \Rightarrow \ ( \mbox{convergent\_lines}(l,n)\mbox{ or }\mbox{unorthogonal\_lines}(m,n) ) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO044-3.p} First outer transitivity property for betweenness






$\mbox{include}(\mbox{'Axioms/GEO002-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO002-2.ax'})$

$\mbox{equidistant}(u,v,u,v)\qquad\mbox{cnf}(d_1,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(w,x,u,v)\qquad\mbox{cnf}(d_2,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(v,u,w,x)\qquad\mbox{cnf}(d_3,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(u,v,x,w)\qquad\mbox{cnf}(\mbox{d4}_1,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(v,u,x,w)\qquad\mbox{cnf}(\mbox{d4}_2,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(w,x,v,u)\qquad\mbox{cnf}(\mbox{d4}_3,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(x,w,u,v)\qquad\mbox{cnf}(\mbox{d4}_4,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(x,w,v,u)\qquad\mbox{cnf}(\mbox{d4}_5,\mbox{axiom})$

$( \mbox{equidistant}(u,v,w,x)\mbox{ and }\mbox{equidistant}(w,x,y,z) ) \ \Rightarrow \ \mbox{equidistant}(u,v,y,z)\qquad\mbox{cnf}(d_5,\mbox{axiom})$

$v = \mbox{extension}(u,v,w,w)\qquad\mbox{cnf}(e_1,\mbox{axiom})$

$y = \mbox{extension}(u,v,w,x) \ \Rightarrow \ \mbox{between}(u,v,y)\qquad\mbox{cnf}(b_0,\mbox{axiom})$

$\mbox{between}(u,v,\mbox{reflection}(u,v))\qquad\mbox{cnf}(\mbox{r2}_1,\mbox{axiom})$

$\mbox{equidistant}(v,\mbox{reflection}(u,v),u,v)\qquad\mbox{cnf}(\mbox{r2}_2,\mbox{axiom})$

$u = v \ \Rightarrow \ v = \mbox{reflection}(u,v)\qquad\mbox{cnf}(\mbox{r3}_1,\mbox{axiom})$

$u = \mbox{reflection}(u,u)\qquad\mbox{cnf}(\mbox{r3}_2,\mbox{axiom})$

$v = \mbox{reflection}(u,v) \ \Rightarrow \ u = v\qquad\mbox{cnf}(r_4,\mbox{axiom})$

$\mbox{equidistant}(u,u,v,v)\qquad\mbox{cnf}(d_7,\mbox{axiom})$

$( \mbox{equidistant}(u,v,u_1,v_1)\mbox{ and }\mbox{equidistant}(v,w,v_1,w_1)\mbox{ and }\mbox{between}(u,v,w)\mbox{ and }\mbox{between}(u_1,v_1,w_1) ) \ \Rightarrow \ \mbox{equidistant}(u,w,u_1,w_1)\qquad\mbox{cnf}(d_8,\mbox{axiom})$

$( \mbox{between}(u,v,w)\mbox{ and }\mbox{between}(u,v,x)\mbox{ and }\mbox{equidistant}(v,w,v,x) ) \ \Rightarrow \ ( u = v\mbox{ or }w = x )\qquad\mbox{cnf}(d_9,\mbox{axiom})$

$\mbox{between}(u,v,w) \ \Rightarrow \ ( u = v\mbox{ or }w = \mbox{extension}(u,v,v,w) )\qquad\mbox{cnf}(\mbox{d10}_1,\mbox{axiom})$

$\mbox{equidistant}(w,x,y,z) \ \Rightarrow \ ( \mbox{extension}(u,v,w,x) = \mbox{extension}(u,v,y,z)\mbox{ or }u = v )\qquad\mbox{cnf}(\mbox{d10}_2,\mbox{axiom})$

$\mbox{extension}(u,v,u,v) = \mbox{extension}(u,v,v,u)\mbox{ or }u = v\qquad\mbox{cnf}(\mbox{d10}_3,\mbox{axiom})$

$\mbox{equidistant}(v,u,v,\mbox{reflection}(\mbox{reflection}(u,v),v))\qquad\mbox{cnf}(r_5,\mbox{axiom})$

$u = \mbox{reflection}(\mbox{reflection}(u,v),v)\qquad\mbox{cnf}(r_6,\mbox{axiom})$

$\mbox{between}(u,v,v)\qquad\mbox{cnf}(t_3,\mbox{axiom})$

$( \mbox{between}(u,w,x)\mbox{ and }u = x ) \ \Rightarrow \ \mbox{between}(v,w,x)\qquad\mbox{cnf}(b_1,\mbox{axiom})$

$\mbox{between}(u,v,w) \ \Rightarrow \ \mbox{between}(w,v,u)\qquad\mbox{cnf}(t_1,\mbox{axiom})$

$\mbox{between}(u,u,v)\qquad\mbox{cnf}(t_2,\mbox{axiom})$

$( \mbox{between}(u,v,w)\mbox{ and }\mbox{between}(v,u,w) ) \ \Rightarrow \ u = v\qquad\mbox{cnf}(b_2,\mbox{axiom})$

$( \mbox{between}(u,v,w)\mbox{ and }\mbox{between}(u,w,v) ) \ \Rightarrow \ v = w\qquad\mbox{cnf}(b_3,\mbox{axiom})$

$( \mbox{between}(u,v,w)\mbox{ and }\mbox{between}(v,u,w) ) \ \Rightarrow \ ( u = v\mbox{ or }v = w )\qquad\mbox{cnf}(\mbox{t6}_1,\mbox{axiom})$

$( \mbox{between}(u,v,w)\mbox{ and }\mbox{between}(u,w,v) ) \ \Rightarrow \ ( u = v\mbox{ or }v = w )\qquad\mbox{cnf}(\mbox{t6}_2,\mbox{axiom})$

$( \mbox{between}(u,v,w)\mbox{ and }\mbox{between}(v,w,x) ) \ \Rightarrow \ \mbox{between}(u,v,w)\qquad\mbox{cnf}(b_4,\mbox{axiom})$

$( \mbox{between}(u,v,w)\mbox{ and }\mbox{between}(u,w,x) ) \ \Rightarrow \ \mbox{between}(v,w,x)\qquad\mbox{cnf}(b_5,\mbox{axiom})$

$\mbox{between}(u,v,w)\qquad\mbox{cnf}(\mbox{v\_between\_u\_and\_w},\mbox{hypothesis})$

$\mbox{between}(v,w,x)\qquad\mbox{cnf}(\mbox{w\_between\_v\_and\_x},\mbox{hypothesis})$

$v\ne w\qquad\mbox{cnf}(\mbox{v\_not\_w},\mbox{hypothesis})$

$\neg\,\mbox{between}(u,w,x)\qquad\mbox{cnf}(\mbox{prove\_w\_between\_u\_and\_x},\mbox{negated\_conjecture})$

\medskip\textbf{GEO140+1.p} Unique oppositely oriented curve 1

For every oriented line there is exactly one uniquely determined
 oriented line with the same underlying curve that orders the
 points in the opposite way.




$\mbox{include}(\mbox{'Axioms/GEO004+0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO004+1.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO004+2.ax'})$

$\forall o,p,q,r{:}\;( \mbox{ordered\_by}(o,p,q) \ \Rightarrow \ ( \mbox{ordered\_by}(o,r,p)\iff \mbox{between}(o,r,p,q) ) )\qquad\mbox{fof}(\mbox{theorem\_4\_23}_1,\mbox{conjecture})$

\medskip\textbf{GEO218+1.p} Transitivity of parallel and orthogonal

If two lines L and M are parallel and a third line N is
 orthogonal to L, then M is orthogonal to N.




$\mbox{include}(\mbox{'Axioms/GEO006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+4.ax'})$

$\forall l,m,n{:}\;( ( \neg\,\mbox{convergent\_lines}(l,m)\mbox{ and }\neg\,\mbox{unorthogonal\_lines}(l,n) ) \ \Rightarrow \ \neg\,\mbox{unorthogonal\_lines}(m,n) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO206+3.p} Point on parallel lines

If the point X is incident with the line Y, and the lines
 Y and Z are parallel, then the line Y is equal to the parallel
 of Z through point X.




$\mbox{include}(\mbox{'Axioms/GEO006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+1.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+2.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+3.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+4.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+5.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+6.ax'})$

$\forall x,y,z{:}\;( ( \mbox{incident\_point\_and\_line}(x,y)\mbox{ and }\mbox{parallel\_lines}(y,z) ) \ \Rightarrow \ \mbox{equal\_lines}(y,\mbox{parallel\_through\_point}(z,x)) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO262+1.p} Lemma for parallel projection preserves or reverses order






$\mbox{include}(\mbox{'Axioms/GEO007+0.ax'})$

$\forall l,m,n,a,b,c{:}\;( ( \mbox{between\_on\_line}(l,a,b,c)\mbox{ and }\mbox{convergent\_lines}(l,m)\mbox{ and }\neg\,\mbox{apart\_point\_and\_line}(b,m)\mbox{ and }\mbox{convergent\_lines}(l,n)\mbox{ and }\mbox{convergent\_lines}(m,n)\mbox{ and }\neg\,\mbox{apart\_point\_and\_line}(b,n) ) \ \Rightarrow \ \mbox{between\_on\_line}(m,\mbox{intersection\_point}(m,\mbox{parallel\_through\_point}(n,a)),b,\mbox{intersection\_point}(m,\mbox{parallel\_through\_point}(n,c))) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO196+2.p} Symmetry of incidence

If the lines X and Y are convergent, U and V are convergent,
 and the intersection point of X and Y is incident with U and V,
 then the intersection point of U and V is incident with X and Y.




$\mbox{include}(\mbox{'Axioms/GEO008+0.ax'})$

$\forall x,y,u,v{:}\;( ( \mbox{convergent\_lines}(x,y)\mbox{ and }\mbox{convergent\_lines}(u,v)\mbox{ and }\neg\,\mbox{apart\_point\_and\_line}(\mbox{intersection\_point}(x,y),u)\mbox{ and }\neg\,\mbox{apart\_point\_and\_line}(\mbox{intersection\_point}(x,y),v) ) \ \Rightarrow \ ( \neg\,\mbox{apart\_point\_and\_line}(\mbox{intersection\_point}(u,v),x)\mbox{ and }\neg\,\mbox{apart\_point\_and\_line}(\mbox{intersection\_point}(u,v),y) ) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO200+1.p} Line equals its converse

If the points X and Y are distinct, then the line connecting
 X and Y is equal to the line connecting Y and X.




$\mbox{include}(\mbox{'Axioms/GEO006+0.ax'})$

$\forall x,y{:}\;( \mbox{distinct\_points}(x,y) \ \Rightarrow \ \neg\,\mbox{distinct\_lines}(\mbox{line\_connecting}(x,y),\mbox{line\_connecting}(y,x)) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO096-1.p} Endpoints of curves are endpoints of sum

If two curves meet, than the endpoints of the sum are exactly
 those endpoints of the two curves that are not meeting-points
 of them.




$\mbox{include}(\mbox{'Axioms/GEO004-0.ax'})$

$\mbox{sk}_{16}\wedge \mbox{sk}_{14}{=}\mbox{sk}_{15}\qquad\mbox{cnf}(\mbox{proposition\_2\_14\_5}_{67},\mbox{negated\_conjecture})$

$\mbox{sk}_{17}\wedge \mbox{sk}_{14}{=}\mbox{sk}_{15} \ \Rightarrow \ \mbox{end\_point}(\mbox{sk}_{17},\mbox{sk}_{14} + \mbox{sk}_{15})\qquad\mbox{cnf}(\mbox{proposition\_2\_14\_5}_{68},\mbox{negated\_conjecture})$

$\mbox{end\_point}(\mbox{sk}_{17},\mbox{sk}_{14} + \mbox{sk}_{15})\mbox{ or }\mbox{end\_point}(\mbox{sk}_{17},\mbox{sk}_{14})\mbox{ or }\mbox{end\_point}(\mbox{sk}_{17},\mbox{sk}_{15})\qquad\mbox{cnf}(\mbox{proposition\_2\_14\_5}_{69},\mbox{negated\_conjecture})$

$\mbox{end\_point}(\mbox{sk}_{17},\mbox{sk}_{14} + \mbox{sk}_{15}) \ \Rightarrow \ \mbox{end\_point}(\mbox{sk}_{17},\mbox{sk}_{14} + \mbox{sk}_{15})\qquad\mbox{cnf}(\mbox{proposition\_2\_14\_5}_{70},\mbox{negated\_conjecture})$

$( \mbox{end\_point}(\mbox{sk}_{17},\mbox{sk}_{14})\mbox{ and }\mbox{sk}_{17}\wedge \mbox{sk}_{14}{=}\mbox{sk}_{15} ) \ \Rightarrow \ \mbox{sk}_{17}\wedge \mbox{sk}_{14}{=}\mbox{sk}_{15}\qquad\mbox{cnf}(\mbox{proposition\_2\_14\_5}_{71},\mbox{negated\_conjecture})$

$\mbox{end\_point}(\mbox{sk}_{17},\mbox{sk}_{14}) \ \Rightarrow \ ( \mbox{sk}_{17}\wedge \mbox{sk}_{14}{=}\mbox{sk}_{15}\mbox{ or }\mbox{end\_point}(\mbox{sk}_{17},\mbox{sk}_{14})\mbox{ or }\mbox{end\_point}(\mbox{sk}_{17},\mbox{sk}_{15}) )\qquad\mbox{cnf}(\mbox{proposition\_2\_14\_5}_{72},\mbox{negated\_conjecture})$

$( \mbox{end\_point}(\mbox{sk}_{17},\mbox{sk}_{15})\mbox{ and }\mbox{sk}_{17}\wedge \mbox{sk}_{14}{=}\mbox{sk}_{15} ) \ \Rightarrow \ \mbox{sk}_{17}\wedge \mbox{sk}_{14}{=}\mbox{sk}_{15}\qquad\mbox{cnf}(\mbox{proposition\_2\_14\_5}_{73},\mbox{negated\_conjecture})$

$\mbox{end\_point}(\mbox{sk}_{17},\mbox{sk}_{15}) \ \Rightarrow \ ( \mbox{sk}_{17}\wedge \mbox{sk}_{14}{=}\mbox{sk}_{15}\mbox{ or }\mbox{end\_point}(\mbox{sk}_{17},\mbox{sk}_{14})\mbox{ or }\mbox{end\_point}(\mbox{sk}_{17},\mbox{sk}_{15}) )\qquad\mbox{cnf}(\mbox{proposition\_2\_14\_5}_{74},\mbox{negated\_conjecture})$

$( \mbox{end\_point}(\mbox{sk}_{17},\mbox{sk}_{14})\mbox{ and }\mbox{end\_point}(\mbox{sk}_{17},\mbox{sk}_{14} + \mbox{sk}_{15}) ) \ \Rightarrow \ \mbox{sk}_{17}\wedge \mbox{sk}_{14}{=}\mbox{sk}_{15}\qquad\mbox{cnf}(\mbox{proposition\_2\_14\_5}_{75},\mbox{negated\_conjecture})$

$( \mbox{end\_point}(\mbox{sk}_{17},\mbox{sk}_{15})\mbox{ and }\mbox{end\_point}(\mbox{sk}_{17},\mbox{sk}_{14} + \mbox{sk}_{15}) ) \ \Rightarrow \ \mbox{sk}_{17}\wedge \mbox{sk}_{14}{=}\mbox{sk}_{15}\qquad\mbox{cnf}(\mbox{proposition\_2\_14\_5}_{76},\mbox{negated\_conjecture})$

\medskip\textbf{GEO126-1.p} Every oriented curve orders some points






$\mbox{include}(\mbox{'Axioms/GEO004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO004-2.ax'})$

$\mbox{ordered\_by}(\mbox{sk}_{25},a,b) \ \Rightarrow \ a = b\qquad\mbox{cnf}(\mbox{theorem\_4\_11}_{133},\mbox{negated\_conjecture})$

\medskip\textbf{GEO218+2.p} Transitivity of parallel and orthogonal

If two lines L and M are parallel and a third line N is
 orthogonal to L, then M is orthogonal to N.




$\mbox{include}(\mbox{'Axioms/GEO008+0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+4.ax'})$

$\forall l,m,n{:}\;( ( \neg\,\mbox{convergent\_lines}(l,m)\mbox{ and }\neg\,\mbox{unorthogonal\_lines}(l,n) ) \ \Rightarrow \ \neg\,\mbox{unorthogonal\_lines}(m,n) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO127-1.p} Incidence on oriented curves can be defined using precedence






$\mbox{include}(\mbox{'Axioms/GEO004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO004-2.ax'})$

$\mbox{incident\_o}(\mbox{sk}_{26},\mbox{sk}_{25})\mbox{ or }\mbox{ordered\_by}(\mbox{sk}_{25},\mbox{sk}_{26},\mbox{sk}_{27})\mbox{ or }\mbox{ordered\_by}(\mbox{sk}_{25},\mbox{sk}_{27},\mbox{sk}_{26})\qquad\mbox{cnf}(\mbox{theorem\_4\_12}_{133},\mbox{negated\_conjecture})$

$\mbox{incident\_o}(\mbox{sk}_{26},\mbox{sk}_{25}) \ \Rightarrow \ \mbox{incident\_o}(\mbox{sk}_{26},\mbox{sk}_{25})\qquad\mbox{cnf}(\mbox{theorem\_4\_12}_{134},\mbox{negated\_conjecture})$

$\mbox{ordered\_by}(\mbox{sk}_{25},\mbox{sk}_{26},a) \ \Rightarrow \ ( \mbox{ordered\_by}(\mbox{sk}_{25},\mbox{sk}_{26},\mbox{sk}_{27})\mbox{ or }\mbox{ordered\_by}(\mbox{sk}_{25},\mbox{sk}_{27},\mbox{sk}_{26}) )\qquad\mbox{cnf}(\mbox{theorem\_4\_12}_{135},\mbox{negated\_conjecture})$

$\mbox{ordered\_by}(\mbox{sk}_{25},a,\mbox{sk}_{26}) \ \Rightarrow \ ( \mbox{ordered\_by}(\mbox{sk}_{25},\mbox{sk}_{26},\mbox{sk}_{27})\mbox{ or }\mbox{ordered\_by}(\mbox{sk}_{25},\mbox{sk}_{27},\mbox{sk}_{26}) )\qquad\mbox{cnf}(\mbox{theorem\_4\_12}_{136},\mbox{negated\_conjecture})$

$\mbox{ordered\_by}(\mbox{sk}_{25},\mbox{sk}_{26},a) \ \Rightarrow \ \neg\,\mbox{incident\_o}(\mbox{sk}_{26},\mbox{sk}_{25})\qquad\mbox{cnf}(\mbox{theorem\_4\_12}_{137},\mbox{negated\_conjecture})$

$\mbox{ordered\_by}(\mbox{sk}_{25},a,\mbox{sk}_{26}) \ \Rightarrow \ \neg\,\mbox{incident\_o}(\mbox{sk}_{26},\mbox{sk}_{25})\qquad\mbox{cnf}(\mbox{theorem\_4\_12}_{138},\mbox{negated\_conjecture})$

\medskip\textbf{GEO158+1.p} Simple curve axioms






$\mbox{include}(\mbox{'Axioms/GEO004+0.ax'})$

\medskip\textbf{GEO035-2.p} A null extension does not extend a line






$\mbox{include}(\mbox{'Axioms/GEO002-0.ax'})$

$v\ne \mbox{extension}(u,v,w,w)\qquad\mbox{cnf}(\mbox{prove\_null\_extension},\mbox{negated\_conjecture})$

\medskip\textbf{GEO161-1.p} Trajectories






$\mbox{include}(\mbox{'Axioms/GEO004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO004-2.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO004-3.ax'})$

\medskip\textbf{GEO187+2.p} Symmetry of incidence

If X and Y are distinct points, U and V are distinct points,
 X and Y are incident with the line connecting U and V, then
 U and V are incident with the line connecting X and Y.




$\mbox{include}(\mbox{'Axioms/GEO008+0.ax'})$

$\forall x,y,u,v{:}\;( ( \mbox{distinct\_points}(x,y)\mbox{ and }\mbox{distinct\_points}(u,v)\mbox{ and }\neg\,\mbox{apart\_point\_and\_line}(x,\mbox{line\_connecting}(u,v))\mbox{ and }\neg\,\mbox{apart\_point\_and\_line}(y,\mbox{line\_connecting}(u,v)) ) \ \Rightarrow \ ( \neg\,\mbox{apart\_point\_and\_line}(u,\mbox{line\_connecting}(x,y))\mbox{ and }\neg\,\mbox{apart\_point\_and\_line}(v,\mbox{line\_connecting}(x,y)) ) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO231+1.p} Oppositely and equally directed lines






$\mbox{include}(\mbox{'Axioms/GEO007+0.ax'})$

$\forall l,m,n{:}\;( ( \neg\,\mbox{unequally\_directed\_lines}(l,\mbox{reverse\_line}(m))\mbox{ and }\neg\,\mbox{unequally\_directed\_lines}(l,n) ) \ \Rightarrow \ \neg\,\mbox{unequally\_directed\_lines}(m,\mbox{reverse\_line}(n)) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO237+1.p} Axiom of Pasch






$\mbox{include}(\mbox{'Axioms/GEO007+0.ax'})$

$\forall a,b,c,l{:}\;( \mbox{apart\_point\_and\_line}(c,l) \ \Rightarrow \ ( \mbox{divides\_points}(l,a,b) \ \Rightarrow \ ( \mbox{divides\_points}(l,a,c)\mbox{ or }\mbox{divides\_points}(l,b,c) ) ) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO217+1.p} Transitivity of parallel

If a line L is parallel to the lines M and N, then M and N
 are parallel.




$\mbox{include}(\mbox{'Axioms/GEO006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+4.ax'})$

$\forall l,m,n{:}\;( ( \neg\,\mbox{convergent\_lines}(l,m)\mbox{ and }\neg\,\mbox{convergent\_lines}(l,n) ) \ \Rightarrow \ \neg\,\mbox{convergent\_lines}(m,n) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO091-1.p} Two points determine subcurve

Two distinct points on an open curve uniquely determine the
 sub-curve connecting these points




$\mbox{include}(\mbox{'Axioms/GEO004-0.ax'})$

$\mbox{part\_of}(\mbox{sk}_{15},\mbox{sk}_{14})\qquad\mbox{cnf}(\mbox{theorem\_2\_13}_{67},\mbox{negated\_conjecture})$

$\mbox{part\_of}(\mbox{sk}_{16},\mbox{sk}_{14})\qquad\mbox{cnf}(\mbox{theorem\_2\_13}_{68},\mbox{negated\_conjecture})$

$\mbox{open}(\mbox{sk}_{14})\qquad\mbox{cnf}(\mbox{theorem\_2\_13}_{69},\mbox{negated\_conjecture})$

$\mbox{sk}_{17}\ne \mbox{sk}_{18}\qquad\mbox{cnf}(\mbox{theorem\_2\_13}_{70},\mbox{negated\_conjecture})$

$\mbox{end\_point}(\mbox{sk}_{17},\mbox{sk}_{15})\qquad\mbox{cnf}(\mbox{theorem\_2\_13}_{71},\mbox{negated\_conjecture})$

$\mbox{end\_point}(\mbox{sk}_{17},\mbox{sk}_{16})\qquad\mbox{cnf}(\mbox{theorem\_2\_13}_{72},\mbox{negated\_conjecture})$

$\mbox{end\_point}(\mbox{sk}_{18},\mbox{sk}_{15})\qquad\mbox{cnf}(\mbox{theorem\_2\_13}_{73},\mbox{negated\_conjecture})$

$\mbox{end\_point}(\mbox{sk}_{18},\mbox{sk}_{16})\qquad\mbox{cnf}(\mbox{theorem\_2\_13}_{74},\mbox{negated\_conjecture})$

$\mbox{sk}_{15}\ne \mbox{sk}_{16}\qquad\mbox{cnf}(\mbox{theorem\_2\_13}_{75},\mbox{negated\_conjecture})$

\medskip\textbf{GEO016-2.p} Equidistance is symmetric within its argument pairs

Show that if the distance from A to B equals the distance
 from C to D, then the distance from B to A equals the
 distance from C to D.




$\mbox{include}(\mbox{'Axioms/GEO002-0.ax'})$

$\mbox{equidistant}(u,v,w,x)\qquad\mbox{cnf}(\mbox{u\_to\_v\_equals\_w\_to\_x},\mbox{hypothesis})$

$\neg\,\mbox{equidistant}(v,u,w,x)\qquad\mbox{cnf}(\mbox{prove\_symmetry},\mbox{negated\_conjecture})$

\medskip\textbf{GEO078-6.p} Every plane contains 3 noncollinear points






$\mbox{include}(\mbox{'Axioms/GEO005-0.ax'})$

$\mbox{plane}(\mbox{a\_plane})\qquad\mbox{cnf}(\mbox{there\_is\_a\_plane},\mbox{hypothesis})$

$( \mbox{point}(x_1)\mbox{ and }\mbox{point}(x_2)\mbox{ and }\mbox{point}(x_3)\mbox{ and }\mbox{point\_on\_plane}(x_1,\mbox{a\_plane})\mbox{ and }\mbox{point\_on\_plane}(x_2,\mbox{a\_plane})\mbox{ and }\mbox{point\_on\_plane}(x_3,\mbox{a\_plane}) ) \ \Rightarrow \ ( \mbox{collinear}(x_1,x_2,x_3)\mbox{ or }x_1 = x_2\mbox{ or }x_1 = x_3\mbox{ or }x_2 = x_3 )\qquad\mbox{cnf}(\mbox{prove\_every\_plane\_contains\_3\_noncollinear\_points},\mbox{negated\_conjecture})$

\medskip\textbf{GEO242+3.p} Lemma on oriented intersection of lines with plane






$\mbox{include}(\mbox{'Axioms/GEO009+0.ax'})$

$\forall a,b,l{:}\;( ( \mbox{incident\_point\_and\_line}(a,l)\mbox{ and }\mbox{distinct\_points}(a,b)\mbox{ and }\mbox{right\_convergent\_lines}(l,\mbox{line\_connecting}(a,b)) ) \ \Rightarrow \ \mbox{right\_apart\_point}(b,l) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO217+3.p} Transitivity of parallel

If a line L is parallel to the lines M and N, then M and N
 are parallel.




$\mbox{include}(\mbox{'Axioms/GEO006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+1.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+2.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+3.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+4.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+5.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+6.ax'})$

$\forall l,m,n{:}\;( ( \mbox{parallel\_lines}(l,m)\mbox{ and }\mbox{parallel\_lines}(l,n) ) \ \Rightarrow \ \mbox{parallel\_lines}(m,n) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO077-4.p} Three points not collinear if not on line






$\mbox{include}(\mbox{'Axioms/GEO003-0.ax'})$

$\mbox{point}(\mbox{point}_1)\qquad\mbox{cnf}(\mbox{point}_1,\mbox{hypothesis})$

$\mbox{point}(\mbox{point}_2)\qquad\mbox{cnf}(\mbox{point}_2,\mbox{hypothesis})$

$\mbox{point}(\mbox{point}_3)\qquad\mbox{cnf}(\mbox{point}_3,\mbox{hypothesis})$

$\mbox{line}(\mbox{a\_line})\qquad\mbox{cnf}(\mbox{line},\mbox{hypothesis})$

$\mbox{on}(\mbox{point}_1,\mbox{a\_line})\qquad\mbox{cnf}(\mbox{point1\_on\_line},\mbox{hypothesis})$

$\mbox{on}(\mbox{point}_2,\mbox{a\_line})\qquad\mbox{cnf}(\mbox{point2\_on\_line},\mbox{hypothesis})$

$\neg\,\mbox{on}(\mbox{point}_3,\mbox{a\_line})\qquad\mbox{cnf}(\mbox{point3\_not\_on\_line},\mbox{hypothesis})$

$\mbox{point}_1\ne \mbox{point}_2\qquad\mbox{cnf}(\mbox{point1\_not\_point}_2,\mbox{hypothesis})$

$\mbox{point}_1\ne \mbox{point}_3\qquad\mbox{cnf}(\mbox{point1\_not\_point}_3,\mbox{hypothesis})$

$\mbox{point}_2\ne \mbox{point}_3\qquad\mbox{cnf}(\mbox{point2\_not\_point}_3,\mbox{hypothesis})$

$\mbox{collinear}(\mbox{point}_1,\mbox{point}_2,\mbox{point}_3)\qquad\mbox{cnf}(\mbox{prove\_points\_noncollinear},\mbox{negated\_conjecture})$

\medskip\textbf{GEO088-1.p} Endpoint of subcurve or curve

If an endpoint of a given curve lies on a sub-curve then it is
 also an endpoint of this sub-curve




$\mbox{include}(\mbox{'Axioms/GEO004-0.ax'})$

$\mbox{part\_of}(\mbox{sk}_{15},\mbox{sk}_{14})\qquad\mbox{cnf}(\mbox{theorem\_2\_10}_{67},\mbox{negated\_conjecture})$

$\mbox{end\_point}(\mbox{sk}_{16},\mbox{sk}_{14})\qquad\mbox{cnf}(\mbox{theorem\_2\_10}_{68},\mbox{negated\_conjecture})$

$\mbox{incident\_c}(\mbox{sk}_{16},\mbox{sk}_{15})\qquad\mbox{cnf}(\mbox{theorem\_2\_10}_{69},\mbox{negated\_conjecture})$

$\neg\,\mbox{end\_point}(\mbox{sk}_{16},\mbox{sk}_{15})\qquad\mbox{cnf}(\mbox{theorem\_2\_10}_{70},\mbox{negated\_conjecture})$

\medskip\textbf{GEO003-3.p} For all points x and y, y is between x and y






$\mbox{include}(\mbox{'Axioms/GEO002-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO002-2.ax'})$

$\mbox{equidistant}(u,v,u,v)\qquad\mbox{cnf}(d_1,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(w,x,u,v)\qquad\mbox{cnf}(d_2,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(v,u,w,x)\qquad\mbox{cnf}(d_3,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(u,v,x,w)\qquad\mbox{cnf}(\mbox{d4}_1,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(v,u,x,w)\qquad\mbox{cnf}(\mbox{d4}_2,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(w,x,v,u)\qquad\mbox{cnf}(\mbox{d4}_3,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(x,w,u,v)\qquad\mbox{cnf}(\mbox{d4}_4,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(x,w,v,u)\qquad\mbox{cnf}(\mbox{d4}_5,\mbox{axiom})$

$( \mbox{equidistant}(u,v,w,x)\mbox{ and }\mbox{equidistant}(w,x,y,z) ) \ \Rightarrow \ \mbox{equidistant}(u,v,y,z)\qquad\mbox{cnf}(d_5,\mbox{axiom})$

$v = \mbox{extension}(u,v,w,w)\qquad\mbox{cnf}(e_1,\mbox{axiom})$

$y = \mbox{extension}(u,v,w,x) \ \Rightarrow \ \mbox{between}(u,v,y)\qquad\mbox{cnf}(b_0,\mbox{axiom})$

$\mbox{between}(u,v,\mbox{reflection}(u,v))\qquad\mbox{cnf}(\mbox{r2}_1,\mbox{axiom})$

$\mbox{equidistant}(v,\mbox{reflection}(u,v),u,v)\qquad\mbox{cnf}(\mbox{r2}_2,\mbox{axiom})$

$u = v \ \Rightarrow \ v = \mbox{reflection}(u,v)\qquad\mbox{cnf}(\mbox{r3}_1,\mbox{axiom})$

$u = \mbox{reflection}(u,u)\qquad\mbox{cnf}(\mbox{r3}_2,\mbox{axiom})$

$v = \mbox{reflection}(u,v) \ \Rightarrow \ u = v\qquad\mbox{cnf}(r_4,\mbox{axiom})$

$\mbox{equidistant}(u,u,v,v)\qquad\mbox{cnf}(d_7,\mbox{axiom})$

$( \mbox{equidistant}(u,v,u_1,v_1)\mbox{ and }\mbox{equidistant}(v,w,v_1,w_1)\mbox{ and }\mbox{between}(u,v,w)\mbox{ and }\mbox{between}(u_1,v_1,w_1) ) \ \Rightarrow \ \mbox{equidistant}(u,w,u_1,w_1)\qquad\mbox{cnf}(d_8,\mbox{axiom})$

$( \mbox{between}(u,v,w)\mbox{ and }\mbox{between}(u,v,x)\mbox{ and }\mbox{equidistant}(v,w,v,x) ) \ \Rightarrow \ ( u = v\mbox{ or }w = x )\qquad\mbox{cnf}(d_9,\mbox{axiom})$

$\mbox{between}(u,v,w) \ \Rightarrow \ ( u = v\mbox{ or }w = \mbox{extension}(u,v,v,w) )\qquad\mbox{cnf}(\mbox{d10}_1,\mbox{axiom})$

$\mbox{equidistant}(w,x,y,z) \ \Rightarrow \ ( \mbox{extension}(u,v,w,x) = \mbox{extension}(u,v,y,z)\mbox{ or }u = v )\qquad\mbox{cnf}(\mbox{d10}_2,\mbox{axiom})$

$\mbox{extension}(u,v,u,v) = \mbox{extension}(u,v,v,u)\mbox{ or }u = v\qquad\mbox{cnf}(\mbox{d10}_3,\mbox{axiom})$

$\mbox{equidistant}(v,u,v,\mbox{reflection}(\mbox{reflection}(u,v),v))\qquad\mbox{cnf}(r_5,\mbox{axiom})$

$u = \mbox{reflection}(\mbox{reflection}(u,v),v)\qquad\mbox{cnf}(r_6,\mbox{axiom})$

$\neg\,\mbox{between}(a,b,b)\qquad\mbox{cnf}(\mbox{prove\_b\_between\_a\_and\_b},\mbox{negated\_conjecture})$

\medskip\textbf{GEO122+1.p} Every curve has a finishing point






$\mbox{include}(\mbox{'Axioms/GEO004+0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO004+1.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO004+2.ax'})$

$\forall o{:}\;\exists p{:}\;\mbox{finish\_point}(p,o)\qquad\mbox{fof}(\mbox{corollary\_4}_8,\mbox{conjecture})$

\medskip\textbf{GEO241+1.p} Lemma on oriented intersection of lines with plane






$\mbox{include}(\mbox{'Axioms/GEO007+0.ax'})$

$\forall a,b,l{:}\;( ( \neg\,\mbox{apart\_point\_and\_line}(a,l)\mbox{ and }\mbox{distinct\_points}(a,b)\mbox{ and }\mbox{left\_convergent\_lines}(l,\mbox{line\_connecting}(a,b)) ) \ \Rightarrow \ \mbox{left\_apart\_point}(b,l) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO147-1.p} Meeting is possible only if there is a common position






$\mbox{include}(\mbox{'Axioms/GEO004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO004-2.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO004-3.ax'})$

$\mbox{connect}(\mbox{sk}_{28},\mbox{sk}_{29},\mbox{sk}_{27})\qquad\mbox{cnf}(\mbox{t13}_{156},\mbox{negated\_conjecture})$

$\mbox{incident\_o}(\mbox{sk}_{27},\mbox{trajectory\_of}(\mbox{sk}_{28})) \ \Rightarrow \ \neg\,\mbox{incident\_o}(\mbox{sk}_{27},\mbox{trajectory\_of}(\mbox{sk}_{29}))\qquad\mbox{cnf}(\mbox{t13}_{157},\mbox{negated\_conjecture})$

\medskip\textbf{GEO226+3.p} Existence of point incident to line

Assume orthogonal geometry. Given a point and a line, to find
 a point incident with the line.




$\mbox{include}(\mbox{'Axioms/GEO006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+1.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+2.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+3.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+4.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+5.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+6.ax'})$

$\forall l,m{:}\;( ( \mbox{line}(l)\mbox{ and }\mbox{line}(m)\mbox{ and }\mbox{convergent\_lines}(l,m) ) \ \Rightarrow \ \exists x{:}\;( \mbox{point}(x) \ \Rightarrow \ ( \mbox{incident\_point\_and\_line}(x,l)\mbox{ and }\mbox{incident\_point\_and\_line}(x,m) ) ) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO133+1.p} Betweenness and precedence property 2






$\mbox{include}(\mbox{'Axioms/GEO004+0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO004+1.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO004+2.ax'})$

$\forall o,p,q,r,s{:}\;( ( \mbox{ordered\_by}(o,p,q)\mbox{ and }r\ne s\mbox{ and }\mbox{incident\_o}(s,o)\mbox{ and }\mbox{between}(o,r,p,q) ) \ \Rightarrow \ ( \mbox{ordered\_by}(o,r,s)\iff \neg\,\mbox{between}(o,s,r,q) ) )\qquad\mbox{fof}(\mbox{theorem\_4\_17}_2,\mbox{conjecture})$

\medskip\textbf{GEO249+1.p} A point in each region formed by parallel lines






$\mbox{include}(\mbox{'Axioms/GEO007+0.ax'})$

$\forall a,b,l{:}\;( ( \mbox{left\_apart\_point}(a,\mbox{reverse\_line}(l))\mbox{ and }\mbox{left\_apart\_point}(b,\mbox{reverse\_line}(\mbox{parallel\_through\_point}(l,a))) ) \ \Rightarrow \ \mbox{left\_apart\_point}(b,\mbox{reverse\_line}(l)) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO078-5.p} Every plane contains 3 noncollinear points






$\mbox{include}(\mbox{'Axioms/GEO003-0.ax'})$

$( \mbox{point}(x_1)\mbox{ and }\mbox{point}(x_2)\mbox{ and }\mbox{point}(x_3)\mbox{ and }\mbox{on}(x_1,y_1)\mbox{ and }\mbox{on}(x_2,y_1)\mbox{ and }\mbox{line}(y_1)\mbox{ and }\mbox{collinear}(x_1,x_2,x_3) ) \ \Rightarrow \ ( x_1 = x_2\mbox{ or }x_1 = x_3\mbox{ or }x_2 = x_3\mbox{ or }\mbox{on}(x_3,y_1) )\qquad\mbox{cnf}(\mbox{points\_not\_collinear},\mbox{axiom})$

$\mbox{plane}(\mbox{a\_plane})\qquad\mbox{cnf}(\mbox{there\_is\_a\_plane},\mbox{hypothesis})$

$( \mbox{point}(x_1)\mbox{ and }\mbox{point}(x_2)\mbox{ and }\mbox{point}(x_3)\mbox{ and }\mbox{on}(x_1,\mbox{a\_plane})\mbox{ and }\mbox{on}(x_2,\mbox{a\_plane})\mbox{ and }\mbox{on}(x_3,\mbox{a\_plane}) ) \ \Rightarrow \ ( \mbox{collinear}(x_1,x_2,x_3)\mbox{ or }x_1 = x_2\mbox{ or }x_1 = x_3\mbox{ or }x_2 = x_3 )\qquad\mbox{cnf}(\mbox{prove\_every\_plane\_contains\_3\_noncollinear\_points},\mbox{negated\_conjecture})$

\medskip\textbf{GEO018-2.p} Corollary 2 to symmetries of equidistance

Show that if the distance from A to B equals the distance
 from C to D, then the distance from B to A equals the
 distance from D to C.




$\mbox{include}(\mbox{'Axioms/GEO002-0.ax'})$

$\mbox{equidistant}(u,v,w,x)\qquad\mbox{cnf}(\mbox{u\_to\_v\_equals\_w\_to\_x},\mbox{hypothesis})$

$\neg\,\mbox{equidistant}(v,u,x,w)\qquad\mbox{cnf}(\mbox{prove\_symmetry},\mbox{negated\_conjecture})$

\medskip\textbf{GEO202+3.p} Diverging lines have equal ends

If the point X is distinct to the points Y and Z, and the lines
 connecting X and Y, and connecting X and Z are convergent, then
 the intersection point of these lines is equal to X.




$\mbox{include}(\mbox{'Axioms/GEO006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+1.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+2.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+3.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+4.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+5.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+6.ax'})$

$\forall x,y,z{:}\;( ( \mbox{distinct\_points}(x,y)\mbox{ and }\mbox{distinct\_points}(x,z)\mbox{ and }\mbox{convergent\_lines}(\mbox{line\_connecting}(x,y),\mbox{line\_connecting}(x,z)) ) \ \Rightarrow \ \mbox{equal\_points}(\mbox{intersection\_point}(\mbox{line\_connecting}(x,y),\mbox{line\_connecting}(x,z)),x) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO250+3.p} A point in each region formed by parallel lines






$\mbox{include}(\mbox{'Axioms/GEO009+0.ax'})$

$\forall a,b,l{:}\;( \mbox{left\_apart\_point}(b,\mbox{parallel\_through\_point}(l,a)) \ \Rightarrow \ \mbox{right\_apart\_point}(a,\mbox{parallel\_through\_point}(l,b)) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO081-1.p} Transitivity of part\_of






$\mbox{include}(\mbox{'Axioms/GEO004-0.ax'})$

$\mbox{part\_of}(\mbox{sk}_{14},\mbox{sk}_{15})\qquad\mbox{cnf}(\mbox{part\_of\_transitivity}_{67},\mbox{negated\_conjecture})$

$\mbox{part\_of}(\mbox{sk}_{15},\mbox{sk}_{16})\qquad\mbox{cnf}(\mbox{part\_of\_transitivity}_{68},\mbox{negated\_conjecture})$

$\neg\,\mbox{part\_of}(\mbox{sk}_{14},\mbox{sk}_{16})\qquad\mbox{cnf}(\mbox{part\_of\_transitivity}_{69},\mbox{negated\_conjecture})$

\medskip\textbf{GEO199+2.p} Corollary to symmetry of incidence

If the lines X, Y, and Z are pairwise convergent, and the
 intersection point of X and Y is incident with Z, then the
 intersection point of Y and X is incident with Z.




$\mbox{include}(\mbox{'Axioms/GEO008+0.ax'})$

$\forall x,y,z{:}\;( ( \mbox{convergent\_lines}(x,y)\mbox{ and }\mbox{convergent\_lines}(z,y)\mbox{ and }\mbox{convergent\_lines}(x,z)\mbox{ and }\neg\,\mbox{apart\_point\_and\_line}(\mbox{intersection\_point}(x,y),z) ) \ \Rightarrow \ \neg\,\mbox{apart\_point\_and\_line}(\mbox{intersection\_point}(y,x),z) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO083-1.p} Sum is monotone, part 1






$\mbox{include}(\mbox{'Axioms/GEO004-0.ax'})$

$\mbox{part\_of}(\mbox{sk}_{15},\mbox{sk}_{16})\qquad\mbox{cnf}(\mbox{corollary\_2\_6\_1}_{67},\mbox{negated\_conjecture})$

$\mbox{sk}_{17}\wedge \mbox{sk}_{14}{=}\mbox{sk}_{15}\qquad\mbox{cnf}(\mbox{corollary\_2\_6\_1}_{68},\mbox{negated\_conjecture})$

$\mbox{sk}_{17}\wedge \mbox{sk}_{14}{=}\mbox{sk}_{16}\qquad\mbox{cnf}(\mbox{corollary\_2\_6\_1}_{69},\mbox{negated\_conjecture})$

$\neg\,\mbox{part\_of}(\mbox{sk}_{14} + \mbox{sk}_{15},\mbox{sk}_{14} + \mbox{sk}_{16})\qquad\mbox{cnf}(\mbox{corollary\_2\_6\_1}_{70},\mbox{negated\_conjecture})$

\medskip\textbf{GEO261+3.p} Lemma for parallel projection preserves or reverses order






$\mbox{include}(\mbox{'Axioms/GEO009+0.ax'})$

$\forall l,m,a,b,c{:}\;( ( \mbox{between\_on\_line}(l,a,b,c)\mbox{ and }\mbox{convergent\_lines}(l,m)\mbox{ and }\mbox{incident\_point\_and\_line}(b,m) ) \ \Rightarrow \ \mbox{divides\_points}(m,a,c) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO197+2.p} Corollary to symmetry of incidence

If the lines X, Y, and Z are pairwise convergent, and the
 intersection point of X and Y is incident with Z, then the
 intersection point of Z and Y is incident with X.




$\mbox{include}(\mbox{'Axioms/GEO008+0.ax'})$

$\forall x,y,z{:}\;( ( \mbox{convergent\_lines}(x,y)\mbox{ and }\mbox{convergent\_lines}(z,y)\mbox{ and }\mbox{convergent\_lines}(x,z)\mbox{ and }\neg\,\mbox{apart\_point\_and\_line}(\mbox{intersection\_point}(x,y),z) ) \ \Rightarrow \ \neg\,\mbox{apart\_point\_and\_line}(\mbox{intersection\_point}(z,y),x) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO011-2.p} The axiom set points are not collinear






$\mbox{include}(\mbox{'Axioms/GEO002-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO002-1.ax'})$

$\mbox{colinear}(\mbox{lower\_dimension\_point}_1,\mbox{lower\_dimension\_point}_2,\mbox{lower\_dimension\_point}_3)\qquad\mbox{cnf}(\mbox{prove\_lower\_dimension\_points\_not\_colinear},\mbox{negated\_conjecture})$

\medskip\textbf{GEO257+3.p} Transitivity of order on a line






$\mbox{include}(\mbox{'Axioms/GEO009+0.ax'})$

$\forall l,a,b,c,d{:}\;( ( \mbox{distinct\_points}(a,c)\mbox{ and }\mbox{distinct\_points}(b,c)\mbox{ and }\mbox{incident\_point\_and\_line}(c,l)\mbox{ and }\mbox{left\_apart\_point}(d,l) ) \ \Rightarrow \ ( ( \mbox{before\_on\_line}(l,a,b)\mbox{ and }\mbox{before\_on\_line}(l,b,c) ) \ \Rightarrow \ \mbox{before\_on\_line}(l,a,c) ) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO004-2.p} Every line segment has a midpoint






$\mbox{include}(\mbox{'Axioms/GEO002-0.ax'})$

$\mbox{between}(a,\mbox{midpoint}(a,b),b) \ \Rightarrow \ \neg\,\mbox{equidistant}(a,\mbox{midpoint}(a,b),b,\mbox{midpoint}(a,b))\qquad\mbox{cnf}(\mbox{prove\_midpoint},\mbox{negated\_conjecture})$

\medskip\textbf{GEO212+3.p} Non-orthogonal lines and a third line

If a line L is not orthogonal to M, then a third line N is
 convergent to L or not orthogonal to M.




$\mbox{include}(\mbox{'Axioms/GEO006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+1.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+2.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+3.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+4.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+5.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+6.ax'})$

$\forall l,m,n{:}\;( \mbox{not\_orthogonal\_lines}(l,m) \ \Rightarrow \ ( \mbox{convergent\_lines}(l,n)\mbox{ or }\mbox{not\_orthogonal\_lines}(m,n) ) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO060-2.p} Reflection is an involution






$\mbox{include}(\mbox{'Axioms/GEO002-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO002-2.ax'})$

$u\ne \mbox{reflection}(\mbox{reflection}(u,v),v)\qquad\mbox{cnf}(\mbox{prove\_\_involution},\mbox{negated\_conjecture})$

\medskip\textbf{GEO236+1.p} Left and right apart leads to distinctness of lines






$\mbox{include}(\mbox{'Axioms/GEO007+0.ax'})$

$\forall a,l,m{:}\;( ( \mbox{left\_apart\_point}(a,l)\mbox{ and }\mbox{left\_apart\_point}(a,\mbox{reverse\_line}(m)) ) \ \Rightarrow \ ( \mbox{unequally\_directed\_lines}(l,m)\mbox{ or }\mbox{distinct\_lines}(l,m) ) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO206+2.p} Point on parallel lines

If the point X is incident with the line Y, and the lines
 Y and Z are parallel, then the line Y is equal to the parallel
 of Z through point X.




$\mbox{include}(\mbox{'Axioms/GEO008+0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+2.ax'})$

$\forall x,y,z{:}\;( ( \neg\,\mbox{apart\_point\_and\_line}(x,y)\mbox{ and }\neg\,\mbox{convergent\_lines}(y,z) ) \ \Rightarrow \ \neg\,\mbox{distinct\_lines}(y,\mbox{parallel\_through\_point}(z,x)) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO244+1.p} Configurations in terms of apartness






$\mbox{include}(\mbox{'Axioms/GEO007+0.ax'})$

$\forall a,b,c{:}\;( \mbox{distinct\_points}(a,b) \ \Rightarrow \ ( \mbox{left\_apart\_point}(c,\mbox{reverse\_line}(\mbox{line\_connecting}(a,b))) \ \Rightarrow \ \mbox{left\_apart\_point}(c,\mbox{line\_connecting}(b,a)) ) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO151-1.p} Object stays still while one moves

If an object is in a position before and after another object
 moves, then it stays in this position while the other one moves




$\mbox{include}(\mbox{'Axioms/GEO004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO004-2.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO004-3.ax'})$

$\mbox{once}(\mbox{at\_the\_same\_time}(\mbox{at}(\mbox{sk}_{31},\mbox{sk}_{27}),\mbox{at}(\mbox{sk}_{32},\mbox{sk}_{28})))\qquad\mbox{cnf}(\mbox{t17}_{156},\mbox{negated\_conjecture})$

$\mbox{once}(\mbox{at\_the\_same\_time}(\mbox{at}(\mbox{sk}_{31},\mbox{sk}_{27}),\mbox{at}(\mbox{sk}_{32},\mbox{sk}_{30})))\qquad\mbox{cnf}(\mbox{t17}_{157},\mbox{negated\_conjecture})$

$\mbox{between\_o}(\mbox{trajectory\_of}(\mbox{sk}_{31}),\mbox{sk}_{28},\mbox{sk}_{29},\mbox{sk}_{30})\qquad\mbox{cnf}(\mbox{t17}_{158},\mbox{negated\_conjecture})$

$\neg\,\mbox{once}(\mbox{at\_the\_same\_time}(\mbox{at}(\mbox{sk}_{31},\mbox{sk}_{27}),\mbox{at}(\mbox{sk}_{31},\mbox{sk}_{29})))\qquad\mbox{cnf}(\mbox{t17}_{159},\mbox{negated\_conjecture})$

\medskip\textbf{GEO018-3.p} Corollary 2 to symmetries of equidistance

Show that if the distance from A to B equals the distance
 from C to D, then the distance from B to A equals the
 distance from D to C.




$\mbox{include}(\mbox{'Axioms/GEO002-0.ax'})$

$\mbox{equidistant}(u,v,u,v)\qquad\mbox{cnf}(d_1,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(w,x,u,v)\qquad\mbox{cnf}(d_2,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(v,u,w,x)\qquad\mbox{cnf}(d_3,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x)\qquad\mbox{cnf}(\mbox{u\_to\_v\_equals\_w\_to\_x},\mbox{hypothesis})$

$\neg\,\mbox{equidistant}(v,u,x,w)\qquad\mbox{cnf}(\mbox{prove\_symmetry},\mbox{negated\_conjecture})$

\medskip\textbf{GEO141-1.p} Unique oppositely oriented curve 2

For every oriented line there is exactly one uniquely determined
 oriented line with the same underlying curve that orders the
 points in the opposite way.




$\mbox{include}(\mbox{'Axioms/GEO004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO004-2.ax'})$

$\mbox{ordered\_by}(\mbox{sk}_{25},\mbox{sk}_{26},\mbox{sk}_{27})\qquad\mbox{cnf}(\mbox{theorem\_4\_23\_2}_{133},\mbox{negated\_conjecture})$

$\mbox{ordered\_by}(\mbox{sk}_{25},\mbox{sk}_{28},\mbox{sk}_{27})\mbox{ or }\mbox{between}(\mbox{sk}_{25},\mbox{sk}_{28},\mbox{sk}_{26},\mbox{sk}_{27})\mbox{ or }\mbox{between}(\mbox{sk}_{25},\mbox{sk}_{26},\mbox{sk}_{28},\mbox{sk}_{27})\mbox{ or }\mbox{sk}_{26} = \mbox{sk}_{28}\qquad\mbox{cnf}(\mbox{theorem\_4\_23\_2}_{134},\mbox{negated\_conjecture})$

$\mbox{ordered\_by}(\mbox{sk}_{25},\mbox{sk}_{28},\mbox{sk}_{27}) \ \Rightarrow \ \mbox{ordered\_by}(\mbox{sk}_{25},\mbox{sk}_{28},\mbox{sk}_{27})\qquad\mbox{cnf}(\mbox{theorem\_4\_23\_2}_{135},\mbox{negated\_conjecture})$

$\mbox{between}(\mbox{sk}_{25},\mbox{sk}_{28},\mbox{sk}_{26},\mbox{sk}_{27}) \ \Rightarrow \ ( \mbox{between}(\mbox{sk}_{25},\mbox{sk}_{28},\mbox{sk}_{26},\mbox{sk}_{27})\mbox{ or }\mbox{between}(\mbox{sk}_{25},\mbox{sk}_{26},\mbox{sk}_{28},\mbox{sk}_{27})\mbox{ or }\mbox{sk}_{26} = \mbox{sk}_{28} )\qquad\mbox{cnf}(\mbox{theorem\_4\_23\_2}_{136},\mbox{negated\_conjecture})$

$\mbox{between}(\mbox{sk}_{25},\mbox{sk}_{26},\mbox{sk}_{28},\mbox{sk}_{27}) \ \Rightarrow \ ( \mbox{between}(\mbox{sk}_{25},\mbox{sk}_{28},\mbox{sk}_{26},\mbox{sk}_{27})\mbox{ or }\mbox{between}(\mbox{sk}_{25},\mbox{sk}_{26},\mbox{sk}_{28},\mbox{sk}_{27})\mbox{ or }\mbox{sk}_{26} = \mbox{sk}_{28} )\qquad\mbox{cnf}(\mbox{theorem\_4\_23\_2}_{137},\mbox{negated\_conjecture})$

$\mbox{sk}_{26} = \mbox{sk}_{28} \ \Rightarrow \ ( \mbox{between}(\mbox{sk}_{25},\mbox{sk}_{28},\mbox{sk}_{26},\mbox{sk}_{27})\mbox{ or }\mbox{between}(\mbox{sk}_{25},\mbox{sk}_{26},\mbox{sk}_{28},\mbox{sk}_{27})\mbox{ or }\mbox{sk}_{26} = \mbox{sk}_{28} )\qquad\mbox{cnf}(\mbox{theorem\_4\_23\_2}_{138},\mbox{negated\_conjecture})$

$\mbox{between}(\mbox{sk}_{25},\mbox{sk}_{28},\mbox{sk}_{26},\mbox{sk}_{27}) \ \Rightarrow \ \neg\,\mbox{ordered\_by}(\mbox{sk}_{25},\mbox{sk}_{28},\mbox{sk}_{27})\qquad\mbox{cnf}(\mbox{theorem\_4\_23\_2}_{139},\mbox{negated\_conjecture})$

$\mbox{between}(\mbox{sk}_{25},\mbox{sk}_{26},\mbox{sk}_{28},\mbox{sk}_{27}) \ \Rightarrow \ \neg\,\mbox{ordered\_by}(\mbox{sk}_{25},\mbox{sk}_{28},\mbox{sk}_{27})\qquad\mbox{cnf}(\mbox{theorem\_4\_23\_2}_{140},\mbox{negated\_conjecture})$

$\mbox{sk}_{26} = \mbox{sk}_{28} \ \Rightarrow \ \neg\,\mbox{ordered\_by}(\mbox{sk}_{25},\mbox{sk}_{28},\mbox{sk}_{27})\qquad\mbox{cnf}(\mbox{theorem\_4\_23\_2}_{141},\mbox{negated\_conjecture})$

\medskip\textbf{GEO198+2.p} Corollary to symmetry of incidence

If the lines X, Y, and Z are pairwise convergent, and the
 intersection point of X and Y is incident with Z, then the
 intersection point of X and Z is incident with Y.




$\mbox{include}(\mbox{'Axioms/GEO008+0.ax'})$

$\forall x,y,z{:}\;( ( \mbox{convergent\_lines}(x,y)\mbox{ and }\mbox{convergent\_lines}(z,y)\mbox{ and }\mbox{convergent\_lines}(x,z)\mbox{ and }\neg\,\mbox{apart\_point\_and\_line}(\mbox{intersection\_point}(x,y),z) ) \ \Rightarrow \ \neg\,\mbox{apart\_point\_and\_line}(\mbox{intersection\_point}(x,z),y) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO111-1.p} Basic property of orderings on linear structures 1

If Q is between P and R wrt. c, then P, Q and R are incident
 with c and are pairwise distinct




$\mbox{include}(\mbox{'Axioms/GEO004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO004-1.ax'})$

$\mbox{between\_c}(\mbox{sk}_{15},\mbox{sk}_{16},\mbox{sk}_{17},\mbox{sk}_{18})\qquad\mbox{cnf}(\mbox{theorem\_3\_8\_1}_{77},\mbox{negated\_conjecture})$

$( \mbox{incident\_c}(\mbox{sk}_{16},\mbox{sk}_{15})\mbox{ and }\mbox{incident\_c}(\mbox{sk}_{17},\mbox{sk}_{15})\mbox{ and }\mbox{incident\_c}(\mbox{sk}_{18},\mbox{sk}_{15}) ) \ \Rightarrow \ ( \mbox{sk}_{16} = \mbox{sk}_{17}\mbox{ or }\mbox{sk}_{17} = \mbox{sk}_{18}\mbox{ or }\mbox{sk}_{16} = \mbox{sk}_{18} )\qquad\mbox{cnf}(\mbox{theorem\_3\_8\_1}_{78},\mbox{negated\_conjecture})$

\medskip\textbf{GEO098+1.p} For closed curves, there are two complementary sub-curves






$\mbox{include}(\mbox{'Axioms/GEO004+0.ax'})$

$\forall c,p,q{:}\;( ( \mbox{closed}(c)\mbox{ and }\mbox{incident\_c}(p,c)\mbox{ and }\mbox{incident\_c}(q,c)\mbox{ and }p\ne q ) \ \Rightarrow \ \exists c_1,c_2{:}\;( p\wedge c_1{=}c_2\mbox{ and }q\wedge c_1{=}c_2\mbox{ and }c = c_1 + c_2 ) )\qquad\mbox{fof}(\mbox{theorem\_2}_{16},\mbox{conjecture})$

\medskip\textbf{GEO213+3.p} Corollary to non-orthogonal lines and a third line

If line L is not orthogonal to line M, then a third line N
 is distinct from L or not orthogonal to M.




$\mbox{include}(\mbox{'Axioms/GEO006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+1.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+2.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+3.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+4.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+5.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+6.ax'})$

$\forall l,m,n{:}\;( \mbox{not\_orthogonal\_lines}(l,m) \ \Rightarrow \ ( \mbox{distinct\_lines}(l,n)\mbox{ or }\mbox{not\_orthogonal\_lines}(m,n) ) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO101+1.p} Intensification of GEO100+1






$\mbox{include}(\mbox{'Axioms/GEO004+0.ax'})$

$\forall c,c_1,p{:}\;( ( \mbox{part\_of}(c_1,c)\mbox{ and }c_1\ne c\mbox{ and }\mbox{open}(c)\mbox{ and }\mbox{end\_point}(p,c_1)\mbox{ and }\mbox{end\_point}(p,c) ) \ \Rightarrow \ \exists q,r,c_2{:}\;( q\wedge c_1{=}c_2\mbox{ and }c = c_1 + c_2\mbox{ and }p\ne q\mbox{ and }q\ne r\mbox{ and }p\ne r\mbox{ and }\mbox{end\_point}(r,c_2)\mbox{ and }\mbox{end\_point}(r,c) ) )\qquad\mbox{fof}(\mbox{corollary\_2}_{19},\mbox{conjecture})$

\medskip\textbf{GEO123-1.p} Every oriented curve orders all points on it






$\mbox{include}(\mbox{'Axioms/GEO004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO004-2.ax'})$

$\mbox{incident\_o}(\mbox{sk}_{26},\mbox{sk}_{25})\qquad\mbox{cnf}(\mbox{theorem\_4\_9}_{133},\mbox{negated\_conjecture})$

$\mbox{incident\_o}(\mbox{sk}_{27},\mbox{sk}_{25})\qquad\mbox{cnf}(\mbox{theorem\_4\_9}_{134},\mbox{negated\_conjecture})$

$\neg\,\mbox{ordered\_by}(\mbox{sk}_{25},\mbox{sk}_{26},\mbox{sk}_{27})\qquad\mbox{cnf}(\mbox{theorem\_4\_9}_{135},\mbox{negated\_conjecture})$

$\mbox{sk}_{26}\ne \mbox{sk}_{27}\qquad\mbox{cnf}(\mbox{theorem\_4\_9}_{136},\mbox{negated\_conjecture})$

$\neg\,\mbox{ordered\_by}(\mbox{sk}_{25},\mbox{sk}_{27},\mbox{sk}_{26})\qquad\mbox{cnf}(\mbox{theorem\_4\_9}_{137},\mbox{negated\_conjecture})$

\medskip\textbf{GEO047-3.p} Corollary to second inner inner transitivity of betweenness

Corollary of second inner transitivity property of betweenness,
 using symmetry.




$\mbox{include}(\mbox{'Axioms/GEO002-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO002-2.ax'})$

$\mbox{equidistant}(u,v,u,v)\qquad\mbox{cnf}(d_1,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(w,x,u,v)\qquad\mbox{cnf}(d_2,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(v,u,w,x)\qquad\mbox{cnf}(d_3,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(u,v,x,w)\qquad\mbox{cnf}(\mbox{d4}_1,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(v,u,x,w)\qquad\mbox{cnf}(\mbox{d4}_2,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(w,x,v,u)\qquad\mbox{cnf}(\mbox{d4}_3,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(x,w,u,v)\qquad\mbox{cnf}(\mbox{d4}_4,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(x,w,v,u)\qquad\mbox{cnf}(\mbox{d4}_5,\mbox{axiom})$

$( \mbox{equidistant}(u,v,w,x)\mbox{ and }\mbox{equidistant}(w,x,y,z) ) \ \Rightarrow \ \mbox{equidistant}(u,v,y,z)\qquad\mbox{cnf}(d_5,\mbox{axiom})$

$v = \mbox{extension}(u,v,w,w)\qquad\mbox{cnf}(e_1,\mbox{axiom})$

$y = \mbox{extension}(u,v,w,x) \ \Rightarrow \ \mbox{between}(u,v,y)\qquad\mbox{cnf}(b_0,\mbox{axiom})$

$\mbox{between}(u,v,\mbox{reflection}(u,v))\qquad\mbox{cnf}(\mbox{r2}_1,\mbox{axiom})$

$\mbox{equidistant}(v,\mbox{reflection}(u,v),u,v)\qquad\mbox{cnf}(\mbox{r2}_2,\mbox{axiom})$

$u = v \ \Rightarrow \ v = \mbox{reflection}(u,v)\qquad\mbox{cnf}(\mbox{r3}_1,\mbox{axiom})$

$u = \mbox{reflection}(u,u)\qquad\mbox{cnf}(\mbox{r3}_2,\mbox{axiom})$

$v = \mbox{reflection}(u,v) \ \Rightarrow \ u = v\qquad\mbox{cnf}(r_4,\mbox{axiom})$

$\mbox{equidistant}(u,u,v,v)\qquad\mbox{cnf}(d_7,\mbox{axiom})$

$( \mbox{equidistant}(u,v,u_1,v_1)\mbox{ and }\mbox{equidistant}(v,w,v_1,w_1)\mbox{ and }\mbox{between}(u,v,w)\mbox{ and }\mbox{between}(u_1,v_1,w_1) ) \ \Rightarrow \ \mbox{equidistant}(u,w,u_1,w_1)\qquad\mbox{cnf}(d_8,\mbox{axiom})$

$( \mbox{between}(u,v,w)\mbox{ and }\mbox{between}(u,v,x)\mbox{ and }\mbox{equidistant}(v,w,v,x) ) \ \Rightarrow \ ( u = v\mbox{ or }w = x )\qquad\mbox{cnf}(d_9,\mbox{axiom})$

$\mbox{between}(u,v,w) \ \Rightarrow \ ( u = v\mbox{ or }w = \mbox{extension}(u,v,v,w) )\qquad\mbox{cnf}(\mbox{d10}_1,\mbox{axiom})$

$\mbox{equidistant}(w,x,y,z) \ \Rightarrow \ ( \mbox{extension}(u,v,w,x) = \mbox{extension}(u,v,y,z)\mbox{ or }u = v )\qquad\mbox{cnf}(\mbox{d10}_2,\mbox{axiom})$

$\mbox{extension}(u,v,u,v) = \mbox{extension}(u,v,v,u)\mbox{ or }u = v\qquad\mbox{cnf}(\mbox{d10}_3,\mbox{axiom})$

$\mbox{equidistant}(v,u,v,\mbox{reflection}(\mbox{reflection}(u,v),v))\qquad\mbox{cnf}(r_5,\mbox{axiom})$

$u = \mbox{reflection}(\mbox{reflection}(u,v),v)\qquad\mbox{cnf}(r_6,\mbox{axiom})$

$\mbox{between}(u,v,v)\qquad\mbox{cnf}(t_3,\mbox{axiom})$

$( \mbox{between}(u,w,x)\mbox{ and }u = x ) \ \Rightarrow \ \mbox{between}(v,w,x)\qquad\mbox{cnf}(b_1,\mbox{axiom})$

$\mbox{between}(u,v,w) \ \Rightarrow \ \mbox{between}(w,v,u)\qquad\mbox{cnf}(t_1,\mbox{axiom})$

$\mbox{between}(u,u,v)\qquad\mbox{cnf}(t_2,\mbox{axiom})$

$( \mbox{between}(u,v,w)\mbox{ and }\mbox{between}(v,u,w) ) \ \Rightarrow \ u = v\qquad\mbox{cnf}(b_2,\mbox{axiom})$

$( \mbox{between}(u,v,w)\mbox{ and }\mbox{between}(u,w,v) ) \ \Rightarrow \ v = w\qquad\mbox{cnf}(b_3,\mbox{axiom})$

$( \mbox{between}(u,v,w)\mbox{ and }\mbox{between}(v,u,w) ) \ \Rightarrow \ ( u = v\mbox{ or }v = w )\qquad\mbox{cnf}(\mbox{t6}_1,\mbox{axiom})$

$( \mbox{between}(u,v,w)\mbox{ and }\mbox{between}(u,w,v) ) \ \Rightarrow \ ( u = v\mbox{ or }v = w )\qquad\mbox{cnf}(\mbox{t6}_2,\mbox{axiom})$

$( \mbox{between}(u,v,w)\mbox{ and }\mbox{between}(v,w,x) ) \ \Rightarrow \ \mbox{between}(u,v,w)\qquad\mbox{cnf}(b_4,\mbox{axiom})$

$( \mbox{between}(u,v,w)\mbox{ and }\mbox{between}(u,w,x) ) \ \Rightarrow \ \mbox{between}(v,w,x)\qquad\mbox{cnf}(b_5,\mbox{axiom})$

$( \mbox{between}(u,v,w)\mbox{ and }\mbox{between}(v,w,x) ) \ \Rightarrow \ ( \mbox{between}(u,w,x)\mbox{ or }v = w )\qquad\mbox{cnf}(b_6,\mbox{axiom})$

$( \mbox{between}(u,v,w)\mbox{ and }\mbox{between}(v,w,x) ) \ \Rightarrow \ ( \mbox{between}(u,v,x)\mbox{ or }v = w )\qquad\mbox{cnf}(b_7,\mbox{axiom})$

$( \mbox{between}(u,v,x)\mbox{ and }\mbox{between}(v,w,x) ) \ \Rightarrow \ \mbox{between}(u,w,x)\qquad\mbox{cnf}(b_8,\mbox{axiom})$

$\mbox{between}(u,v,w)\qquad\mbox{cnf}(\mbox{v\_between\_u\_and\_w},\mbox{hypothesis})$

$\mbox{between}(u,w,x)\qquad\mbox{cnf}(\mbox{w\_between\_u\_and\_x},\mbox{hypothesis})$

$\neg\,\mbox{between}(u,v,x)\qquad\mbox{cnf}(\mbox{prove\_v\_between\_u\_and\_x},\mbox{negated\_conjecture})$

\medskip\textbf{GEO221+3.p} Lemma on orthogonality

If a point B is incident with the orthogonal to a line L
 through point A, then this orthogonal is equal to the
 orthogonal to L through B.




$\mbox{include}(\mbox{'Axioms/GEO006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+1.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+2.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+3.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+4.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+5.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+6.ax'})$

$\forall a,b,l{:}\;( \mbox{incident\_point\_and\_line}(b,\mbox{orthogonal\_through\_point}(l,a)) \ \Rightarrow \ \mbox{equal\_lines}(\mbox{orthogonal\_through\_point}(l,a),\mbox{orthogonal\_through\_point}(l,b)) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO269+3.p} Equally directed opposite reveresed lines






$\mbox{include}(\mbox{'Axioms/GEO009+0.ax'})$

$\forall l{:}\;\mbox{equally\_directed\_opposite\_lines}(l,\mbox{reverse\_line}(l))\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO181+2.p} Triangle axiom 2

If X and Y are distinct points and Z is apart from the line
 connecting X and Y, then Y is apart from the line connecting
 X and Z.




$\mbox{include}(\mbox{'Axioms/GEO008+0.ax'})$

$\forall x,y,z{:}\;( \mbox{distinct\_points}(x,y) \ \Rightarrow \ ( \mbox{apart\_point\_and\_line}(z,\mbox{line\_connecting}(x,y)) \ \Rightarrow \ \mbox{apart\_point\_and\_line}(y,\mbox{line\_connecting}(x,z)) ) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO233+1.p} Reverse is idempotent for direction






$\mbox{include}(\mbox{'Axioms/GEO007+0.ax'})$

$\forall l{:}\;\neg\,\mbox{unequally\_directed\_lines}(\mbox{reverse\_line}(\mbox{reverse\_line}(l)),l)\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO131+1.p} Betweenness and precedence for three points, corollary






$\mbox{include}(\mbox{'Axioms/GEO004+0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO004+1.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO004+2.ax'})$

$\forall p,q,r,o{:}\;( \mbox{between}(o,p,q,r) \ \Rightarrow \ ( \mbox{ordered\_by}(o,p,q)\iff \mbox{ordered\_by}(o,p,r) ) )\qquad\mbox{fof}(\mbox{corollary\_4}_{16},\mbox{conjecture})$

\medskip\textbf{GEO188+1.p} Collary to symmetry of incidence

If X, Y, and Z are pairwise distinct, and Z is incident with
 the line connecting X and Y, then X is incident with the line
 connecting Z and Y.




$\mbox{include}(\mbox{'Axioms/GEO006+0.ax'})$

$\forall x,y,z{:}\;( ( \mbox{distinct\_points}(x,y)\mbox{ and }\mbox{distinct\_points}(x,z)\mbox{ and }\mbox{distinct\_points}(y,z)\mbox{ and }\neg\,\mbox{apart\_point\_and\_line}(z,\mbox{line\_connecting}(x,y)) ) \ \Rightarrow \ \neg\,\mbox{apart\_point\_and\_line}(x,\mbox{line\_connecting}(z,y)) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO150-1.p} Objects cannot be at two places simultaneously






$\mbox{include}(\mbox{'Axioms/GEO004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO004-2.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO004-3.ax'})$

$\mbox{once}(\mbox{at\_the\_same\_time}(\mbox{at}(\mbox{sk}_{29},\mbox{sk}_{27}),\mbox{at}(\mbox{sk}_{29},\mbox{sk}_{28})))\qquad\mbox{cnf}(\mbox{t16}_{156},\mbox{negated\_conjecture})$

$\mbox{sk}_{27}\ne \mbox{sk}_{28}\qquad\mbox{cnf}(\mbox{t16}_{157},\mbox{negated\_conjecture})$

\medskip\textbf{GEO123+1.p} Every oriented curve orders all points on it






$\mbox{include}(\mbox{'Axioms/GEO004+0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO004+1.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO004+2.ax'})$

$\forall o,p,q{:}\;( ( \mbox{incident\_o}(p,o)\mbox{ and }\mbox{incident\_o}(q,o) ) \ \Rightarrow \ ( \mbox{ordered\_by}(o,p,q)\mbox{ or }p = q\mbox{ or }\mbox{ordered\_by}(o,q,p) ) )\qquad\mbox{fof}(\mbox{theorem\_4}_9,\mbox{conjecture})$

\medskip\textbf{GEO105-1.p} If subcurves meet at an endpoint then there's a meeting

If two sub-curves of an open curve meet at a point and this
 point is an endpoint for another sub-curve then this sub-curve
 meets one of the former sub-curves at this point.




$\mbox{include}(\mbox{'Axioms/GEO004-0.ax'})$

$\mbox{part\_of}(\mbox{sk}_{15},\mbox{sk}_{14})\qquad\mbox{cnf}(\mbox{proposition\_2\_23}_{67},\mbox{negated\_conjecture})$

$\mbox{part\_of}(\mbox{sk}_{16},\mbox{sk}_{14})\qquad\mbox{cnf}(\mbox{proposition\_2\_23}_{68},\mbox{negated\_conjecture})$

$\mbox{part\_of}(\mbox{sk}_{17},\mbox{sk}_{14})\qquad\mbox{cnf}(\mbox{proposition\_2\_23}_{69},\mbox{negated\_conjecture})$

$\mbox{sk}_{18}\wedge \mbox{sk}_{15}{=}\mbox{sk}_{16}\qquad\mbox{cnf}(\mbox{proposition\_2\_23}_{70},\mbox{negated\_conjecture})$

$\mbox{end\_point}(\mbox{sk}_{18},\mbox{sk}_{17})\qquad\mbox{cnf}(\mbox{proposition\_2\_23}_{71},\mbox{negated\_conjecture})$

$\mbox{open}(\mbox{sk}_{14})\qquad\mbox{cnf}(\mbox{proposition\_2\_23}_{72},\mbox{negated\_conjecture})$

$\neg\,\mbox{sk}_{18}\wedge \mbox{sk}_{15}{=}\mbox{sk}_{17}\qquad\mbox{cnf}(\mbox{proposition\_2\_23}_{73},\mbox{negated\_conjecture})$

$\neg\,\mbox{sk}_{18}\wedge \mbox{sk}_{16}{=}\mbox{sk}_{17}\qquad\mbox{cnf}(\mbox{proposition\_2\_23}_{74},\mbox{negated\_conjecture})$

\medskip\textbf{GEO182+3.p} Triangle axiom 3

If X and Y are distinct points and Z is apart from the line
 connecting X and Y, then Z is apart from the line connecting
 Y and X.




$\mbox{include}(\mbox{'Axioms/GEO006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+1.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+2.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+3.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+4.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+5.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+6.ax'})$

$\forall x,y,z{:}\;( \mbox{distinct\_points}(x,y) \ \Rightarrow \ ( \mbox{apart\_point\_and\_line}(z,\mbox{line\_connecting}(x,y)) \ \Rightarrow \ \mbox{apart\_point\_and\_line}(z,\mbox{line\_connecting}(y,x)) ) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO183+2.p} Lemma on symmetry and apartness

If X and Y are distinct points, then they are incident with
 the line that is equal to the line connecting the points.




$\mbox{include}(\mbox{'Axioms/GEO008+0.ax'})$

$\forall x,y,u,v{:}\;( ( \mbox{distinct\_points}(x,y)\mbox{ and }\mbox{convergent\_lines}(u,v)\mbox{ and }\neg\,\mbox{distinct\_lines}(u,\mbox{line\_connecting}(x,y)) ) \ \Rightarrow \ ( \neg\,\mbox{apart\_point\_and\_line}(x,u)\mbox{ and }\neg\,\mbox{apart\_point\_and\_line}(y,u) ) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO180+1.p} Triangle axiom 1

If X and Y are distinct points and Z is apart from the line
 connecting X and Y, then X is apart from the line connecting
 Z and Y.




$\mbox{include}(\mbox{'Axioms/GEO006+0.ax'})$

$\forall x,y,z{:}\;( \mbox{distinct\_points}(x,y) \ \Rightarrow \ ( \mbox{apart\_point\_and\_line}(z,\mbox{line\_connecting}(x,y)) \ \Rightarrow \ \mbox{apart\_point\_and\_line}(x,\mbox{line\_connecting}(z,y)) ) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO210+1.p} Uniqueness of orthogonality

If the point A is incident with line L, and the line L is
 orthogonal to M, then L is equal to the orthogonal to M
 through A.




$\mbox{include}(\mbox{'Axioms/GEO006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+2.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+3.ax'})$

$\forall a,l,m{:}\;( ( \neg\,\mbox{apart\_point\_and\_line}(a,l)\mbox{ and }\neg\,\mbox{unorthogonal\_lines}(l,m) ) \ \Rightarrow \ \neg\,\mbox{distinct\_lines}(l,\mbox{orthogonal\_through\_point}(m,a)) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO234+1.p} Unequally directed lines






$\mbox{include}(\mbox{'Axioms/GEO007+0.ax'})$

$\forall l,m,n{:}\;( \mbox{unequally\_directed\_lines}(l,\mbox{reverse\_line}(m)) \ \Rightarrow \ ( \mbox{unequally\_directed\_lines}(l,n)\mbox{ or }\mbox{unequally\_directed\_lines}(m,\mbox{reverse\_line}(n)) ) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO109-1.p} Every endpoint of an open curve is not between any other points






$\mbox{include}(\mbox{'Axioms/GEO004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO004-1.ax'})$

$\mbox{open}(\mbox{sk}_{15})\qquad\mbox{cnf}(\mbox{theorem\_3\_6}_{77},\mbox{negated\_conjecture})$

$\mbox{end\_point}(\mbox{sk}_{16},\mbox{sk}_{15})\mbox{ or }\mbox{incident\_c}(\mbox{sk}_{16},\mbox{sk}_{15})\qquad\mbox{cnf}(\mbox{theorem\_3\_6}_{78},\mbox{negated\_conjecture})$

$\mbox{between\_c}(\mbox{sk}_{15},a,\mbox{sk}_{16},b) \ \Rightarrow \ \mbox{end\_point}(\mbox{sk}_{16},\mbox{sk}_{15})\qquad\mbox{cnf}(\mbox{theorem\_3\_6}_{79},\mbox{negated\_conjecture})$

$\mbox{end\_point}(\mbox{sk}_{16},\mbox{sk}_{15}) \ \Rightarrow \ \mbox{end\_point}(\mbox{sk}_{16},\mbox{sk}_{15})\qquad\mbox{cnf}(\mbox{theorem\_3\_6}_{80},\mbox{negated\_conjecture})$

$\mbox{incident\_c}(\mbox{sk}_{16},\mbox{sk}_{15}) \ \Rightarrow \ ( \mbox{between\_c}(\mbox{sk}_{15},\mbox{sk}_{17},\mbox{sk}_{16},\mbox{sk}_{18})\mbox{ or }\mbox{incident\_c}(\mbox{sk}_{16},\mbox{sk}_{15}) )\qquad\mbox{cnf}(\mbox{theorem\_3\_6}_{81},\mbox{negated\_conjecture})$

$( \mbox{incident\_c}(\mbox{sk}_{16},\mbox{sk}_{15})\mbox{ and }\mbox{between\_c}(\mbox{sk}_{15},a,\mbox{sk}_{16},b) ) \ \Rightarrow \ \mbox{between\_c}(\mbox{sk}_{15},\mbox{sk}_{17},\mbox{sk}_{16},\mbox{sk}_{18})\qquad\mbox{cnf}(\mbox{theorem\_3\_6}_{82},\mbox{negated\_conjecture})$

$( \mbox{incident\_c}(\mbox{sk}_{16},\mbox{sk}_{15})\mbox{ and }\mbox{end\_point}(\mbox{sk}_{16},\mbox{sk}_{15}) ) \ \Rightarrow \ \mbox{between\_c}(\mbox{sk}_{15},\mbox{sk}_{17},\mbox{sk}_{16},\mbox{sk}_{18})\qquad\mbox{cnf}(\mbox{theorem\_3\_6}_{83},\mbox{negated\_conjecture})$

\medskip\textbf{GEO109+1.p} Every endpoint of an open curve is not between any other points






$\mbox{include}(\mbox{'Axioms/GEO004+0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO004+1.ax'})$

$\forall c,p{:}\;( \mbox{open}(c) \ \Rightarrow \ ( \mbox{end\_point}(p,c)\iff ( \mbox{incident\_c}(p,c)\mbox{ and }\neg\,\exists q,r{:}\;\mbox{between\_c}(c,q,p,r) ) ) )\qquad\mbox{fof}(\mbox{theorem\_3}_6,\mbox{conjecture})$

\medskip\textbf{GEO149+1.p} Condition for meeting at two points

Objects can meet at two points only if they are ordered in the
 same way on both trajectories




$\mbox{include}(\mbox{'Axioms/GEO004+0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO004+1.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO004+2.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO004+3.ax'})$

$\forall p,q,x,y{:}\;( ( \mbox{connect}(x,y,p)\mbox{ and }\mbox{connect}(x,y,q)\mbox{ and }\mbox{ordered\_by}(\mbox{trajectory\_of}(x),p,q) ) \ \Rightarrow \ \mbox{ordered\_by}(\mbox{trajectory\_of}(y),p,q) )\qquad\mbox{fof}(t_{15},\mbox{conjecture})$

\medskip\textbf{GEO115+1.p} Basic property of orderings on linear structures 5

If Q is between P and R wrt. c and Q' another point distinct
 from Q and lying on c then Q is either between P and Q' or
 between Q' and R wrt. c.




$\mbox{include}(\mbox{'Axioms/GEO004+0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO004+1.ax'})$

$\forall c,p,q,r,\mbox{qpp}{:}\;( ( \mbox{between\_c}(c,p,q,r)\mbox{ and }\mbox{incident\_c}(\mbox{qpp},c)\mbox{ and }q\ne \mbox{qpp} ) \ \Rightarrow \ ( \mbox{between\_c}(c,p,q,\mbox{qpp})\mbox{ or }\mbox{between\_c}(c,\mbox{qpp},q,r) ) )\qquad\mbox{fof}(\mbox{theorem\_3\_8}_5,\mbox{conjecture})$

\medskip\textbf{GEO263+1.p} Parallel projection preserves or reverses order






$\mbox{include}(\mbox{'Axioms/GEO007+0.ax'})$

$\forall l,m,n,a,b,c{:}\;( ( \mbox{between\_on\_line}(l,a,b,c)\mbox{ and }\mbox{convergent\_lines}(l,m)\mbox{ and }\mbox{convergent\_lines}(l,n)\mbox{ and }\mbox{convergent\_lines}(m,n) ) \ \Rightarrow \ \mbox{between\_on\_line}(m,\mbox{intersection\_point}(m,\mbox{parallel\_through\_point}(n,a)),\mbox{intersection\_point}(m,\mbox{parallel\_through\_point}(n,b)),\mbox{intersection\_point}(m,\mbox{parallel\_through\_point}(n,c))) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO114+1.p} Basic property of orderings on linear structures 4

If P, Q and R are distinct and on c then one of the points is
 between the others wrt. c.




$\mbox{include}(\mbox{'Axioms/GEO004+0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO004+1.ax'})$

$\forall c,p,q,r{:}\;( ( \mbox{incident\_c}(p,c)\mbox{ and }\mbox{incident\_c}(q,c)\mbox{ and }\mbox{incident\_c}(r,c)\mbox{ and }p\ne q\mbox{ and }q\ne r\mbox{ and }p\ne r ) \ \Rightarrow \ ( \mbox{between\_c}(c,p,q,r)\mbox{ or }\mbox{between\_c}(c,q,p,r)\mbox{ or }\mbox{between\_c}(c,p,r,q) ) )\qquad\mbox{fof}(\mbox{theorem\_3\_8}_4,\mbox{conjecture})$

\medskip\textbf{GEO190+2.p} Collary to symmetry of incidence

If X, Y, and Z are pairwise distinct, and Z is incident with
 the line connecting X and Y, then Z is incident with the line
 connecting Y and X.




$\mbox{include}(\mbox{'Axioms/GEO008+0.ax'})$

$\forall x,y,z{:}\;( ( \mbox{distinct\_points}(x,y)\mbox{ and }\mbox{distinct\_points}(x,z)\mbox{ and }\mbox{distinct\_points}(y,z)\mbox{ and }\neg\,\mbox{apart\_point\_and\_line}(z,\mbox{line\_connecting}(x,y)) ) \ \Rightarrow \ \neg\,\mbox{apart\_point\_and\_line}(z,\mbox{line\_connecting}(y,x)) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO259+1.p} Betweeness






$\mbox{include}(\mbox{'Axioms/GEO007+0.ax'})$

$\forall l,a,b,c{:}\;( ( \mbox{line}(l)\mbox{ and }\mbox{distinct\_points}(a,c)\mbox{ and }\mbox{distinct\_points}(b,c)\mbox{ and }\mbox{distinct\_points}(a,b) ) \ \Rightarrow \ \neg\,\mbox{between\_on\_line}(l,a,b,c)\mbox{ and }\mbox{between\_on\_line}(l,b,a,c) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO173+1.p} Lemma on symmetry and apartness

If two points are distinct and a line U is distinct from the
 line connecting the points, then U is apart from at least one
 of these points.




$\mbox{include}(\mbox{'Axioms/GEO006+0.ax'})$

$\forall x,y,u,v{:}\;( ( \mbox{distinct\_points}(x,y)\mbox{ and }\mbox{convergent\_lines}(u,v)\mbox{ and }\mbox{distinct\_lines}(u,\mbox{line\_connecting}(x,y)) ) \ \Rightarrow \ ( \mbox{apart\_point\_and\_line}(x,u)\mbox{ or }\mbox{apart\_point\_and\_line}(y,u) ) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO220+2.p} Transitivity of orthogonal

If a line L is orthogonal to line M and line N is orthogonal
 to L, then M and N are parallel.




$\mbox{include}(\mbox{'Axioms/GEO008+0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+4.ax'})$

$\forall l,m,n{:}\;( ( \neg\,\mbox{unorthogonal\_lines}(l,m)\mbox{ and }\neg\,\mbox{unorthogonal\_lines}(l,n) ) \ \Rightarrow \ \neg\,\mbox{convergent\_lines}(m,n) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO218+3.p} Transitivity of parallel and orthogonal

If two lines L and M are parallel and a third line N is
 orthogonal to L, then M is orthogonal to N.




$\mbox{include}(\mbox{'Axioms/GEO006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+1.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+2.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+3.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+4.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+5.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+6.ax'})$

$\forall l,m,n{:}\;( ( \mbox{parallel\_lines}(l,m)\mbox{ and }\mbox{orthogonal\_lines}(l,n) ) \ \Rightarrow \ \mbox{orthogonal\_lines}(m,n) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO058-2.p} U is the only fixed point of reflection(U,V)






$\mbox{include}(\mbox{'Axioms/GEO002-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO002-2.ax'})$

$v = \mbox{reflection}(u,v)\qquad\mbox{cnf}(\mbox{v\_equals\_reflection},\mbox{hypothesis})$

$u\ne v\qquad\mbox{cnf}(\mbox{prove\_u\_equals\_v},\mbox{negated\_conjecture})$

\medskip\textbf{GEO156-1.p} Insertion axioms for the GEO003 geometry axioms






$\mbox{include}(\mbox{'Axioms/GEO002-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO002-3.ax'})$

\medskip\textbf{GEO113+1.p} Basic property of orderings on linear structures 3

If c is open and Q is between P and R wrt. c, then P is not
 between Q and R wrt. c




$\mbox{include}(\mbox{'Axioms/GEO004+0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO004+1.ax'})$

$\forall c,p,q,r{:}\;( ( \mbox{open}(c)\mbox{ and }\mbox{between\_c}(c,p,q,r) ) \ \Rightarrow \ \neg\,\mbox{between\_c}(c,q,p,r) )\qquad\mbox{fof}(\mbox{theorem\_3\_8}_3,\mbox{conjecture})$

\medskip\textbf{GEO089+1.p} Inner points of a sub-curve of a curve are inner points






$\mbox{include}(\mbox{'Axioms/GEO004+0.ax'})$

$\forall c,p{:}\;( \exists \mbox{cpp}{:}\;( \mbox{part\_of}(\mbox{cpp},c)\mbox{ and }\mbox{inner\_point}(p,\mbox{cpp}) ) \ \Rightarrow \ \mbox{inner\_point}(p,c) )\qquad\mbox{fof}(\mbox{corollary\_2}_{11},\mbox{conjecture})$

\medskip\textbf{GEO219+1.p} Transitivity of orthogonal and parallel

If line L is orthogonal to M and a line N is parallel to L,
 then M is orthogonal to N.




$\mbox{include}(\mbox{'Axioms/GEO006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+4.ax'})$

$\forall l,m,n{:}\;( ( \neg\,\mbox{unorthogonal\_lines}(l,m)\mbox{ and }\neg\,\mbox{convergent\_lines}(l,n) ) \ \Rightarrow \ \neg\,\mbox{unorthogonal\_lines}(m,n) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO020-2.p} Corollary 4 to symmetries of equidistance

Show that if the distance from A to B equals the distance
 from C to D, then the distance from D to C equals the
 distance from A to B.




$\mbox{include}(\mbox{'Axioms/GEO002-0.ax'})$

$\mbox{equidistant}(u,v,w,x)\qquad\mbox{cnf}(\mbox{u\_to\_v\_equals\_w\_to\_x},\mbox{hypothesis})$

$\neg\,\mbox{equidistant}(x,w,u,v)\qquad\mbox{cnf}(\mbox{prove\_symmetry},\mbox{negated\_conjecture})$

\medskip\textbf{GEO170+1.p} Uniqueness of constructed lines

If two distinct points are incident with a line, then this line
 is equivalent with the connecting line of these points.




$\mbox{include}(\mbox{'Axioms/GEO006+0.ax'})$

$\forall x,y,z{:}\;( ( \mbox{distinct\_points}(x,y)\mbox{ and }\neg\,\mbox{apart\_point\_and\_line}(x,z)\mbox{ and }\neg\,\mbox{apart\_point\_and\_line}(y,z) ) \ \Rightarrow \ \neg\,\mbox{distinct\_lines}(z,\mbox{line\_connecting}(x,y)) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO175+1.p} Lemma on symmetry and apartness

If two lines are convergent and a point is distinct from the
 intersection point then this point is apart from at least one of
 these lines.




$\mbox{include}(\mbox{'Axioms/GEO006+0.ax'})$

$\forall x,y,u,v{:}\;( ( \mbox{distinct\_points}(x,y)\mbox{ and }\mbox{convergent\_lines}(u,v)\mbox{ and }\mbox{distinct\_points}(x,\mbox{intersection\_point}(u,v)) ) \ \Rightarrow \ ( \mbox{apart\_point\_and\_line}(x,u)\mbox{ or }\mbox{apart\_point\_and\_line}(x,v) ) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO027-3.p} Corollary 1 to unique extension






$\mbox{include}(\mbox{'Axioms/GEO002-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO002-2.ax'})$

$\mbox{equidistant}(u,v,u,v)\qquad\mbox{cnf}(d_1,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(w,x,u,v)\qquad\mbox{cnf}(d_2,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(v,u,w,x)\qquad\mbox{cnf}(d_3,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(u,v,x,w)\qquad\mbox{cnf}(\mbox{d4}_1,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(v,u,x,w)\qquad\mbox{cnf}(\mbox{d4}_2,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(w,x,v,u)\qquad\mbox{cnf}(\mbox{d4}_3,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(x,w,u,v)\qquad\mbox{cnf}(\mbox{d4}_4,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(x,w,v,u)\qquad\mbox{cnf}(\mbox{d4}_5,\mbox{axiom})$

$( \mbox{equidistant}(u,v,w,x)\mbox{ and }\mbox{equidistant}(w,x,y,z) ) \ \Rightarrow \ \mbox{equidistant}(u,v,y,z)\qquad\mbox{cnf}(d_5,\mbox{axiom})$

$v = \mbox{extension}(u,v,w,w)\qquad\mbox{cnf}(e_1,\mbox{axiom})$

$y = \mbox{extension}(u,v,w,x) \ \Rightarrow \ \mbox{between}(u,v,y)\qquad\mbox{cnf}(b_0,\mbox{axiom})$

$\mbox{between}(u,v,\mbox{reflection}(u,v))\qquad\mbox{cnf}(\mbox{r2}_1,\mbox{axiom})$

$\mbox{equidistant}(v,\mbox{reflection}(u,v),u,v)\qquad\mbox{cnf}(\mbox{r2}_2,\mbox{axiom})$

$u = v \ \Rightarrow \ v = \mbox{reflection}(u,v)\qquad\mbox{cnf}(\mbox{r3}_1,\mbox{axiom})$

$u = \mbox{reflection}(u,u)\qquad\mbox{cnf}(\mbox{r3}_2,\mbox{axiom})$

$v = \mbox{reflection}(u,v) \ \Rightarrow \ u = v\qquad\mbox{cnf}(r_4,\mbox{axiom})$

$\mbox{equidistant}(u,u,v,v)\qquad\mbox{cnf}(d_7,\mbox{axiom})$

$( \mbox{equidistant}(u,v,u_1,v_1)\mbox{ and }\mbox{equidistant}(v,w,v_1,w_1)\mbox{ and }\mbox{between}(u,v,w)\mbox{ and }\mbox{between}(u_1,v_1,w_1) ) \ \Rightarrow \ \mbox{equidistant}(u,w,u_1,w_1)\qquad\mbox{cnf}(d_8,\mbox{axiom})$

$( \mbox{between}(u,v,w)\mbox{ and }\mbox{between}(u,v,x)\mbox{ and }\mbox{equidistant}(v,w,v,x) ) \ \Rightarrow \ ( u = v\mbox{ or }w = x )\qquad\mbox{cnf}(d_9,\mbox{axiom})$

$\mbox{between}(u,v,w)\qquad\mbox{cnf}(\mbox{v\_between\_u\_and\_w},\mbox{hypothesis})$

$u\ne v\qquad\mbox{cnf}(\mbox{u\_not\_v},\mbox{hypothesis})$

$w\ne \mbox{extension}(u,v,v,w)\qquad\mbox{cnf}(\mbox{prove\_w\_is\_an\_extension},\mbox{negated\_conjecture})$

\medskip\textbf{GEO174+3.p} Lemma on symmetry and apartness

If two points are distinct and a line U is apart from at least
 one of these points, then this line is distinct from the line
 connecting these points




$\mbox{include}(\mbox{'Axioms/GEO006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+1.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+2.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+3.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+4.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+5.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+6.ax'})$

$\forall x,y,u,v{:}\;( ( \mbox{distinct\_points}(x,y)\mbox{ and }\mbox{convergent\_lines}(u,v)\mbox{ and }( \mbox{apart\_point\_and\_line}(x,u)\mbox{ or }\mbox{apart\_point\_and\_line}(y,u) ) ) \ \Rightarrow \ \mbox{distinct\_lines}(u,\mbox{line\_connecting}(x,y)) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO213+1.p} Corollary to non-orthogonal lines and a third line

If line L is not orthogonal to line M, then a third line N
 is distinct from L or not orthogonal to M.




$\mbox{include}(\mbox{'Axioms/GEO006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+2.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+3.ax'})$

$\forall l,m,n{:}\;( \mbox{unorthogonal\_lines}(l,m) \ \Rightarrow \ ( \mbox{distinct\_lines}(l,n)\mbox{ or }\mbox{unorthogonal\_lines}(m,n) ) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO176+3.p} Lemma on symmetry and apartness

If two lines are convergent and a point is apart from at least
 one of these lines, then this point is distinct from the
 intersection point of these lines.




$\mbox{include}(\mbox{'Axioms/GEO006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+1.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+2.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+3.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+4.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+5.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+6.ax'})$

$\forall x,y,u,v{:}\;( ( \mbox{distinct\_points}(x,y)\mbox{ and }\mbox{convergent\_lines}(u,v)\mbox{ and }( \mbox{apart\_point\_and\_line}(x,u)\mbox{ or }\mbox{apart\_point\_and\_line}(x,v) ) ) \ \Rightarrow \ \mbox{distinct\_points}(x,\mbox{intersection\_point}(u,v)) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO012-1.p} Collinearity for 4 points

If any three distinct points x, y, and z are collinear and
 a fourth point w is collinear with x and y, then
 w is collinear with x and z and also with x and y.




$\mbox{include}(\mbox{'Axioms/GEO001-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO001-1.ax'})$

$a\ne b\qquad\mbox{cnf}(\mbox{a\_not\_b},\mbox{hypothesis})$

$a\ne c\qquad\mbox{cnf}(\mbox{a\_not\_c},\mbox{hypothesis})$

$b\ne c\qquad\mbox{cnf}(\mbox{b\_not\_c},\mbox{hypothesis})$

$\mbox{colinear}(a,b,c)\qquad\mbox{cnf}(\mbox{abc\_colinear},\mbox{hypothesis})$

$\mbox{colinear}(a,b,d)\qquad\mbox{cnf}(\mbox{abd\_colinear},\mbox{hypothesis})$

$\mbox{colinear}(a,c,d) \ \Rightarrow \ \neg\,\mbox{colinear}(b,c,d)\qquad\mbox{cnf}(\mbox{prove\_colinearity},\mbox{negated\_conjecture})$

\medskip\textbf{GEO075-2.p} Show reflexivity for equidistance is dependent

All of the axioms in GEO003.ax are known to be independent
 except A1 and A7. Tarski and his students have been unable
 to establish their status.




$( \mbox{equidistant}(x,y,z,v)\mbox{ and }\mbox{equidistant}(x,y,v_2,w) ) \ \Rightarrow \ \mbox{equidistant}(z,v,v_2,w)\qquad\mbox{cnf}(\mbox{transitivity\_for\_equidistance},\mbox{axiom})$

$\mbox{equidistant}(x,y,z,z) \ \Rightarrow \ x = y\qquad\mbox{cnf}(\mbox{identity\_for\_equidistance},\mbox{axiom})$

$\mbox{between}(x,y,\mbox{extension}(x,y,w,v))\qquad\mbox{cnf}(\mbox{segment\_construction}_1,\mbox{axiom})$

$\mbox{equidistant}(y,\mbox{extension}(x,y,w,v),w,v)\qquad\mbox{cnf}(\mbox{segment\_construction}_2,\mbox{axiom})$

$( \mbox{equidistant}(x,y,x_1,y_1)\mbox{ and }\mbox{equidistant}(y,z,y_1,z_1)\mbox{ and }\mbox{equidistant}(x,v,x_1,v_1)\mbox{ and }\mbox{equidistant}(y,v,y_1,v_1)\mbox{ and }\mbox{between}(x,y,z)\mbox{ and }\mbox{between}(x_1,y_1,z_1) ) \ \Rightarrow \ ( x = y\mbox{ or }\mbox{equidistant}(z,v,z_1,v_1) )\qquad\mbox{cnf}(\mbox{outer\_five\_segment},\mbox{axiom})$

$\mbox{between}(x,y,x) \ \Rightarrow \ x = y\qquad\mbox{cnf}(\mbox{identity\_for\_betweeness},\mbox{axiom})$

$( \mbox{between}(u,v,w)\mbox{ and }\mbox{between}(y,x,w) ) \ \Rightarrow \ \mbox{between}(v,\mbox{inner\_pasch}(u,v,w,x,y),y)\qquad\mbox{cnf}(\mbox{inner\_pasch}_1,\mbox{axiom})$

$( \mbox{between}(u,v,w)\mbox{ and }\mbox{between}(y,x,w) ) \ \Rightarrow \ \mbox{between}(x,\mbox{inner\_pasch}(u,v,w,x,y),u)\qquad\mbox{cnf}(\mbox{inner\_pasch}_2,\mbox{axiom})$

$\neg\,\mbox{between}(\mbox{lower\_dimension\_point}_1,\mbox{lower\_dimension\_point}_2,\mbox{lower\_dimension\_point}_3)\qquad\mbox{cnf}(\mbox{lower\_dimension}_1,\mbox{axiom})$

$\neg\,\mbox{between}(\mbox{lower\_dimension\_point}_2,\mbox{lower\_dimension\_point}_3,\mbox{lower\_dimension\_point}_1)\qquad\mbox{cnf}(\mbox{lower\_dimension}_2,\mbox{axiom})$

$\neg\,\mbox{between}(\mbox{lower\_dimension\_point}_3,\mbox{lower\_dimension\_point}_1,\mbox{lower\_dimension\_point}_2)\qquad\mbox{cnf}(\mbox{lower\_dimension}_3,\mbox{axiom})$

$( \mbox{equidistant}(x,w,x,v)\mbox{ and }\mbox{equidistant}(y,w,y,v)\mbox{ and }\mbox{equidistant}(z,w,z,v) ) \ \Rightarrow \ ( \mbox{between}(x,y,z)\mbox{ or }\mbox{between}(y,z,x)\mbox{ or }\mbox{between}(z,x,y)\mbox{ or }w = v )\qquad\mbox{cnf}(\mbox{upper\_dimension},\mbox{axiom})$

$( \mbox{between}(u,w,y)\mbox{ and }\mbox{between}(v,w,x) ) \ \Rightarrow \ ( u = w\mbox{ or }\mbox{between}(u,v,\mbox{euclid}_1(u,v,w,x,y)) )\qquad\mbox{cnf}(\mbox{euclid}_1,\mbox{axiom})$

$( \mbox{between}(u,w,y)\mbox{ and }\mbox{between}(v,w,x) ) \ \Rightarrow \ ( u = w\mbox{ or }\mbox{between}(u,x,\mbox{euclid}_2(u,v,w,x,y)) )\qquad\mbox{cnf}(\mbox{euclid}_2,\mbox{axiom})$

$( \mbox{between}(u,w,y)\mbox{ and }\mbox{between}(v,w,x) ) \ \Rightarrow \ ( u = w\mbox{ or }\mbox{between}(\mbox{euclid}_1(u,v,w,x,y),y,\mbox{euclid}_2(u,v,w,x,y)) )\qquad\mbox{cnf}(\mbox{euclid}_3,\mbox{axiom})$

$( \mbox{equidistant}(u,v,u,v_1)\mbox{ and }\mbox{equidistant}(u,x,u,x_1)\mbox{ and }\mbox{between}(u,v,x)\mbox{ and }\mbox{between}(v,w,x) ) \ \Rightarrow \ \mbox{between}(v_1,\mbox{continuous}(u,v,v_1,w,x,x_1),x_1)\qquad\mbox{cnf}(\mbox{continuity}_1,\mbox{axiom})$

$( \mbox{equidistant}(u,v,u,v_1)\mbox{ and }\mbox{equidistant}(u,x,u,x_1)\mbox{ and }\mbox{between}(u,v,x)\mbox{ and }\mbox{between}(v,w,x) ) \ \Rightarrow \ \mbox{equidistant}(u,w,u,\mbox{continuous}(u,v,v_1,w,x,x_1))\qquad\mbox{cnf}(\mbox{continuity}_2,\mbox{axiom})$

$\neg\,\mbox{equidistant}(u,v,v,u)\qquad\mbox{cnf}(\mbox{prove\_reflexivity},\mbox{negated\_conjecture})$

\medskip\textbf{GEO056-3.p} Corollary 1 to null extension






$\mbox{include}(\mbox{'Axioms/GEO002-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO002-2.ax'})$

$\mbox{equidistant}(u,v,u,v)\qquad\mbox{cnf}(d_1,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(w,x,u,v)\qquad\mbox{cnf}(d_2,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(v,u,w,x)\qquad\mbox{cnf}(d_3,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(u,v,x,w)\qquad\mbox{cnf}(\mbox{d4}_1,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(v,u,x,w)\qquad\mbox{cnf}(\mbox{d4}_2,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(w,x,v,u)\qquad\mbox{cnf}(\mbox{d4}_3,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(x,w,u,v)\qquad\mbox{cnf}(\mbox{d4}_4,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(x,w,v,u)\qquad\mbox{cnf}(\mbox{d4}_5,\mbox{axiom})$

$( \mbox{equidistant}(u,v,w,x)\mbox{ and }\mbox{equidistant}(w,x,y,z) ) \ \Rightarrow \ \mbox{equidistant}(u,v,y,z)\qquad\mbox{cnf}(d_5,\mbox{axiom})$

$v = \mbox{extension}(u,v,w,w)\qquad\mbox{cnf}(e_1,\mbox{axiom})$

$y = \mbox{extension}(u,v,w,x) \ \Rightarrow \ \mbox{between}(u,v,y)\qquad\mbox{cnf}(b_0,\mbox{axiom})$

$\mbox{between}(u,v,\mbox{reflection}(u,v))\qquad\mbox{cnf}(\mbox{r2}_1,\mbox{axiom})$

$\mbox{equidistant}(v,\mbox{reflection}(u,v),u,v)\qquad\mbox{cnf}(\mbox{r2}_2,\mbox{axiom})$

$u = v\qquad\mbox{cnf}(\mbox{u\_equals\_v},\mbox{hypothesis})$

$v\ne \mbox{reflection}(u,v)\qquad\mbox{cnf}(\mbox{prove\_v\_equals\_reflection},\mbox{negated\_conjecture})$

\medskip\textbf{GEO232+1.p} A line is not oppositely directed to itself






$\mbox{include}(\mbox{'Axioms/GEO007+0.ax'})$

$\forall l{:}\;( \mbox{line}(l) \ \Rightarrow \ \mbox{unequally\_directed\_lines}(l,\mbox{reverse\_line}(l)) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO099+1.p} Open subcurves can be complemented to form the sum

Every open sub-curve of a closed curve can be complemented by
 another curve so that their sum constitute the closed curve.




$\mbox{include}(\mbox{'Axioms/GEO004+0.ax'})$

$\forall c,c_1,p,q{:}\;( ( \mbox{closed}(c)\mbox{ and }\mbox{part\_of}(c_1,c)\mbox{ and }\mbox{end\_point}(p,c_1)\mbox{ and }\mbox{end\_point}(q,c_1)\mbox{ and }p\ne q ) \ \Rightarrow \ \exists c_2{:}\;( p\wedge c_1{=}c_2\mbox{ and }q\wedge c_1{=}c_2\mbox{ and }c = c_1 + c_2 ) )\qquad\mbox{fof}(\mbox{theorem\_2}_{17},\mbox{conjecture})$

\medskip\textbf{GEO184+3.p} Lemma on symmetry and apartness

If X and Y are distinct points, then the line that is incident
 with both points is equal to the line connecting them.




$\mbox{include}(\mbox{'Axioms/GEO006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+1.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+2.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+3.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+4.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+5.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+6.ax'})$

$\forall x,y,u,v{:}\;( ( \mbox{distinct\_points}(x,y)\mbox{ and }\mbox{convergent\_lines}(u,v)\mbox{ and }\mbox{incident\_point\_and\_line}(x,u)\mbox{ and }\mbox{incident\_point\_and\_line}(y,u) ) \ \Rightarrow \ \mbox{equal\_lines}(u,\mbox{line\_connecting}(x,y)) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO223+2.p} Corollary to uniqueness of parallels

The parallel to line L through a point A is equal to the line,
 that is orthogonal to the orthogonal to L through A, and goes
 through A as well.




$\mbox{include}(\mbox{'Axioms/GEO008+0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+2.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+3.ax'})$

$\forall a,l{:}\;\neg\,\mbox{distinct\_lines}(\mbox{parallel\_through\_point}(l,a),\mbox{orthogonal\_through\_point}(\mbox{orthogonal\_through\_point}(l,a),a))\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO258+3.p} Betweeness






$\mbox{include}(\mbox{'Axioms/GEO009+0.ax'})$

$\forall l,a,b,c{:}\;( \mbox{between\_on\_line}(l,a,b,c) \ \Rightarrow \ \mbox{between\_on\_line}(l,c,b,a) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO216+2.p} A line is not orthogonal to itself

A Line is not orthogonal to itself.




$\mbox{include}(\mbox{'Axioms/GEO008+0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+4.ax'})$

$\forall l{:}\;\neg\,\neg\,\mbox{unorthogonal\_lines}(l,l)\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO091+1.p} Two points determine subcurve

Two distinct points on an open curve uniquely determine the
 sub-curve connecting these points




$\mbox{include}(\mbox{'Axioms/GEO004+0.ax'})$

$\forall c,c_1,c_2{:}\;( ( \mbox{part\_of}(c_1,c)\mbox{ and }\mbox{part\_of}(c_2,c)\mbox{ and }\mbox{open}(c)\mbox{ and }\exists p,q{:}\;( p\ne q\mbox{ and }\mbox{end\_point}(p,c_1)\mbox{ and }\mbox{end\_point}(p,c_2)\mbox{ and }\mbox{end\_point}(q,c_1)\mbox{ and }\mbox{end\_point}(q,c_2) ) ) \ \Rightarrow \ c_1 = c_2 )\qquad\mbox{fof}(\mbox{theorem\_2}_{13},\mbox{conjecture})$

\medskip\textbf{GEO196+3.p} Symmetry of incidence

If the lines X and Y are convergent, U and V are convergent,
 and the intersection point of X and Y is incident with U and V,
 then the intersection point of U and V is incident with X and Y.




$\mbox{include}(\mbox{'Axioms/GEO006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+1.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+2.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+3.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+4.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+5.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+6.ax'})$

$\forall x,y,u,v{:}\;( ( \mbox{convergent\_lines}(x,y)\mbox{ and }\mbox{convergent\_lines}(u,v)\mbox{ and }\mbox{incident\_point\_and\_line}(\mbox{intersection\_point}(x,y),u)\mbox{ and }\mbox{incident\_point\_and\_line}(\mbox{intersection\_point}(x,y),v) ) \ \Rightarrow \ ( \mbox{incident\_point\_and\_line}(\mbox{intersection\_point}(u,v),x)\mbox{ and }\mbox{incident\_point\_and\_line}(\mbox{intersection\_point}(u,v),y) ) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO154-1.p} Colinearity axioms for the GEO001 geometry axioms






$\mbox{include}(\mbox{'Axioms/GEO001-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO001-1.ax'})$

\medskip\textbf{GEO112+1.p} Basic property of orderings on linear structures 2

If Q is between P and R wrt. c, then Q is between R and P wrt. c




$\mbox{include}(\mbox{'Axioms/GEO004+0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO004+1.ax'})$

$\forall c,p,q,r{:}\;( \mbox{between\_c}(c,p,q,r) \ \Rightarrow \ \mbox{between\_c}(c,r,q,p) )\qquad\mbox{fof}(\mbox{theorem\_3\_8}_2,\mbox{conjecture})$

\medskip\textbf{GEO242+1.p} Lemma on oriented intersection of lines with plane






$\mbox{include}(\mbox{'Axioms/GEO007+0.ax'})$

$\forall a,b,l{:}\;( ( \neg\,\mbox{apart\_point\_and\_line}(a,l)\mbox{ and }\mbox{distinct\_points}(a,b)\mbox{ and }\mbox{left\_convergent\_lines}(l,\mbox{reverse\_line}(\mbox{line\_connecting}(a,b))) ) \ \Rightarrow \ \mbox{left\_apart\_point}(b,\mbox{reverse\_line}(l)) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO103+1.p} Common endpoint of subcurves and another point means inclusion

If two sub-curves of an open curve have a common endpoint and
 another point in common, then one of the two sub-curves is
 included in the other.




$\mbox{include}(\mbox{'Axioms/GEO004+0.ax'})$

$\forall c,c_1,c_2,p,q{:}\;( ( \mbox{open}(c)\mbox{ and }\mbox{part\_of}(c_1,c)\mbox{ and }\mbox{part\_of}(c_2,c)\mbox{ and }\mbox{end\_point}(p,c_1)\mbox{ and }\mbox{end\_point}(p,c_2)\mbox{ and }p\ne q\mbox{ and }\mbox{incident\_c}(q,c_1)\mbox{ and }\mbox{incident\_c}(q,c_2) ) \ \Rightarrow \ ( \mbox{part\_of}(c_1,c_2)\mbox{ or }\mbox{part\_of}(c_2,c_1) ) )\qquad\mbox{fof}(\mbox{corollary\_2}_{21},\mbox{conjecture})$

\medskip\textbf{GEO039-3.p} Corollary the identity axiom for betweenness






$\mbox{include}(\mbox{'Axioms/GEO002-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO002-2.ax'})$

$\mbox{equidistant}(u,v,u,v)\qquad\mbox{cnf}(d_1,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(w,x,u,v)\qquad\mbox{cnf}(d_2,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(v,u,w,x)\qquad\mbox{cnf}(d_3,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(u,v,x,w)\qquad\mbox{cnf}(\mbox{d4}_1,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(v,u,x,w)\qquad\mbox{cnf}(\mbox{d4}_2,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(w,x,v,u)\qquad\mbox{cnf}(\mbox{d4}_3,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(x,w,u,v)\qquad\mbox{cnf}(\mbox{d4}_4,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(x,w,v,u)\qquad\mbox{cnf}(\mbox{d4}_5,\mbox{axiom})$

$( \mbox{equidistant}(u,v,w,x)\mbox{ and }\mbox{equidistant}(w,x,y,z) ) \ \Rightarrow \ \mbox{equidistant}(u,v,y,z)\qquad\mbox{cnf}(d_5,\mbox{axiom})$

$v = \mbox{extension}(u,v,w,w)\qquad\mbox{cnf}(e_1,\mbox{axiom})$

$y = \mbox{extension}(u,v,w,x) \ \Rightarrow \ \mbox{between}(u,v,y)\qquad\mbox{cnf}(b_0,\mbox{axiom})$

$\mbox{between}(u,v,\mbox{reflection}(u,v))\qquad\mbox{cnf}(\mbox{r2}_1,\mbox{axiom})$

$\mbox{equidistant}(v,\mbox{reflection}(u,v),u,v)\qquad\mbox{cnf}(\mbox{r2}_2,\mbox{axiom})$

$u = v \ \Rightarrow \ v = \mbox{reflection}(u,v)\qquad\mbox{cnf}(\mbox{r3}_1,\mbox{axiom})$

$u = \mbox{reflection}(u,u)\qquad\mbox{cnf}(\mbox{r3}_2,\mbox{axiom})$

$v = \mbox{reflection}(u,v) \ \Rightarrow \ u = v\qquad\mbox{cnf}(r_4,\mbox{axiom})$

$\mbox{equidistant}(u,u,v,v)\qquad\mbox{cnf}(d_7,\mbox{axiom})$

$( \mbox{equidistant}(u,v,u_1,v_1)\mbox{ and }\mbox{equidistant}(v,w,v_1,w_1)\mbox{ and }\mbox{between}(u,v,w)\mbox{ and }\mbox{between}(u_1,v_1,w_1) ) \ \Rightarrow \ \mbox{equidistant}(u,w,u_1,w_1)\qquad\mbox{cnf}(d_8,\mbox{axiom})$

$( \mbox{between}(u,v,w)\mbox{ and }\mbox{between}(u,v,x)\mbox{ and }\mbox{equidistant}(v,w,v,x) ) \ \Rightarrow \ ( u = v\mbox{ or }w = x )\qquad\mbox{cnf}(d_9,\mbox{axiom})$

$\mbox{between}(u,v,w) \ \Rightarrow \ ( u = v\mbox{ or }w = \mbox{extension}(u,v,v,w) )\qquad\mbox{cnf}(\mbox{d10}_1,\mbox{axiom})$

$\mbox{equidistant}(w,x,y,z) \ \Rightarrow \ ( \mbox{extension}(u,v,w,x) = \mbox{extension}(u,v,y,z)\mbox{ or }u = v )\qquad\mbox{cnf}(\mbox{d10}_2,\mbox{axiom})$

$\mbox{extension}(u,v,u,v) = \mbox{extension}(u,v,v,u)\mbox{ or }u = v\qquad\mbox{cnf}(\mbox{d10}_3,\mbox{axiom})$

$\mbox{equidistant}(v,u,v,\mbox{reflection}(\mbox{reflection}(u,v),v))\qquad\mbox{cnf}(r_5,\mbox{axiom})$

$u = \mbox{reflection}(\mbox{reflection}(u,v),v)\qquad\mbox{cnf}(r_6,\mbox{axiom})$

$\mbox{between}(u,v,v)\qquad\mbox{cnf}(t_3,\mbox{axiom})$

$\mbox{between}(u,w,x)\qquad\mbox{cnf}(\mbox{w\_between\_u\_and\_x},\mbox{hypothesis})$

$u = x\qquad\mbox{cnf}(\mbox{u\_is\_x},\mbox{hypothesis})$

$\neg\,\mbox{between}(v,w,x)\qquad\mbox{cnf}(\mbox{prove\_corollary},\mbox{negated\_conjecture})$

\medskip\textbf{GEO150+1.p} Objects cannot be at two places simultaneously






$\mbox{include}(\mbox{'Axioms/GEO004+0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO004+1.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO004+2.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO004+3.ax'})$

$\forall p,q,x{:}\;( \mbox{once}(\mbox{at\_the\_same\_time}(\mbox{at}(x,p),\mbox{at}(x,q))) \ \Rightarrow \ p = q )\qquad\mbox{fof}(t_{16},\mbox{conjecture})$

\medskip\textbf{GEO002-4.p} For all points x and y, x is between x and y






$( \mbox{between}(x,y,v)\mbox{ and }\mbox{between}(y,z,v) ) \ \Rightarrow \ \mbox{between}(x,y,z)\qquad\mbox{cnf}(\mbox{transitivity\_for\_betweeness},\mbox{axiom})$

$\mbox{equidistant}(x,y,z,z) \ \Rightarrow \ x{=}y\qquad\mbox{cnf}(\mbox{identity\_for\_equidistance},\mbox{axiom})$

$( \mbox{between}(x,w,v)\mbox{ and }\mbox{between}(y,v,z) ) \ \Rightarrow \ \mbox{between}(x,\mbox{outer\_pasch}(w,x,y,z,v),y)\qquad\mbox{cnf}(\mbox{outer\_pasch}_1,\mbox{axiom})$

$( \mbox{between}(x,w,v)\mbox{ and }\mbox{between}(y,v,z) ) \ \Rightarrow \ \mbox{between}(z,w,\mbox{outer\_pasch}(w,x,y,z,v))\qquad\mbox{cnf}(\mbox{outer\_pasch}_2,\mbox{axiom})$

$\mbox{between}(x,y,\mbox{extension}(x,y,w,v))\qquad\mbox{cnf}(\mbox{segment\_construction}_1,\mbox{axiom})$

$\mbox{equidistant}(y,\mbox{extension}(x,y,w,v),w,v)\qquad\mbox{cnf}(\mbox{segment\_construction}_2,\mbox{axiom})$

$( x{=}y\mbox{ and }\mbox{between}(w,z,x) ) \ \Rightarrow \ \mbox{between}(w,z,y)\qquad\mbox{cnf}(\mbox{between\_substitution}_3,\mbox{axiom})$

$\neg\,\mbox{between}(a,a,b)\qquad\mbox{cnf}(\mbox{prove\_a\_between\_a\_and\_b},\mbox{negated\_conjecture})$

\medskip\textbf{GEO011-5.p} The axiom set points are not collinear






$\mbox{include}(\mbox{'Axioms/GEO002-1.ax'})$

$\mbox{equal\_distance}(\mbox{distance}(x,y),\mbox{distance}(y,x))\qquad\mbox{cnf}(\mbox{reflexivity\_for\_equidistance},\mbox{axiom})$

$( \mbox{equal\_distance}(\mbox{distance}(x,y),\mbox{distance}(z,v))\mbox{ and }\mbox{equal\_distance}(\mbox{distance}(x,y),\mbox{distance}(v_2,w)) ) \ \Rightarrow \ \mbox{equal\_distance}(\mbox{distance}(z,v),\mbox{distance}(v_2,w))\qquad\mbox{cnf}(\mbox{transitivity\_for\_equidistance},\mbox{axiom})$

$\mbox{equal\_distance}(\mbox{distance}(x,y),\mbox{distance}(z,z)) \ \Rightarrow \ x = y\qquad\mbox{cnf}(\mbox{identity\_for\_equidistance},\mbox{axiom})$

$\mbox{between}(x,y,\mbox{extension}(x,y,w,v))\qquad\mbox{cnf}(\mbox{segment\_construction}_1,\mbox{axiom})$

$\mbox{equal\_distance}(\mbox{distance}(y,\mbox{extension}(x,y,w,v)),\mbox{distance}(w,v))\qquad\mbox{cnf}(\mbox{segment\_construction}_2,\mbox{axiom})$

$( \mbox{equal\_distance}(\mbox{distance}(x,y),\mbox{distance}(x_1,y_1))\mbox{ and }\mbox{equal\_distance}(\mbox{distance}(y,z),\mbox{distance}(y_1,z_1))\mbox{ and }\mbox{equal\_distance}(\mbox{distance}(x,v),\mbox{distance}(x_1,v_1))\mbox{ and }\mbox{equal\_distance}(\mbox{distance}(y,v),\mbox{distance}(y_1,v_1))\mbox{ and }\mbox{between}(x,y,z)\mbox{ and }\mbox{between}(x_1,y_1,z_1) ) \ \Rightarrow \ ( x = y\mbox{ or }\mbox{equal\_distance}(\mbox{distance}(z,v),\mbox{distance}(z_1,v_1)) )\qquad\mbox{cnf}(\mbox{outer\_five\_segment},\mbox{axiom})$

$\mbox{between}(x,y,x) \ \Rightarrow \ x = y\qquad\mbox{cnf}(\mbox{identity\_for\_betweeness},\mbox{axiom})$

$( \mbox{between}(u,v,w)\mbox{ and }\mbox{between}(y,x,w) ) \ \Rightarrow \ \mbox{between}(v,\mbox{inner\_pasch}(u,v,w,x,y),y)\qquad\mbox{cnf}(\mbox{inner\_pasch}_1,\mbox{axiom})$

$( \mbox{between}(u,v,w)\mbox{ and }\mbox{between}(y,x,w) ) \ \Rightarrow \ \mbox{between}(x,\mbox{inner\_pasch}(u,v,w,x,y),u)\qquad\mbox{cnf}(\mbox{inner\_pasch}_2,\mbox{axiom})$

$\neg\,\mbox{between}(\mbox{lower\_dimension\_point}_1,\mbox{lower\_dimension\_point}_2,\mbox{lower\_dimension\_point}_3)\qquad\mbox{cnf}(\mbox{lower\_dimension}_1,\mbox{axiom})$

$\neg\,\mbox{between}(\mbox{lower\_dimension\_point}_2,\mbox{lower\_dimension\_point}_3,\mbox{lower\_dimension\_point}_1)\qquad\mbox{cnf}(\mbox{lower\_dimension}_2,\mbox{axiom})$

$\neg\,\mbox{between}(\mbox{lower\_dimension\_point}_3,\mbox{lower\_dimension\_point}_1,\mbox{lower\_dimension\_point}_2)\qquad\mbox{cnf}(\mbox{lower\_dimension}_3,\mbox{axiom})$

$\mbox{colinear}(\mbox{lower\_dimension\_point}_1,\mbox{lower\_dimension\_point}_2,\mbox{lower\_dimension\_point}_3)\qquad\mbox{cnf}(\mbox{prove\_lower\_dimension\_points\_not\_colinear},\mbox{negated\_conjecture})$

\medskip\textbf{GEO207+2.p} Irreflexivity of line convergence

A line is not convergent to itself.




$\mbox{include}(\mbox{'Axioms/GEO008+0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+2.ax'})$

$\forall x{:}\;\neg\,\mbox{convergent\_lines}(x,x)\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO225+3.p} Existence of line joining distinct points

When there are two distinct points, then a line connecting
 them can be constructed.




$\mbox{include}(\mbox{'Axioms/GEO006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+1.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+2.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+3.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+4.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+5.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+6.ax'})$

$\forall a,b{:}\;( ( \mbox{point}(a)\mbox{ and }\mbox{point}(b)\mbox{ and }\mbox{distinct\_points}(a,b) ) \ \Rightarrow \ \exists x{:}\;( \mbox{line}(x) \ \Rightarrow \ ( \mbox{incident\_point\_and\_line}(a,x)\mbox{ and }\mbox{incident\_point\_and\_line}(b,x) ) ) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO113-1.p} Basic property of orderings on linear structures 3

If c is open and Q is between P and R wrt. c, then P is not
 between Q and R wrt. c




$\mbox{include}(\mbox{'Axioms/GEO004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO004-1.ax'})$

$\mbox{open}(\mbox{sk}_{15})\qquad\mbox{cnf}(\mbox{theorem\_3\_8\_3}_{77},\mbox{negated\_conjecture})$

$\mbox{between\_c}(\mbox{sk}_{15},\mbox{sk}_{16},\mbox{sk}_{17},\mbox{sk}_{18})\qquad\mbox{cnf}(\mbox{theorem\_3\_8\_3}_{78},\mbox{negated\_conjecture})$

$\mbox{between\_c}(\mbox{sk}_{15},\mbox{sk}_{17},\mbox{sk}_{16},\mbox{sk}_{18})\qquad\mbox{cnf}(\mbox{theorem\_3\_8\_3}_{79},\mbox{negated\_conjecture})$

\medskip\textbf{GEO046-3.p} Second inner transitivity property of betweenness






$\mbox{include}(\mbox{'Axioms/GEO002-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO002-2.ax'})$

$\mbox{equidistant}(u,v,u,v)\qquad\mbox{cnf}(d_1,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(w,x,u,v)\qquad\mbox{cnf}(d_2,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(v,u,w,x)\qquad\mbox{cnf}(d_3,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(u,v,x,w)\qquad\mbox{cnf}(\mbox{d4}_1,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(v,u,x,w)\qquad\mbox{cnf}(\mbox{d4}_2,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(w,x,v,u)\qquad\mbox{cnf}(\mbox{d4}_3,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(x,w,u,v)\qquad\mbox{cnf}(\mbox{d4}_4,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(x,w,v,u)\qquad\mbox{cnf}(\mbox{d4}_5,\mbox{axiom})$

$( \mbox{equidistant}(u,v,w,x)\mbox{ and }\mbox{equidistant}(w,x,y,z) ) \ \Rightarrow \ \mbox{equidistant}(u,v,y,z)\qquad\mbox{cnf}(d_5,\mbox{axiom})$

$v = \mbox{extension}(u,v,w,w)\qquad\mbox{cnf}(e_1,\mbox{axiom})$

$y = \mbox{extension}(u,v,w,x) \ \Rightarrow \ \mbox{between}(u,v,y)\qquad\mbox{cnf}(b_0,\mbox{axiom})$

$\mbox{between}(u,v,\mbox{reflection}(u,v))\qquad\mbox{cnf}(\mbox{r2}_1,\mbox{axiom})$

$\mbox{equidistant}(v,\mbox{reflection}(u,v),u,v)\qquad\mbox{cnf}(\mbox{r2}_2,\mbox{axiom})$

$u = v \ \Rightarrow \ v = \mbox{reflection}(u,v)\qquad\mbox{cnf}(\mbox{r3}_1,\mbox{axiom})$

$u = \mbox{reflection}(u,u)\qquad\mbox{cnf}(\mbox{r3}_2,\mbox{axiom})$

$v = \mbox{reflection}(u,v) \ \Rightarrow \ u = v\qquad\mbox{cnf}(r_4,\mbox{axiom})$

$\mbox{equidistant}(u,u,v,v)\qquad\mbox{cnf}(d_7,\mbox{axiom})$

$( \mbox{equidistant}(u,v,u_1,v_1)\mbox{ and }\mbox{equidistant}(v,w,v_1,w_1)\mbox{ and }\mbox{between}(u,v,w)\mbox{ and }\mbox{between}(u_1,v_1,w_1) ) \ \Rightarrow \ \mbox{equidistant}(u,w,u_1,w_1)\qquad\mbox{cnf}(d_8,\mbox{axiom})$

$( \mbox{between}(u,v,w)\mbox{ and }\mbox{between}(u,v,x)\mbox{ and }\mbox{equidistant}(v,w,v,x) ) \ \Rightarrow \ ( u = v\mbox{ or }w = x )\qquad\mbox{cnf}(d_9,\mbox{axiom})$

$\mbox{between}(u,v,w) \ \Rightarrow \ ( u = v\mbox{ or }w = \mbox{extension}(u,v,v,w) )\qquad\mbox{cnf}(\mbox{d10}_1,\mbox{axiom})$

$\mbox{equidistant}(w,x,y,z) \ \Rightarrow \ ( \mbox{extension}(u,v,w,x) = \mbox{extension}(u,v,y,z)\mbox{ or }u = v )\qquad\mbox{cnf}(\mbox{d10}_2,\mbox{axiom})$

$\mbox{extension}(u,v,u,v) = \mbox{extension}(u,v,v,u)\mbox{ or }u = v\qquad\mbox{cnf}(\mbox{d10}_3,\mbox{axiom})$

$\mbox{equidistant}(v,u,v,\mbox{reflection}(\mbox{reflection}(u,v),v))\qquad\mbox{cnf}(r_5,\mbox{axiom})$

$u = \mbox{reflection}(\mbox{reflection}(u,v),v)\qquad\mbox{cnf}(r_6,\mbox{axiom})$

$\mbox{between}(u,v,v)\qquad\mbox{cnf}(t_3,\mbox{axiom})$

$( \mbox{between}(u,w,x)\mbox{ and }u = x ) \ \Rightarrow \ \mbox{between}(v,w,x)\qquad\mbox{cnf}(b_1,\mbox{axiom})$

$\mbox{between}(u,v,w) \ \Rightarrow \ \mbox{between}(w,v,u)\qquad\mbox{cnf}(t_1,\mbox{axiom})$

$\mbox{between}(u,u,v)\qquad\mbox{cnf}(t_2,\mbox{axiom})$

$( \mbox{between}(u,v,w)\mbox{ and }\mbox{between}(v,u,w) ) \ \Rightarrow \ u = v\qquad\mbox{cnf}(b_2,\mbox{axiom})$

$( \mbox{between}(u,v,w)\mbox{ and }\mbox{between}(u,w,v) ) \ \Rightarrow \ v = w\qquad\mbox{cnf}(b_3,\mbox{axiom})$

$( \mbox{between}(u,v,w)\mbox{ and }\mbox{between}(v,u,w) ) \ \Rightarrow \ ( u = v\mbox{ or }v = w )\qquad\mbox{cnf}(\mbox{t6}_1,\mbox{axiom})$

$( \mbox{between}(u,v,w)\mbox{ and }\mbox{between}(u,w,v) ) \ \Rightarrow \ ( u = v\mbox{ or }v = w )\qquad\mbox{cnf}(\mbox{t6}_2,\mbox{axiom})$

$( \mbox{between}(u,v,w)\mbox{ and }\mbox{between}(v,w,x) ) \ \Rightarrow \ \mbox{between}(u,v,w)\qquad\mbox{cnf}(b_4,\mbox{axiom})$

$( \mbox{between}(u,v,w)\mbox{ and }\mbox{between}(u,w,x) ) \ \Rightarrow \ \mbox{between}(v,w,x)\qquad\mbox{cnf}(b_5,\mbox{axiom})$

$( \mbox{between}(u,v,w)\mbox{ and }\mbox{between}(v,w,x) ) \ \Rightarrow \ ( \mbox{between}(u,w,x)\mbox{ or }v = w )\qquad\mbox{cnf}(b_6,\mbox{axiom})$

$( \mbox{between}(u,v,w)\mbox{ and }\mbox{between}(v,w,x) ) \ \Rightarrow \ ( \mbox{between}(u,v,x)\mbox{ or }v = w )\qquad\mbox{cnf}(b_7,\mbox{axiom})$

$\mbox{between}(u,v,x)\qquad\mbox{cnf}(\mbox{v\_between\_u\_and\_x},\mbox{hypothesis})$

$\mbox{between}(v,w,x)\qquad\mbox{cnf}(\mbox{w\_between\_v\_and\_x},\mbox{hypothesis})$

$\neg\,\mbox{between}(u,w,x)\qquad\mbox{cnf}(\mbox{prove\_w\_between\_u\_and\_x},\mbox{negated\_conjecture})$

\medskip\textbf{GEO186+3.p} Lemma on symmetry and apartness

If the lines U and V are convergent, then the point that is
 incident with both lines is equal to the intersection point
 of both lines.




$\mbox{include}(\mbox{'Axioms/GEO006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+1.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+2.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+3.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+4.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+5.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+6.ax'})$

$\forall x,y,u,v{:}\;( ( \mbox{distinct\_points}(x,y)\mbox{ and }\mbox{convergent\_lines}(u,v)\mbox{ and }\mbox{incident\_point\_and\_line}(x,u)\mbox{ and }\mbox{incident\_point\_and\_line}(x,v) ) \ \Rightarrow \ \mbox{equal\_points}(x,\mbox{intersection\_point}(u,v)) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO265+3.p} Equally directed opposite and reversed lines






$\mbox{include}(\mbox{'Axioms/GEO009+0.ax'})$

$\forall l,m{:}\;( \mbox{equally\_directed\_lines}(l,\mbox{reverse\_line}(m)) \ \Rightarrow \ \mbox{equally\_directed\_opposite\_lines}(l,m) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO153-1.p} Tarski geometry axioms






$\mbox{include}(\mbox{'Axioms/GEO001-0.ax'})$

\medskip\textbf{GEO101-1.p} Intensification of GEO100+1






$\mbox{include}(\mbox{'Axioms/GEO004-0.ax'})$

$\mbox{part\_of}(\mbox{sk}_{15},\mbox{sk}_{14})\qquad\mbox{cnf}(\mbox{corollary\_2\_19}_{67},\mbox{negated\_conjecture})$

$\mbox{sk}_{15}\ne \mbox{sk}_{14}\qquad\mbox{cnf}(\mbox{corollary\_2\_19}_{68},\mbox{negated\_conjecture})$

$\mbox{open}(\mbox{sk}_{14})\qquad\mbox{cnf}(\mbox{corollary\_2\_19}_{69},\mbox{negated\_conjecture})$

$\mbox{end\_point}(\mbox{sk}_{16},\mbox{sk}_{15})\qquad\mbox{cnf}(\mbox{corollary\_2\_19}_{70},\mbox{negated\_conjecture})$

$\mbox{end\_point}(\mbox{sk}_{16},\mbox{sk}_{14})\qquad\mbox{cnf}(\mbox{corollary\_2\_19}_{71},\mbox{negated\_conjecture})$

$( a\wedge \mbox{sk}_{15}{=}b\mbox{ and }\mbox{sk}_{14} = \mbox{sk}_{15} + b\mbox{ and }\mbox{end\_point}(c,b)\mbox{ and }\mbox{end\_point}(c,\mbox{sk}_{14}) ) \ \Rightarrow \ ( \mbox{sk}_{16} = a\mbox{ or }a = c\mbox{ or }\mbox{sk}_{16} = c )\qquad\mbox{cnf}(\mbox{corollary\_2\_19}_{72},\mbox{negated\_conjecture})$

\medskip\textbf{GEO085-1.p} Every open curve has at least two endpoints






$\mbox{include}(\mbox{'Axioms/GEO004-0.ax'})$

$\mbox{open}(\mbox{sk}_{14})\qquad\mbox{cnf}(\mbox{theorem\_2\_7\_1}_{67},\mbox{negated\_conjecture})$

$( \mbox{end\_point}(a,\mbox{sk}_{14})\mbox{ and }\mbox{end\_point}(b,\mbox{sk}_{14}) ) \ \Rightarrow \ a = b\qquad\mbox{cnf}(\mbox{theorem\_2\_7\_1}_{68},\mbox{negated\_conjecture})$

\medskip\textbf{GEO117+1.p} Precedence on oriented curves is irreflexive






$\mbox{include}(\mbox{'Axioms/GEO004+0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO004+1.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO004+2.ax'})$

$\forall o,p{:}\;\neg\,\mbox{ordered\_by}(o,p,p)\qquad\mbox{fof}(\mbox{theorem\_4}_4,\mbox{conjecture})$

\medskip\textbf{GEO069-2.p} A property of collinearity






$\mbox{include}(\mbox{'Axioms/GEO002-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO002-1.ax'})$

$u\ne v\qquad\mbox{cnf}(\mbox{u\_not\_v},\mbox{hypothesis})$

$\mbox{colinear}(w,v,u)\qquad\mbox{cnf}(\mbox{wvu\_colinear},\mbox{hypothesis})$

$\mbox{colinear}(x,v,u)\qquad\mbox{cnf}(\mbox{xvu\_colinear},\mbox{hypothesis})$

$\mbox{colinear}(x,w,u) \ \Rightarrow \ \neg\,\mbox{colinear}(x,w,v)\qquad\mbox{cnf}(\mbox{prove\_xwu\_and\_xwv\_colinear},\mbox{negated\_conjecture})$

\medskip\textbf{GEO153-2.p} Tarski geometry axioms






$\mbox{include}(\mbox{'Axioms/GEO002-0.ax'})$

\medskip\textbf{GEO140-1.p} Unique oppositely oriented curve 1

For every oriented line there is exactly one uniquely determined
 oriented line with the same underlying curve that orders the
 points in the opposite way.




$\mbox{include}(\mbox{'Axioms/GEO004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO004-2.ax'})$

$\mbox{ordered\_by}(\mbox{sk}_{25},\mbox{sk}_{26},\mbox{sk}_{27})\qquad\mbox{cnf}(\mbox{theorem\_4\_23\_1}_{133},\mbox{negated\_conjecture})$

$\mbox{ordered\_by}(\mbox{sk}_{25},\mbox{sk}_{28},\mbox{sk}_{26})\mbox{ or }\mbox{between}(\mbox{sk}_{25},\mbox{sk}_{28},\mbox{sk}_{26},\mbox{sk}_{27})\qquad\mbox{cnf}(\mbox{theorem\_4\_23\_1}_{134},\mbox{negated\_conjecture})$

$\mbox{ordered\_by}(\mbox{sk}_{25},\mbox{sk}_{28},\mbox{sk}_{26}) \ \Rightarrow \ \mbox{ordered\_by}(\mbox{sk}_{25},\mbox{sk}_{28},\mbox{sk}_{26})\qquad\mbox{cnf}(\mbox{theorem\_4\_23\_1}_{135},\mbox{negated\_conjecture})$

$\mbox{between}(\mbox{sk}_{25},\mbox{sk}_{28},\mbox{sk}_{26},\mbox{sk}_{27}) \ \Rightarrow \ \mbox{between}(\mbox{sk}_{25},\mbox{sk}_{28},\mbox{sk}_{26},\mbox{sk}_{27})\qquad\mbox{cnf}(\mbox{theorem\_4\_23\_1}_{136},\mbox{negated\_conjecture})$

$\mbox{between}(\mbox{sk}_{25},\mbox{sk}_{28},\mbox{sk}_{26},\mbox{sk}_{27}) \ \Rightarrow \ \neg\,\mbox{ordered\_by}(\mbox{sk}_{25},\mbox{sk}_{28},\mbox{sk}_{26})\qquad\mbox{cnf}(\mbox{theorem\_4\_23\_1}_{137},\mbox{negated\_conjecture})$

\medskip\textbf{GEO238+1.p} Strengthened axiom of Pasch






$\mbox{include}(\mbox{'Axioms/GEO007+0.ax'})$

$\forall a,b,c,l{:}\;( ( \mbox{divides\_points}(l,a,b)\mbox{ and }\mbox{divides\_points}(l,a,c) ) \ \Rightarrow \ \neg\,\mbox{divides\_points}(l,b,c) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO001-2.p} Betweenness is symmetric in its outer arguments






$\mbox{include}(\mbox{'Axioms/GEO002-0.ax'})$

$\mbox{between}(a,b,c)\qquad\mbox{cnf}(\mbox{b\_between\_a\_and\_c},\mbox{hypothesis})$

$\neg\,\mbox{between}(c,b,a)\qquad\mbox{cnf}(\mbox{prove\_b\_between\_c\_and\_a},\mbox{negated\_conjecture})$

\medskip\textbf{GEO071-2.p} Corollary 1 to non-collinear points theorem






$\mbox{include}(\mbox{'Axioms/GEO002-0.ax'})$

$\mbox{equidistant}(u,v,w,x)\qquad\mbox{cnf}(\mbox{u\_to\_v\_equals\_w\_to\_x},\mbox{hypothesis})$

$\mbox{equidistant}(v,w,x,u)\qquad\mbox{cnf}(\mbox{v\_to\_w\_equals\_x\_to\_u},\mbox{hypothesis})$

$\mbox{equidistant}(u,w,v,x)\qquad\mbox{cnf}(\mbox{u\_to\_w\_equals\_v\_to\_x},\mbox{hypothesis})$

$\mbox{between}(u,y,w)\qquad\mbox{cnf}(\mbox{y\_between\_u\_and\_w},\mbox{hypothesis})$

$\mbox{between}(v,y,x)\qquad\mbox{cnf}(\mbox{y\_between\_v\_and\_x},\mbox{hypothesis})$

$u\ne v\qquad\mbox{cnf}(\mbox{u\_not\_v},\mbox{hypothesis})$

$x\ne u\qquad\mbox{cnf}(\mbox{x\_not\_u},\mbox{hypothesis})$

$u\ne w\qquad\mbox{cnf}(\mbox{prove\_u\_equals\_w},\mbox{negated\_conjecture})$

\medskip\textbf{GEO246+3.p} Configurations in terms of apartness






$\mbox{include}(\mbox{'Axioms/GEO009+0.ax'})$

$\forall a,b,c{:}\;( \mbox{distinct\_points}(a,b) \ \Rightarrow \ ( \mbox{right\_apart\_point}(c,\mbox{line\_connecting}(a,b)) \ \Rightarrow \ \mbox{left\_apart\_point}(b,\mbox{line\_connecting}(a,c)) ) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO214+3.p} Corollary to non-orthogonal lines and a third line

If the line L is not orthogonal to M, then M is orthogonal to L.




$\mbox{include}(\mbox{'Axioms/GEO006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+1.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+2.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+3.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+4.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+5.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+6.ax'})$

$\forall l,m{:}\;( \mbox{not\_orthogonal\_lines}(l,m) \ \Rightarrow \ \mbox{not\_orthogonal\_lines}(m,l) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO082-1.p} Antisymmetry of part\_of






$\mbox{include}(\mbox{'Axioms/GEO004-0.ax'})$

$\mbox{part\_of}(\mbox{sk}_{14},\mbox{sk}_{15})\qquad\mbox{cnf}(\mbox{part\_of\_antisymmetry}_{67},\mbox{negated\_conjecture})$

$\mbox{part\_of}(\mbox{sk}_{15},\mbox{sk}_{14})\qquad\mbox{cnf}(\mbox{part\_of\_antisymmetry}_{68},\mbox{negated\_conjecture})$

$\mbox{sk}_{14}\ne \mbox{sk}_{15}\qquad\mbox{cnf}(\mbox{part\_of\_antisymmetry}_{69},\mbox{negated\_conjecture})$

\medskip\textbf{GEO179+1.p} Lemma on symmetry and apartness

If two points X and Y are distinct and a point Z is apart from
 the line connecting X and Y, then the line connecting X and Y
 is distinct from the line connecting Z and X and distinct from
 the line connecting Z and Y.




$\mbox{include}(\mbox{'Axioms/GEO006+0.ax'})$

$\forall x,y,z{:}\;( ( \mbox{distinct\_points}(x,y)\mbox{ and }\mbox{apart\_point\_and\_line}(z,\mbox{line\_connecting}(x,y)) ) \ \Rightarrow \ ( \mbox{distinct\_lines}(\mbox{line\_connecting}(x,y),\mbox{line\_connecting}(z,x))\mbox{ and }\mbox{distinct\_lines}(\mbox{line\_connecting}(x,y),\mbox{line\_connecting}(z,y)) ) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO103-1.p} Common endpoint of subcurves and another point means inclusion

If two sub-curves of an open curve have a common endpoint and
 another point in common, then one of the two sub-curves is
 included in the other.




$\mbox{include}(\mbox{'Axioms/GEO004-0.ax'})$

$\mbox{open}(\mbox{sk}_{14})\qquad\mbox{cnf}(\mbox{corollary\_2\_21}_{67},\mbox{negated\_conjecture})$

$\mbox{part\_of}(\mbox{sk}_{15},\mbox{sk}_{14})\qquad\mbox{cnf}(\mbox{corollary\_2\_21}_{68},\mbox{negated\_conjecture})$

$\mbox{part\_of}(\mbox{sk}_{16},\mbox{sk}_{14})\qquad\mbox{cnf}(\mbox{corollary\_2\_21}_{69},\mbox{negated\_conjecture})$

$\mbox{end\_point}(\mbox{sk}_{17},\mbox{sk}_{15})\qquad\mbox{cnf}(\mbox{corollary\_2\_21}_{70},\mbox{negated\_conjecture})$

$\mbox{end\_point}(\mbox{sk}_{17},\mbox{sk}_{16})\qquad\mbox{cnf}(\mbox{corollary\_2\_21}_{71},\mbox{negated\_conjecture})$

$\mbox{sk}_{17}\ne \mbox{sk}_{18}\qquad\mbox{cnf}(\mbox{corollary\_2\_21}_{72},\mbox{negated\_conjecture})$

$\mbox{incident\_c}(\mbox{sk}_{18},\mbox{sk}_{15})\qquad\mbox{cnf}(\mbox{corollary\_2\_21}_{73},\mbox{negated\_conjecture})$

$\mbox{incident\_c}(\mbox{sk}_{18},\mbox{sk}_{16})\qquad\mbox{cnf}(\mbox{corollary\_2\_21}_{74},\mbox{negated\_conjecture})$

$\neg\,\mbox{part\_of}(\mbox{sk}_{15},\mbox{sk}_{16})\qquad\mbox{cnf}(\mbox{corollary\_2\_21}_{75},\mbox{negated\_conjecture})$

$\neg\,\mbox{part\_of}(\mbox{sk}_{16},\mbox{sk}_{15})\qquad\mbox{cnf}(\mbox{corollary\_2\_21}_{76},\mbox{negated\_conjecture})$

\medskip\textbf{GEO116-1.p} Open curve betweenness property for three points

If P, Q and R are points on an open curve c then Q is not between
 P and R wrt. c, iff P is between R and Q wrt. c or R is between
 Q and P wrt. c or at least two of the points are identical.




$\mbox{include}(\mbox{'Axioms/GEO004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO004-2.ax'})$

$\mbox{open}(\mbox{sk}_{15})\qquad\mbox{cnf}(\mbox{corolary\_3\_9}_{77},\mbox{negated\_conjecture})$

$\mbox{incident\_c}(\mbox{sk}_{16},\mbox{sk}_{15})\qquad\mbox{cnf}(\mbox{corolary\_3\_9}_{78},\mbox{negated\_conjecture})$

$\mbox{incident\_c}(\mbox{sk}_{17},\mbox{sk}_{15})\qquad\mbox{cnf}(\mbox{corolary\_3\_9}_{79},\mbox{negated\_conjecture})$

$\mbox{incident\_c}(\mbox{sk}_{18},\mbox{sk}_{15})\qquad\mbox{cnf}(\mbox{corolary\_3\_9}_{80},\mbox{negated\_conjecture})$

$\mbox{between\_c}(\mbox{sk}_{15},\mbox{sk}_{16},\mbox{sk}_{17},\mbox{sk}_{18}) \ \Rightarrow \ ( \mbox{between\_c}(\mbox{sk}_{15},\mbox{sk}_{18},\mbox{sk}_{16},\mbox{sk}_{17})\mbox{ or }\mbox{between\_c}(\mbox{sk}_{15},\mbox{sk}_{17},\mbox{sk}_{18},\mbox{sk}_{16})\mbox{ or }\mbox{sk}_{18} = \mbox{sk}_{17}\mbox{ or }\mbox{sk}_{18} = \mbox{sk}_{16}\mbox{ or }\mbox{sk}_{16} = \mbox{sk}_{17} )\qquad\mbox{cnf}(\mbox{corolary\_3\_9}_{81},\mbox{negated\_conjecture})$

$\mbox{between\_c}(\mbox{sk}_{15},\mbox{sk}_{16},\mbox{sk}_{17},\mbox{sk}_{18}) \ \Rightarrow \ \mbox{between\_c}(\mbox{sk}_{15},\mbox{sk}_{16},\mbox{sk}_{17},\mbox{sk}_{18})\qquad\mbox{cnf}(\mbox{corolary\_3\_9}_{82},\mbox{negated\_conjecture})$

$\mbox{between\_c}(\mbox{sk}_{15},\mbox{sk}_{18},\mbox{sk}_{16},\mbox{sk}_{17}) \ \Rightarrow \ ( \mbox{between\_c}(\mbox{sk}_{15},\mbox{sk}_{18},\mbox{sk}_{16},\mbox{sk}_{17})\mbox{ or }\mbox{between\_c}(\mbox{sk}_{15},\mbox{sk}_{17},\mbox{sk}_{18},\mbox{sk}_{16})\mbox{ or }\mbox{sk}_{18} = \mbox{sk}_{17}\mbox{ or }\mbox{sk}_{18} = \mbox{sk}_{16}\mbox{ or }\mbox{sk}_{16} = \mbox{sk}_{17} )\qquad\mbox{cnf}(\mbox{corolary\_3\_9}_{83},\mbox{negated\_conjecture})$

$\mbox{between\_c}(\mbox{sk}_{15},\mbox{sk}_{17},\mbox{sk}_{18},\mbox{sk}_{16}) \ \Rightarrow \ ( \mbox{between\_c}(\mbox{sk}_{15},\mbox{sk}_{18},\mbox{sk}_{16},\mbox{sk}_{17})\mbox{ or }\mbox{between\_c}(\mbox{sk}_{15},\mbox{sk}_{17},\mbox{sk}_{18},\mbox{sk}_{16})\mbox{ or }\mbox{sk}_{18} = \mbox{sk}_{17}\mbox{ or }\mbox{sk}_{18} = \mbox{sk}_{16}\mbox{ or }\mbox{sk}_{16} = \mbox{sk}_{17} )\qquad\mbox{cnf}(\mbox{corolary\_3\_9}_{84},\mbox{negated\_conjecture})$

$\mbox{sk}_{18} = \mbox{sk}_{17} \ \Rightarrow \ ( \mbox{between\_c}(\mbox{sk}_{15},\mbox{sk}_{18},\mbox{sk}_{16},\mbox{sk}_{17})\mbox{ or }\mbox{between\_c}(\mbox{sk}_{15},\mbox{sk}_{17},\mbox{sk}_{18},\mbox{sk}_{16})\mbox{ or }\mbox{sk}_{18} = \mbox{sk}_{17}\mbox{ or }\mbox{sk}_{18} = \mbox{sk}_{16}\mbox{ or }\mbox{sk}_{16} = \mbox{sk}_{17} )\qquad\mbox{cnf}(\mbox{corolary\_3\_9}_{85},\mbox{negated\_conjecture})$

$\mbox{sk}_{18} = \mbox{sk}_{16} \ \Rightarrow \ ( \mbox{between\_c}(\mbox{sk}_{15},\mbox{sk}_{18},\mbox{sk}_{16},\mbox{sk}_{17})\mbox{ or }\mbox{between\_c}(\mbox{sk}_{15},\mbox{sk}_{17},\mbox{sk}_{18},\mbox{sk}_{16})\mbox{ or }\mbox{sk}_{18} = \mbox{sk}_{17}\mbox{ or }\mbox{sk}_{18} = \mbox{sk}_{16}\mbox{ or }\mbox{sk}_{16} = \mbox{sk}_{17} )\qquad\mbox{cnf}(\mbox{corolary\_3\_9}_{86},\mbox{negated\_conjecture})$

$\mbox{sk}_{16} = \mbox{sk}_{17} \ \Rightarrow \ ( \mbox{between\_c}(\mbox{sk}_{15},\mbox{sk}_{18},\mbox{sk}_{16},\mbox{sk}_{17})\mbox{ or }\mbox{between\_c}(\mbox{sk}_{15},\mbox{sk}_{17},\mbox{sk}_{18},\mbox{sk}_{16})\mbox{ or }\mbox{sk}_{18} = \mbox{sk}_{17}\mbox{ or }\mbox{sk}_{18} = \mbox{sk}_{16}\mbox{ or }\mbox{sk}_{16} = \mbox{sk}_{17} )\qquad\mbox{cnf}(\mbox{corolary\_3\_9}_{87},\mbox{negated\_conjecture})$

$\mbox{between\_c}(\mbox{sk}_{15},\mbox{sk}_{18},\mbox{sk}_{16},\mbox{sk}_{17}) \ \Rightarrow \ \mbox{between\_c}(\mbox{sk}_{15},\mbox{sk}_{16},\mbox{sk}_{17},\mbox{sk}_{18})\qquad\mbox{cnf}(\mbox{corolary\_3\_9}_{88},\mbox{negated\_conjecture})$

$\mbox{between\_c}(\mbox{sk}_{15},\mbox{sk}_{17},\mbox{sk}_{18},\mbox{sk}_{16}) \ \Rightarrow \ \mbox{between\_c}(\mbox{sk}_{15},\mbox{sk}_{16},\mbox{sk}_{17},\mbox{sk}_{18})\qquad\mbox{cnf}(\mbox{corolary\_3\_9}_{89},\mbox{negated\_conjecture})$

$\mbox{sk}_{18} = \mbox{sk}_{17} \ \Rightarrow \ \mbox{between\_c}(\mbox{sk}_{15},\mbox{sk}_{16},\mbox{sk}_{17},\mbox{sk}_{18})\qquad\mbox{cnf}(\mbox{corolary\_3\_9}_{90},\mbox{negated\_conjecture})$

$\mbox{sk}_{18} = \mbox{sk}_{16} \ \Rightarrow \ \mbox{between\_c}(\mbox{sk}_{15},\mbox{sk}_{16},\mbox{sk}_{17},\mbox{sk}_{18})\qquad\mbox{cnf}(\mbox{corolary\_3\_9}_{91},\mbox{negated\_conjecture})$

$\mbox{sk}_{16} = \mbox{sk}_{17} \ \Rightarrow \ \mbox{between\_c}(\mbox{sk}_{15},\mbox{sk}_{16},\mbox{sk}_{17},\mbox{sk}_{18})\qquad\mbox{cnf}(\mbox{corolary\_3\_9}_{92},\mbox{negated\_conjecture})$

\medskip\textbf{GEO062-2.p} Insertion identity






$\mbox{include}(\mbox{'Axioms/GEO002-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO002-3.ax'})$

$\mbox{between}(u,v,w)\qquad\mbox{cnf}(\mbox{v\_between\_u\_and\_w},\mbox{hypothesis})$

$v\ne \mbox{insertion}(u,w,u,v)\qquad\mbox{cnf}(\mbox{prove\_v\_equals\_insertion},\mbox{negated\_conjecture})$

\medskip\textbf{GEO083+1.p} Sum is monotone, part 1






$\mbox{include}(\mbox{'Axioms/GEO004+0.ax'})$

$\forall c_1,c_2,c_3,p{:}\;( ( \mbox{part\_of}(c_2,c_3)\mbox{ and }p\wedge c_1{=}c_2\mbox{ and }p\wedge c_1{=}c_3 ) \ \Rightarrow \ \mbox{part\_of}(c_1 + c_2,c_1 + c_3) )\qquad\mbox{fof}(\mbox{corollary\_2\_6}_1,\mbox{conjecture})$

\medskip\textbf{GEO206+1.p} Point on parallel lines

If the point X is incident with the line Y, and the lines
 Y and Z are parallel, then the line Y is equal to the parallel
 of Z through point X.




$\mbox{include}(\mbox{'Axioms/GEO006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+2.ax'})$

$\forall x,y,z{:}\;( ( \neg\,\mbox{apart\_point\_and\_line}(x,y)\mbox{ and }\neg\,\mbox{convergent\_lines}(y,z) ) \ \Rightarrow \ \neg\,\mbox{distinct\_lines}(y,\mbox{parallel\_through\_point}(z,x)) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO219+3.p} Transitivity of orthogonal and parallel

If line L is orthogonal to M and a line N is parallel to L,
 then M is orthogonal to N.




$\mbox{include}(\mbox{'Axioms/GEO006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+1.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+2.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+3.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+4.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+5.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+6.ax'})$

$\forall l,m,n{:}\;( ( \mbox{orthogonal\_lines}(l,m)\mbox{ and }\mbox{parallel\_lines}(l,n) ) \ \Rightarrow \ \mbox{orthogonal\_lines}(m,n) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO222+1.p} Parallel to orthogonal to orthogonal

A line L is parallel to the line, that is orthogonal to the
 orthogonal to L through A, and goes through A as well.




$\mbox{include}(\mbox{'Axioms/GEO006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+2.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+3.ax'})$

$\forall a,l{:}\;\neg\,\mbox{convergent\_lines}(l,\mbox{orthogonal\_through\_point}(\mbox{orthogonal\_through\_point}(l,a),a))\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO045-3.p} Second outer transitivity property of betweenness






$\mbox{include}(\mbox{'Axioms/GEO002-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO002-2.ax'})$

$\mbox{equidistant}(u,v,u,v)\qquad\mbox{cnf}(d_1,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(w,x,u,v)\qquad\mbox{cnf}(d_2,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(v,u,w,x)\qquad\mbox{cnf}(d_3,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(u,v,x,w)\qquad\mbox{cnf}(\mbox{d4}_1,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(v,u,x,w)\qquad\mbox{cnf}(\mbox{d4}_2,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(w,x,v,u)\qquad\mbox{cnf}(\mbox{d4}_3,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(x,w,u,v)\qquad\mbox{cnf}(\mbox{d4}_4,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(x,w,v,u)\qquad\mbox{cnf}(\mbox{d4}_5,\mbox{axiom})$

$( \mbox{equidistant}(u,v,w,x)\mbox{ and }\mbox{equidistant}(w,x,y,z) ) \ \Rightarrow \ \mbox{equidistant}(u,v,y,z)\qquad\mbox{cnf}(d_5,\mbox{axiom})$

$v = \mbox{extension}(u,v,w,w)\qquad\mbox{cnf}(e_1,\mbox{axiom})$

$y = \mbox{extension}(u,v,w,x) \ \Rightarrow \ \mbox{between}(u,v,y)\qquad\mbox{cnf}(b_0,\mbox{axiom})$

$\mbox{between}(u,v,\mbox{reflection}(u,v))\qquad\mbox{cnf}(\mbox{r2}_1,\mbox{axiom})$

$\mbox{equidistant}(v,\mbox{reflection}(u,v),u,v)\qquad\mbox{cnf}(\mbox{r2}_2,\mbox{axiom})$

$u = v \ \Rightarrow \ v = \mbox{reflection}(u,v)\qquad\mbox{cnf}(\mbox{r3}_1,\mbox{axiom})$

$u = \mbox{reflection}(u,u)\qquad\mbox{cnf}(\mbox{r3}_2,\mbox{axiom})$

$v = \mbox{reflection}(u,v) \ \Rightarrow \ u = v\qquad\mbox{cnf}(r_4,\mbox{axiom})$

$\mbox{equidistant}(u,u,v,v)\qquad\mbox{cnf}(d_7,\mbox{axiom})$

$( \mbox{equidistant}(u,v,u_1,v_1)\mbox{ and }\mbox{equidistant}(v,w,v_1,w_1)\mbox{ and }\mbox{between}(u,v,w)\mbox{ and }\mbox{between}(u_1,v_1,w_1) ) \ \Rightarrow \ \mbox{equidistant}(u,w,u_1,w_1)\qquad\mbox{cnf}(d_8,\mbox{axiom})$

$( \mbox{between}(u,v,w)\mbox{ and }\mbox{between}(u,v,x)\mbox{ and }\mbox{equidistant}(v,w,v,x) ) \ \Rightarrow \ ( u = v\mbox{ or }w = x )\qquad\mbox{cnf}(d_9,\mbox{axiom})$

$\mbox{between}(u,v,w) \ \Rightarrow \ ( u = v\mbox{ or }w = \mbox{extension}(u,v,v,w) )\qquad\mbox{cnf}(\mbox{d10}_1,\mbox{axiom})$

$\mbox{equidistant}(w,x,y,z) \ \Rightarrow \ ( \mbox{extension}(u,v,w,x) = \mbox{extension}(u,v,y,z)\mbox{ or }u = v )\qquad\mbox{cnf}(\mbox{d10}_2,\mbox{axiom})$

$\mbox{extension}(u,v,u,v) = \mbox{extension}(u,v,v,u)\mbox{ or }u = v\qquad\mbox{cnf}(\mbox{d10}_3,\mbox{axiom})$

$\mbox{equidistant}(v,u,v,\mbox{reflection}(\mbox{reflection}(u,v),v))\qquad\mbox{cnf}(r_5,\mbox{axiom})$

$u = \mbox{reflection}(\mbox{reflection}(u,v),v)\qquad\mbox{cnf}(r_6,\mbox{axiom})$

$\mbox{between}(u,v,v)\qquad\mbox{cnf}(t_3,\mbox{axiom})$

$( \mbox{between}(u,w,x)\mbox{ and }u = x ) \ \Rightarrow \ \mbox{between}(v,w,x)\qquad\mbox{cnf}(b_1,\mbox{axiom})$

$\mbox{between}(u,v,w) \ \Rightarrow \ \mbox{between}(w,v,u)\qquad\mbox{cnf}(t_1,\mbox{axiom})$

$\mbox{between}(u,u,v)\qquad\mbox{cnf}(t_2,\mbox{axiom})$

$( \mbox{between}(u,v,w)\mbox{ and }\mbox{between}(v,u,w) ) \ \Rightarrow \ u = v\qquad\mbox{cnf}(b_2,\mbox{axiom})$

$( \mbox{between}(u,v,w)\mbox{ and }\mbox{between}(u,w,v) ) \ \Rightarrow \ v = w\qquad\mbox{cnf}(b_3,\mbox{axiom})$

$( \mbox{between}(u,v,w)\mbox{ and }\mbox{between}(v,u,w) ) \ \Rightarrow \ ( u = v\mbox{ or }v = w )\qquad\mbox{cnf}(\mbox{t6}_1,\mbox{axiom})$

$( \mbox{between}(u,v,w)\mbox{ and }\mbox{between}(u,w,v) ) \ \Rightarrow \ ( u = v\mbox{ or }v = w )\qquad\mbox{cnf}(\mbox{t6}_2,\mbox{axiom})$

$( \mbox{between}(u,v,w)\mbox{ and }\mbox{between}(v,w,x) ) \ \Rightarrow \ \mbox{between}(u,v,w)\qquad\mbox{cnf}(b_4,\mbox{axiom})$

$( \mbox{between}(u,v,w)\mbox{ and }\mbox{between}(u,w,x) ) \ \Rightarrow \ \mbox{between}(v,w,x)\qquad\mbox{cnf}(b_5,\mbox{axiom})$

$( \mbox{between}(u,v,w)\mbox{ and }\mbox{between}(v,w,x) ) \ \Rightarrow \ ( \mbox{between}(u,w,x)\mbox{ or }v = w )\qquad\mbox{cnf}(b_6,\mbox{axiom})$

$\mbox{between}(u,v,w)\qquad\mbox{cnf}(\mbox{v\_between\_u\_and\_w},\mbox{hypothesis})$

$\mbox{between}(v,w,x)\qquad\mbox{cnf}(\mbox{w\_between\_v\_and\_x},\mbox{hypothesis})$

$v\ne w\qquad\mbox{cnf}(\mbox{v\_not\_w},\mbox{hypothesis})$

$\neg\,\mbox{between}(u,v,x)\qquad\mbox{cnf}(\mbox{prove\_v\_between\_u\_and\_x},\mbox{negated\_conjecture})$

\medskip\textbf{GEO035-3.p} A null extension does not extend a line






$\mbox{include}(\mbox{'Axioms/GEO002-0.ax'})$

$\mbox{equidistant}(u,v,u,v)\qquad\mbox{cnf}(d_1,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(w,x,u,v)\qquad\mbox{cnf}(d_2,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(v,u,w,x)\qquad\mbox{cnf}(d_3,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(u,v,x,w)\qquad\mbox{cnf}(\mbox{d4}_1,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(v,u,x,w)\qquad\mbox{cnf}(\mbox{d4}_2,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(w,x,v,u)\qquad\mbox{cnf}(\mbox{d4}_3,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(x,w,u,v)\qquad\mbox{cnf}(\mbox{d4}_4,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(x,w,v,u)\qquad\mbox{cnf}(\mbox{d4}_5,\mbox{axiom})$

$( \mbox{equidistant}(u,v,w,x)\mbox{ and }\mbox{equidistant}(w,x,y,z) ) \ \Rightarrow \ \mbox{equidistant}(u,v,y,z)\qquad\mbox{cnf}(d_5,\mbox{axiom})$

$v\ne \mbox{extension}(u,v,w,w)\qquad\mbox{cnf}(\mbox{prove\_null\_extension},\mbox{negated\_conjecture})$

\medskip\textbf{GEO264+3.p} Triangle divides plane into seven regions






$\mbox{include}(\mbox{'Axioms/GEO009+0.ax'})$

$\forall a,b,c,d{:}\;( \mbox{left\_apart\_point}(c,\mbox{line\_connecting}(a,b)) \ \Rightarrow \ ( ( \mbox{right\_apart\_point}(d,\mbox{line\_connecting}(b,c))\mbox{ and }\mbox{right\_apart\_point}(d,\mbox{line\_connecting}(c,a)) ) \ \Rightarrow \ \mbox{left\_apart\_point}(d,\mbox{line\_connecting}(a,b)) ) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO178+1.p} Lemma on symmetry and apartness

If two points X and Y are distinct and a point Z is apart from
 the line connecting X and Y, then Z and X are distinct, and
 Z and Y are distinct.




$\mbox{include}(\mbox{'Axioms/GEO006+0.ax'})$

$\forall x,y,z{:}\;( ( \mbox{distinct\_points}(x,y)\mbox{ and }\mbox{apart\_point\_and\_line}(z,\mbox{line\_connecting}(x,y)) ) \ \Rightarrow \ ( \mbox{distinct\_points}(z,x)\mbox{ and }\mbox{distinct\_points}(z,y) ) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO055-3.p} Corollary 3 to the segment construction axiom






$\mbox{include}(\mbox{'Axioms/GEO002-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO002-2.ax'})$

$\mbox{equidistant}(u,v,u,v)\qquad\mbox{cnf}(d_1,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(w,x,u,v)\qquad\mbox{cnf}(d_2,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(v,u,w,x)\qquad\mbox{cnf}(d_3,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(u,v,x,w)\qquad\mbox{cnf}(\mbox{d4}_1,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(v,u,x,w)\qquad\mbox{cnf}(\mbox{d4}_2,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(w,x,v,u)\qquad\mbox{cnf}(\mbox{d4}_3,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(x,w,u,v)\qquad\mbox{cnf}(\mbox{d4}_4,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(x,w,v,u)\qquad\mbox{cnf}(\mbox{d4}_5,\mbox{axiom})$

$( \mbox{equidistant}(u,v,w,x)\mbox{ and }\mbox{equidistant}(w,x,y,z) ) \ \Rightarrow \ \mbox{equidistant}(u,v,y,z)\qquad\mbox{cnf}(d_5,\mbox{axiom})$

$v = \mbox{extension}(u,v,w,w)\qquad\mbox{cnf}(e_1,\mbox{axiom})$

$y = \mbox{extension}(u,v,w,x) \ \Rightarrow \ \mbox{between}(u,v,y)\qquad\mbox{cnf}(b_0,\mbox{axiom})$

$\neg\,\mbox{equidistant}(v,\mbox{reflection}(u,v),u,v)\qquad\mbox{cnf}(\mbox{prove\_equidistance},\mbox{negated\_conjecture})$

\medskip\textbf{GEO241+3.p} Lemma on oriented intersection of lines with plane






$\mbox{include}(\mbox{'Axioms/GEO009+0.ax'})$

$\forall a,b,l{:}\;( ( \mbox{incident\_point\_and\_line}(a,l)\mbox{ and }\mbox{distinct\_points}(a,b)\mbox{ and }\mbox{left\_convergent\_lines}(l,\mbox{line\_connecting}(a,b)) ) \ \Rightarrow \ \mbox{left\_apart\_point}(b,l) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO031-2.p} Second inner five-segment theorem






$\mbox{include}(\mbox{'Axioms/GEO002-0.ax'})$

$\mbox{equidistant}(u,v,u_1,v_1)\qquad\mbox{cnf}(\mbox{u\_to\_v\_equals\_u1\_to\_v}_1,\mbox{hypothesis})$

$\mbox{equidistant}(u,w,u_1,w_1)\qquad\mbox{cnf}(\mbox{u\_to\_w\_equals\_u1\_to\_w}_1,\mbox{hypothesis})$

$\mbox{equidistant}(u,x,u_1,x_1)\qquad\mbox{cnf}(\mbox{u\_to\_x\_equals\_u1\_to\_x}_1,\mbox{hypothesis})$

$\mbox{equidistant}(w,x,w_1,x_1)\qquad\mbox{cnf}(\mbox{w\_to\_x\_equals\_w1\_to\_x}_1,\mbox{hypothesis})$

$\mbox{between}(u,v,w)\qquad\mbox{cnf}(\mbox{v\_between\_u\_and\_w},\mbox{hypothesis})$

$\mbox{between}(u_1,v_1,w_1)\qquad\mbox{cnf}(\mbox{v1\_between\_u1\_and\_w}_1,\mbox{hypothesis})$

$\neg\,\mbox{equidistant}(v,x,v_1,x_1)\qquad\mbox{cnf}(\mbox{prove\_v\_to\_x\_equals\_v1\_to\_x}_1,\mbox{negated\_conjecture})$

\medskip\textbf{GEO102+1.p} Common endpoint of subcurves means inclusion

If two sub-curves of one curve have a common endpoint and
 include a sub-curve starting at this endpoint, then one of the
 two sub-curves is included in the other.




$\mbox{include}(\mbox{'Axioms/GEO004+0.ax'})$

$\forall c_1,c_2{:}\;( ( \exists c_3,p{:}\;( \mbox{part\_of}(c_3,c_1)\mbox{ and }\mbox{part\_of}(c_3,c_2)\mbox{ and }\mbox{end\_point}(p,c_1)\mbox{ and }\mbox{end\_point}(p,c_2)\mbox{ and }\mbox{end\_point}(p,c_3) )\mbox{ and }\exists c{:}\;( \mbox{part\_of}(c_1,c)\mbox{ and }\mbox{part\_of}(c_2,c) ) ) \ \Rightarrow \ ( \mbox{part\_of}(c_1,c_2)\mbox{ or }\mbox{part\_of}(c_2,c_1) ) )\qquad\mbox{fof}(\mbox{theorem\_2}_{20},\mbox{conjecture})$

\medskip\textbf{GEO198+1.p} Corollary to symmetry of incidence

If the lines X, Y, and Z are pairwise convergent, and the
 intersection point of X and Y is incident with Z, then the
 intersection point of X and Z is incident with Y.




$\mbox{include}(\mbox{'Axioms/GEO006+0.ax'})$

$\forall x,y,z{:}\;( ( \mbox{convergent\_lines}(x,y)\mbox{ and }\mbox{convergent\_lines}(z,y)\mbox{ and }\mbox{convergent\_lines}(x,z)\mbox{ and }\neg\,\mbox{apart\_point\_and\_line}(\mbox{intersection\_point}(x,y),z) ) \ \Rightarrow \ \neg\,\mbox{apart\_point\_and\_line}(\mbox{intersection\_point}(x,z),y) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO106-1.p} Two common endpoints means identical or sum to whole

If two sub-curves have two common endpoints then they are
 identical or their sum is the whole curve.




$\mbox{include}(\mbox{'Axioms/GEO004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO004-1.ax'})$

$\mbox{end\_point}(\mbox{sk}_{17},\mbox{sk}_{15})\qquad\mbox{cnf}(\mbox{theorem\_2\_24}_{67},\mbox{negated\_conjecture})$

$\mbox{end\_point}(\mbox{sk}_{18},\mbox{sk}_{15})\qquad\mbox{cnf}(\mbox{theorem\_2\_24}_{68},\mbox{negated\_conjecture})$

$\mbox{end\_point}(\mbox{sk}_{17},\mbox{sk}_{16})\qquad\mbox{cnf}(\mbox{theorem\_2\_24}_{69},\mbox{negated\_conjecture})$

$\mbox{end\_point}(\mbox{sk}_{18},\mbox{sk}_{16})\qquad\mbox{cnf}(\mbox{theorem\_2\_24}_{70},\mbox{negated\_conjecture})$

$\mbox{sk}_{17}\ne \mbox{sk}_{18}\qquad\mbox{cnf}(\mbox{theorem\_2\_24}_{71},\mbox{negated\_conjecture})$

$\mbox{closed}(\mbox{sk}_{14})\qquad\mbox{cnf}(\mbox{theorem\_2\_24}_{72},\mbox{negated\_conjecture})$

$\mbox{part\_of}(\mbox{sk}_{15},\mbox{sk}_{14})\qquad\mbox{cnf}(\mbox{theorem\_2\_24}_{73},\mbox{negated\_conjecture})$

$\mbox{part\_of}(\mbox{sk}_{16},\mbox{sk}_{14})\qquad\mbox{cnf}(\mbox{theorem\_2\_24}_{74},\mbox{negated\_conjecture})$

$\mbox{sk}_{15}\ne \mbox{sk}_{16}\qquad\mbox{cnf}(\mbox{theorem\_2\_24}_{75},\mbox{negated\_conjecture})$

$\mbox{sk}_{14}\ne \mbox{sk}_{15} + \mbox{sk}_{16}\qquad\mbox{cnf}(\mbox{theorem\_2\_24}_{76},\mbox{negated\_conjecture})$

\medskip\textbf{GEO258+1.p} Betweeness






$\mbox{include}(\mbox{'Axioms/GEO007+0.ax'})$

$\forall l,a,b,c{:}\;( \mbox{between\_on\_line}(l,a,b,c) \ \Rightarrow \ \mbox{between\_on\_line}(l,c,b,a) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO116+1.p} Open curve betweenness property for three points

If P, Q and R are points on an open curve c then Q is not between
 P and R wrt. c, iff P is between R and Q wrt. c or R is between
 Q and P wrt. c or at least two of the points are identical.




$\mbox{include}(\mbox{'Axioms/GEO004+0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO004+1.ax'})$

$\forall c,p,q,r{:}\;( ( \mbox{open}(c)\mbox{ and }\mbox{incident\_c}(p,c)\mbox{ and }\mbox{incident\_c}(q,c)\mbox{ and }\mbox{incident\_c}(r,c) ) \ \Rightarrow \ ( \neg\,\mbox{between\_c}(c,p,q,r)\iff ( \mbox{between\_c}(c,r,p,q)\mbox{ or }\mbox{between\_c}(c,q,r,p)\mbox{ or }r = q\mbox{ or }r = p\mbox{ or }p = q ) ) )\qquad\mbox{fof}(\mbox{corolary\_3}_9,\mbox{conjecture})$

\medskip\textbf{GEO204+3.p} Distinct points and equal lines

If the points X and Y are distinct, and the points Y and Z are
 equal, then X and Z are distinct, and the line connecting X
 and Y is equivalent to the line connecting X and Z.




$\mbox{include}(\mbox{'Axioms/GEO006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+1.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+2.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+3.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+4.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+5.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+6.ax'})$

$\forall x,y,z{:}\;( ( \mbox{distinct\_points}(x,y)\mbox{ and }\mbox{equal\_points}(y,z) ) \ \Rightarrow \ ( \mbox{distinct\_points}(x,z)\mbox{ and }\mbox{equal\_lines}(\mbox{line\_connecting}(x,y),\mbox{line\_connecting}(x,z)) ) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO209+3.p} Pont and three parallel lines

If the point A is apart from the line L, but incident with the
 lines M and N, and L is parallel to M and N, then M and N are
 equal.




$\mbox{include}(\mbox{'Axioms/GEO006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+1.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+2.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+3.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+4.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+5.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+6.ax'})$

$\forall a,l,m,n{:}\;( ( \mbox{apart\_point\_and\_line}(a,l)\mbox{ and }\mbox{incident\_point\_and\_line}(a,m)\mbox{ and }\mbox{incident\_point\_and\_line}(a,n)\mbox{ and }\mbox{parallel\_lines}(m,l)\mbox{ and }\mbox{parallel\_lines}(n,l) ) \ \Rightarrow \ \mbox{equal\_lines}(m,n) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO104-1.p} Subcurves with common endpoint meet or include

If two sub-curves of a given open curve have a common endpoint
 then the sub-curves meet or one is included in the other.




$\mbox{include}(\mbox{'Axioms/GEO004-0.ax'})$

$\mbox{end\_point}(\mbox{sk}_{17},\mbox{sk}_{15})\qquad\mbox{cnf}(\mbox{theorem\_2\_22}_{67},\mbox{negated\_conjecture})$

$\mbox{end\_point}(\mbox{sk}_{17},\mbox{sk}_{16})\qquad\mbox{cnf}(\mbox{theorem\_2\_22}_{68},\mbox{negated\_conjecture})$

$\mbox{part\_of}(\mbox{sk}_{15},\mbox{sk}_{14})\qquad\mbox{cnf}(\mbox{theorem\_2\_22}_{69},\mbox{negated\_conjecture})$

$\mbox{part\_of}(\mbox{sk}_{16},\mbox{sk}_{14})\qquad\mbox{cnf}(\mbox{theorem\_2\_22}_{70},\mbox{negated\_conjecture})$

$\mbox{open}(\mbox{sk}_{14})\qquad\mbox{cnf}(\mbox{theorem\_2\_22}_{71},\mbox{negated\_conjecture})$

$\neg\,\mbox{sk}_{17}\wedge \mbox{sk}_{15}{=}\mbox{sk}_{16}\qquad\mbox{cnf}(\mbox{theorem\_2\_22}_{72},\mbox{negated\_conjecture})$

$\neg\,\mbox{part\_of}(\mbox{sk}_{15},\mbox{sk}_{16})\qquad\mbox{cnf}(\mbox{theorem\_2\_22}_{73},\mbox{negated\_conjecture})$

$\neg\,\mbox{part\_of}(\mbox{sk}_{16},\mbox{sk}_{15})\qquad\mbox{cnf}(\mbox{theorem\_2\_22}_{74},\mbox{negated\_conjecture})$

\medskip\textbf{GEO226+2.p} Existence of point incident to line

Assume orthogonal geometry. Given a point and a line, to find
 a point incident with the line.




$\mbox{include}(\mbox{'Axioms/GEO008+0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+5.ax'})$

$\forall l,m{:}\;( ( \mbox{line}(l)\mbox{ and }\mbox{line}(m)\mbox{ and }\mbox{convergent\_lines}(l,m) ) \ \Rightarrow \ \exists x{:}\;( \mbox{point}(x) \ \Rightarrow \ ( \neg\,\mbox{apart\_point\_and\_line}(x,l)\mbox{ and }\neg\,\mbox{apart\_point\_and\_line}(x,m) ) ) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO221+1.p} Lemma on orthogonality

If a point B is incident with the orthogonal to a line L
 through point A, then this orthogonal is equal to the
 orthogonal to L through B.




$\mbox{include}(\mbox{'Axioms/GEO006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+2.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+3.ax'})$

$\forall a,b,l{:}\;( \neg\,\mbox{apart\_point\_and\_line}(b,\mbox{orthogonal\_through\_point}(l,a)) \ \Rightarrow \ \neg\,\mbox{distinct\_lines}(\mbox{orthogonal\_through\_point}(l,a),\mbox{orthogonal\_through\_point}(l,b)) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO097-1.p} A subcurves connects any two points on a curve

For any two points on a curve there is a sub-curve that connects
 these two points, that is to say these points are the endpoints
 of the sub-curve.




$\mbox{include}(\mbox{'Axioms/GEO004-0.ax'})$

$\mbox{sk}_{14}\ne \mbox{sk}_{15}\qquad\mbox{cnf}(\mbox{theorem\_2\_15}_{67},\mbox{negated\_conjecture})$

$\mbox{incident\_c}(\mbox{sk}_{14},\mbox{sk}_{16})\qquad\mbox{cnf}(\mbox{theorem\_2\_15}_{68},\mbox{negated\_conjecture})$

$\mbox{incident\_c}(\mbox{sk}_{15},\mbox{sk}_{16})\qquad\mbox{cnf}(\mbox{theorem\_2\_15}_{69},\mbox{negated\_conjecture})$

$( \mbox{part\_of}(a,\mbox{sk}_{16})\mbox{ and }\mbox{end\_point}(\mbox{sk}_{14},a) ) \ \Rightarrow \ \neg\,\mbox{end\_point}(\mbox{sk}_{15},a)\qquad\mbox{cnf}(\mbox{theorem\_2\_15}_{70},\mbox{negated\_conjecture})$

\medskip\textbf{GEO152+1.p} Ordered meeting places

If three objects meet in pairs such that the meeting place of
 x and z precedes that of x and y on the trajectory of x and the
 meeting place of x and y precedes that of y and z on t(y), then
 the meet-ing place of y and z does not precede that of x and z
 on t(z)




$\mbox{include}(\mbox{'Axioms/GEO004+0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO004+1.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO004+2.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO004+3.ax'})$

$\forall p,q,r,x,y,z{:}\;( ( \mbox{connect}(x,z,p)\mbox{ and }\mbox{connect}(x,y,q)\mbox{ and }\mbox{connect}(y,z,r)\mbox{ and }\mbox{ordered\_by}(\mbox{trajectory\_of}(x),p,q)\mbox{ and }\mbox{ordered\_by}(\mbox{trajectory\_of}(y),q,r) ) \ \Rightarrow \ \neg\,\mbox{ordered\_by}(\mbox{trajectory\_of}(z),r,p) )\qquad\mbox{fof}(t_{18},\mbox{conjecture})$

\medskip\textbf{GEO235+3.p} Left and right apart leads to distinctness of points






$\mbox{include}(\mbox{'Axioms/GEO009+0.ax'})$

$\forall a,b,l{:}\;( ( \mbox{left\_apart\_point}(a,l)\mbox{ and }\mbox{right\_apart\_point}(b,l) ) \ \Rightarrow \ \mbox{distinct\_points}(a,b) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO182+2.p} Triangle axiom 3

If X and Y are distinct points and Z is apart from the line
 connecting X and Y, then Z is apart from the line connecting
 Y and X.




$\mbox{include}(\mbox{'Axioms/GEO008+0.ax'})$

$\forall x,y,z{:}\;( \mbox{distinct\_points}(x,y) \ \Rightarrow \ ( \mbox{apart\_point\_and\_line}(z,\mbox{line\_connecting}(x,y)) \ \Rightarrow \ \mbox{apart\_point\_and\_line}(z,\mbox{line\_connecting}(y,x)) ) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO126+1.p} Every oriented curve orders some points






$\mbox{include}(\mbox{'Axioms/GEO004+0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO004+1.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO004+2.ax'})$

$\forall o{:}\;\exists p,q{:}\;( \mbox{ordered\_by}(o,p,q)\mbox{ and }p\ne q )\qquad\mbox{fof}(\mbox{theorem\_4}_{11},\mbox{conjecture})$

\medskip\textbf{GEO032-2.p} Equal difference between pairs of equal length line segments






$\mbox{include}(\mbox{'Axioms/GEO002-0.ax'})$

$\mbox{between}(u,v,w)\qquad\mbox{cnf}(\mbox{v\_between\_u\_and\_w},\mbox{hypothesis})$

$\mbox{between}(u_1,v_1,w_1)\qquad\mbox{cnf}(\mbox{v1\_between\_u1\_and\_w}_1,\mbox{hypothesis})$

$\mbox{equidistant}(u,v,u_1,v_1)\qquad\mbox{cnf}(\mbox{u\_to\_v\_equals\_u1\_to\_v}_1,\mbox{hypothesis})$

$\mbox{equidistant}(u,w,u_1,w_1)\qquad\mbox{cnf}(\mbox{u\_to\_w\_equals\_u1\_to\_w}_1,\mbox{hypothesis})$

$\neg\,\mbox{equidistant}(v,w,v_1,w_1)\qquad\mbox{cnf}(\mbox{v\_to\_w\_equals\_v1\_to\_w}_1,\mbox{negated\_conjecture})$

\medskip\textbf{GEO107+1.p} Equivalence of betweenness definitions 1 and 2






$\mbox{include}(\mbox{'Axioms/GEO004+0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO004+1.ax'})$

$\forall c,p,q,r{:}\;( \mbox{between\_c}_2(c,p,q,r)\iff ( p\ne q\mbox{ and }p\ne r\mbox{ and }q\ne r\mbox{ and }\exists c_1,c_2{:}\;( q\wedge c_1{=}c_2\mbox{ and }\mbox{part\_of}(c_1,c)\mbox{ and }\mbox{part\_of}(c_2,c)\mbox{ and }\mbox{end\_point}(p,c_1)\mbox{ and }\mbox{end\_point}(r,c_2) ) ) )\qquad\mbox{fof}(\mbox{between\_c2\_defn},\mbox{hypothesis})$

$\forall c,p,q,r{:}\;( \mbox{between\_c}(c,p,q,r)\iff \mbox{between\_c}_2(c,p,q,r) )\qquad\mbox{fof}(\mbox{theorem\_3}_3,\mbox{conjecture})$

\medskip\textbf{GEO114-1.p} Basic property of orderings on linear structures 4

If P, Q and R are distinct and on c then one of the points is
 between the others wrt. c.




$\mbox{include}(\mbox{'Axioms/GEO004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO004-1.ax'})$

$\mbox{incident\_c}(\mbox{sk}_{16},\mbox{sk}_{15})\qquad\mbox{cnf}(\mbox{theorem\_3\_8\_4}_{77},\mbox{negated\_conjecture})$

$\mbox{incident\_c}(\mbox{sk}_{17},\mbox{sk}_{15})\qquad\mbox{cnf}(\mbox{theorem\_3\_8\_4}_{78},\mbox{negated\_conjecture})$

$\mbox{incident\_c}(\mbox{sk}_{18},\mbox{sk}_{15})\qquad\mbox{cnf}(\mbox{theorem\_3\_8\_4}_{79},\mbox{negated\_conjecture})$

$\mbox{sk}_{16}\ne \mbox{sk}_{17}\qquad\mbox{cnf}(\mbox{theorem\_3\_8\_4}_{80},\mbox{negated\_conjecture})$

$\mbox{sk}_{17}\ne \mbox{sk}_{18}\qquad\mbox{cnf}(\mbox{theorem\_3\_8\_4}_{81},\mbox{negated\_conjecture})$

$\mbox{sk}_{16}\ne \mbox{sk}_{18}\qquad\mbox{cnf}(\mbox{theorem\_3\_8\_4}_{82},\mbox{negated\_conjecture})$

$\neg\,\mbox{between\_c}(\mbox{sk}_{15},\mbox{sk}_{16},\mbox{sk}_{17},\mbox{sk}_{18})\qquad\mbox{cnf}(\mbox{theorem\_3\_8\_4}_{83},\mbox{negated\_conjecture})$

$\neg\,\mbox{between\_c}(\mbox{sk}_{15},\mbox{sk}_{17},\mbox{sk}_{16},\mbox{sk}_{18})\qquad\mbox{cnf}(\mbox{theorem\_3\_8\_4}_{84},\mbox{negated\_conjecture})$

$\neg\,\mbox{between\_c}(\mbox{sk}_{15},\mbox{sk}_{16},\mbox{sk}_{18},\mbox{sk}_{17})\qquad\mbox{cnf}(\mbox{theorem\_3\_8\_4}_{85},\mbox{negated\_conjecture})$

\medskip\textbf{GEO250+1.p} A point in each region formed by parallel lines






$\mbox{include}(\mbox{'Axioms/GEO007+0.ax'})$

$\forall a,b,l{:}\;( \mbox{left\_apart\_point}(b,\mbox{parallel\_through\_point}(l,a)) \ \Rightarrow \ \mbox{left\_apart\_point}(a,\mbox{reverse\_line}(\mbox{parallel\_through\_point}(l,b))) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO043-2.p} Corollary to first inner transitivity property of betweenness

Corollary of first inner transitivity property
 of betweenness, using symmetry.




$\mbox{include}(\mbox{'Axioms/GEO002-0.ax'})$

$\mbox{between}(u,v,w)\qquad\mbox{cnf}(\mbox{v\_between\_u\_and\_w},\mbox{hypothesis})$

$\mbox{between}(u,w,x)\qquad\mbox{cnf}(\mbox{w\_between\_u\_and\_x},\mbox{hypothesis})$

$\neg\,\mbox{between}(v,w,x)\qquad\mbox{cnf}(\mbox{prove\_w\_between\_v\_and\_x},\mbox{negated\_conjecture})$

\medskip\textbf{GEO036-2.p} The 3 axiom set points are distinct






$\mbox{include}(\mbox{'Axioms/GEO002-0.ax'})$

$\mbox{lower\_dimension\_point}_1 = \mbox{lower\_dimension\_point}_2\mbox{ or }\mbox{lower\_dimension\_point}_2 = \mbox{lower\_dimension\_point}_3\mbox{ or }\mbox{lower\_dimension\_point}_1 = \mbox{lower\_dimension\_point}_3\qquad\mbox{cnf}(\mbox{prove\_axioms\_points\_are\_distinct},\mbox{negated\_conjecture})$

\medskip\textbf{GEO192+1.p} Lemma on symmetry and apartness

If the lines X and Y are convergent, and the intersection
 point of X and Y is apart from a line Z, then Z is distinct
 from X and Y.




$\mbox{include}(\mbox{'Axioms/GEO006+0.ax'})$

$\forall x,y,z{:}\;( \mbox{convergent\_lines}(x,y) \ \Rightarrow \ ( \mbox{apart\_point\_and\_line}(\mbox{intersection\_point}(x,y),z) \ \Rightarrow \ ( \mbox{distinct\_lines}(x,z)\mbox{ and }\mbox{distinct\_lines}(y,z) ) ) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO130+1.p} Betweenness and precedence for three points






$\mbox{include}(\mbox{'Axioms/GEO004+0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO004+1.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO004+2.ax'})$

$\forall o,p,q,r{:}\;( \mbox{between}(o,p,q,r) \ \Rightarrow \ ( \mbox{ordered\_by}(o,p,q)\iff \mbox{ordered\_by}(o,q,r) ) )\qquad\mbox{fof}(\mbox{theorem\_4}_{15},\mbox{conjecture})$

\medskip\textbf{GEO148+1.p} No meeting if someone has already passed

A point can only be a meeting point of two moving objects if
 it is not the case that one object already passed through it
 when the other object was still moving towards it




$\mbox{include}(\mbox{'Axioms/GEO004+0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO004+1.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO004+2.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO004+3.ax'})$

$\forall p,x,y{:}\;( \mbox{connect}(x,y,p) \ \Rightarrow \ \forall q_1,q_2{:}\;( ( \mbox{ordered\_by}(\mbox{trajectory\_of}(y),q_2,p)\mbox{ and }\mbox{ordered\_by}(\mbox{trajectory\_of}(x),p,q_1) ) \ \Rightarrow \ \neg\,\mbox{once}(\mbox{at\_the\_same\_time}(\mbox{at}(x,q_1),\mbox{at}(y,q_2))) ) )\qquad\mbox{fof}(t_{14},\mbox{conjecture})$

\medskip\textbf{GEO174+2.p} Lemma on symmetry and apartness

If two points are distinct and a line U is apart from at least
 one of these points, then this line is distinct from the line
 connecting these points




$\mbox{include}(\mbox{'Axioms/GEO008+0.ax'})$

$\forall x,y,u,v{:}\;( ( \mbox{distinct\_points}(x,y)\mbox{ and }\mbox{convergent\_lines}(u,v)\mbox{ and }( \mbox{apart\_point\_and\_line}(x,u)\mbox{ or }\mbox{apart\_point\_and\_line}(y,u) ) ) \ \Rightarrow \ \mbox{distinct\_lines}(u,\mbox{line\_connecting}(x,y)) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO171+1.p} Two convergent lines are distinct






$\mbox{include}(\mbox{'Axioms/GEO006+0.ax'})$

$\forall x,y{:}\;( \mbox{convergent\_lines}(x,y) \ \Rightarrow \ \mbox{distinct\_lines}(x,y) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO264+1.p} Traingle divides plane into seven regions






$\mbox{include}(\mbox{'Axioms/GEO007+0.ax'})$

$\forall a,b,c,d{:}\;( \mbox{left\_apart\_point}(c,\mbox{line\_connecting}(a,b)) \ \Rightarrow \ ( ( \mbox{left\_apart\_point}(d,\mbox{reverse\_line}(\mbox{line\_connecting}(b,c)))\mbox{ and }\mbox{left\_apart\_point}(d,\mbox{reverse\_line}(\mbox{line\_connecting}(c,a))) ) \ \Rightarrow \ \mbox{left\_apart\_point}(d,\mbox{line\_connecting}(a,b)) ) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO169+2.p} Reduction of 8 cases to 2 in Cronheim's proof of Hessenberg






$\forall a{:}\;( ( \mbox{incident}(p_3,a)\mbox{ and }\mbox{incident}(p_1,a)\mbox{ and }\mbox{incident}(p_2,a) ) \ \Rightarrow \ \mbox{goal} )\qquad\mbox{fof}(\mbox{goal\_normal},\mbox{axiom})$

$( \mbox{incident}(a_1,\mbox{b2b}_3)\mbox{ and }\mbox{incident}(a_2,\mbox{b3b}_1)\mbox{ and }\mbox{incident}(a_3,\mbox{b1b}_2) ) \ \Rightarrow \ \mbox{goal}\qquad\mbox{fof}(\mbox{t\_a\_in\_b},\mbox{axiom})$

$( \mbox{incident}(b_1,\mbox{a2a}_3)\mbox{ and }\mbox{incident}(b_2,\mbox{a3a}_1)\mbox{ and }\mbox{incident}(b_3,\mbox{a1a}_2) ) \ \Rightarrow \ \mbox{goal}\qquad\mbox{fof}(\mbox{t\_b\_in\_a},\mbox{axiom})$

$\mbox{incident}(a_1,\mbox{b2b}_3)\mbox{ or }\mbox{incident}(b_3,\mbox{a1a}_2)\qquad\mbox{fof}(\mbox{gap}_1,\mbox{axiom})$

$\mbox{incident}(a_2,\mbox{b3b}_1)\mbox{ or }\mbox{incident}(b_1,\mbox{a2a}_3)\qquad\mbox{fof}(\mbox{gap}_2,\mbox{axiom})$

$\mbox{incident}(a_3,\mbox{b1b}_2)\mbox{ or }\mbox{incident}(b_2,\mbox{a3a}_1)\qquad\mbox{fof}(\mbox{gap}_3,\mbox{axiom})$

$\mbox{incident}(a_1,\mbox{a1a}_2)\qquad\mbox{fof}(\mbox{ia1a}_2,\mbox{axiom})$

$\mbox{incident}(a_2,\mbox{a1a}_2)\qquad\mbox{fof}(\mbox{ia2a}_1,\mbox{axiom})$

$\mbox{incident}(a_2,\mbox{a2a}_3)\qquad\mbox{fof}(\mbox{ia2a}_3,\mbox{axiom})$

$\mbox{incident}(a_3,\mbox{a2a}_3)\qquad\mbox{fof}(\mbox{ia3a}_2,\mbox{axiom})$

$\mbox{incident}(a_3,\mbox{a3a}_1)\qquad\mbox{fof}(\mbox{ia3a}_1,\mbox{axiom})$

$\mbox{incident}(a_1,\mbox{a3a}_1)\qquad\mbox{fof}(\mbox{ia1a}_3,\mbox{axiom})$

$\mbox{incident}(b_1,\mbox{b1b}_2)\qquad\mbox{fof}(\mbox{ib1b}_2,\mbox{axiom})$

$\mbox{incident}(b_2,\mbox{b1b}_2)\qquad\mbox{fof}(\mbox{ib2b}_1,\mbox{axiom})$

$\mbox{incident}(b_2,\mbox{b2b}_3)\qquad\mbox{fof}(\mbox{ib2b}_3,\mbox{axiom})$

$\mbox{incident}(b_3,\mbox{b2b}_3)\qquad\mbox{fof}(\mbox{ib3b}_2,\mbox{axiom})$

$\mbox{incident}(b_3,\mbox{b3b}_1)\qquad\mbox{fof}(\mbox{ib3b}_1,\mbox{axiom})$

$\mbox{incident}(b_1,\mbox{b3b}_1)\qquad\mbox{fof}(\mbox{ib1b}_3,\mbox{axiom})$

$\mbox{incident}(s,s_1)\qquad\mbox{fof}(\mbox{iss}_1,\mbox{axiom})$

$\mbox{incident}(s,s_2)\qquad\mbox{fof}(\mbox{iss}_2,\mbox{axiom})$

$\mbox{incident}(s,s_3)\qquad\mbox{fof}(\mbox{iss}_3,\mbox{axiom})$

$\mbox{incident}(a_1,s_1)\qquad\mbox{fof}(\mbox{ia1s}_1,\mbox{axiom})$

$\mbox{incident}(a_2,s_2)\qquad\mbox{fof}(\mbox{ia2s}_2,\mbox{axiom})$

$\mbox{incident}(a_3,s_3)\qquad\mbox{fof}(\mbox{ia3s}_3,\mbox{axiom})$

$\mbox{incident}(b_1,s_1)\qquad\mbox{fof}(\mbox{ib1s}_1,\mbox{axiom})$

$\mbox{incident}(b_2,s_2)\qquad\mbox{fof}(\mbox{ib2s}_2,\mbox{axiom})$

$\mbox{incident}(b_3,s_3)\qquad\mbox{fof}(\mbox{ib3s}_3,\mbox{axiom})$

$\mbox{incident}(p_3,\mbox{a1a}_2)\qquad\mbox{fof}(\mbox{ip3a},\mbox{axiom})$

$\mbox{incident}(p_3,\mbox{b1b}_2)\qquad\mbox{fof}(\mbox{ip3b},\mbox{axiom})$

$\mbox{incident}(p_1,\mbox{a2a}_3)\qquad\mbox{fof}(\mbox{ip1a},\mbox{axiom})$

$\mbox{incident}(p_1,\mbox{b2b}_3)\qquad\mbox{fof}(\mbox{ip1b},\mbox{axiom})$

$\mbox{incident}(p_2,\mbox{a3a}_1)\qquad\mbox{fof}(\mbox{ip2a},\mbox{axiom})$

$\mbox{incident}(p_2,\mbox{b3b}_1)\qquad\mbox{fof}(\mbox{ip2b},\mbox{axiom})$

$\forall a,b{:}\;( \mbox{incident}(a,b) \ \Rightarrow \ \mbox{point}(a) )\qquad\mbox{fof}(\mbox{sort\_point},\mbox{axiom})$

$\forall a,b{:}\;( \mbox{incident}(a,b) \ \Rightarrow \ \mbox{line}(b) )\qquad\mbox{fof}(\mbox{sort\_line},\mbox{axiom})$

$a_1 = a_2 \ \Rightarrow \ \mbox{goal}\qquad\mbox{fof}(\mbox{diff\_a1\_a}_2,\mbox{axiom})$

$a_2 = a_3 \ \Rightarrow \ \mbox{goal}\qquad\mbox{fof}(\mbox{diff\_a2\_a}_3,\mbox{axiom})$

$a_3 = a_1 \ \Rightarrow \ \mbox{goal}\qquad\mbox{fof}(\mbox{diff\_a3\_a}_1,\mbox{axiom})$

$b_1 = b_2 \ \Rightarrow \ \mbox{goal}\qquad\mbox{fof}(\mbox{diff\_b1\_b}_2,\mbox{axiom})$

$b_2 = b_3 \ \Rightarrow \ \mbox{goal}\qquad\mbox{fof}(\mbox{diff\_b2\_b}_3,\mbox{axiom})$

$b_3 = b_1 \ \Rightarrow \ \mbox{goal}\qquad\mbox{fof}(\mbox{diff\_b3\_b}_1,\mbox{axiom})$

$\mbox{a1a}_2 = \mbox{b1b}_2 \ \Rightarrow \ \mbox{goal}\qquad\mbox{fof}(\mbox{not}_{12},\mbox{axiom})$

$\mbox{a2a}_3 = \mbox{b2b}_3 \ \Rightarrow \ \mbox{goal}\qquad\mbox{fof}(\mbox{not}_{23},\mbox{axiom})$

$\mbox{a3a}_1 = \mbox{b3b}_1 \ \Rightarrow \ \mbox{goal}\qquad\mbox{fof}(\mbox{not}_{31},\mbox{axiom})$

$\forall a{:}\;a = a\qquad\mbox{fof}(\mbox{reflexivity\_of\_equal},\mbox{axiom})$

$\forall a,b{:}\;( a = b \ \Rightarrow \ b = a )\qquad\mbox{fof}(\mbox{symmetry\_of\_equal},\mbox{axiom})$

$\forall a,b,c{:}\;( ( a = b\mbox{ and }b = c ) \ \Rightarrow \ a = c )\qquad\mbox{fof}(\mbox{transitivity\_of\_equal},\mbox{axiom})$

$\forall a,b,c{:}\;( ( a = b\mbox{ and }\mbox{incident}(b,c) ) \ \Rightarrow \ \mbox{incident}(a,c) )\qquad\mbox{fof}(\mbox{point\_congruence},\mbox{axiom})$

$\forall a,b,c{:}\;( ( \mbox{incident}(a,b)\mbox{ and }b = c ) \ \Rightarrow \ \mbox{incident}(a,c) )\qquad\mbox{fof}(\mbox{line\_congruence},\mbox{axiom})$

$\forall a,b,c,d{:}\;( ( \mbox{incident}(a,c)\mbox{ and }\mbox{incident}(a,d)\mbox{ and }\mbox{incident}(b,c)\mbox{ and }\mbox{incident}(b,d) ) \ \Rightarrow \ ( a = b\mbox{ or }c = d ) )\qquad\mbox{fof}(\mbox{unique},\mbox{axiom})$

$\forall a,b{:}\;( ( \mbox{point}(a)\mbox{ and }\mbox{point}(b) ) \ \Rightarrow \ \exists c{:}\;( \mbox{incident}(a,c)\mbox{ and }\mbox{incident}(b,c) ) )\qquad\mbox{fof}(\mbox{join},\mbox{axiom})$

$\forall a,b{:}\;( ( \mbox{line}(a)\mbox{ and }\mbox{line}(b) ) \ \Rightarrow \ \exists c{:}\;( \mbox{incident}(c,a)\mbox{ and }\mbox{incident}(c,b) ) )\qquad\mbox{fof}(\mbox{meet},\mbox{axiom})$

$\mbox{goal}\qquad\mbox{fof}(\mbox{goal\_to\_be\_proved},\mbox{conjecture})$

\medskip\textbf{GEO029-2.p} Corollary 3 to unique extension






$\mbox{include}(\mbox{'Axioms/GEO002-0.ax'})$

$u\ne v\qquad\mbox{cnf}(\mbox{u\_not\_v},\mbox{hypothesis})$

$\mbox{extension}(u,v,u,v)\ne \mbox{extension}(u,v,v,u)\qquad\mbox{cnf}(\mbox{prove\_equal\_extensions},\mbox{negated\_conjecture})$

\medskip\textbf{GEO006-1.p} Betweenness for 3 points on a line

For any three distinct points x, y, and z, if y is between
 x and z, then both x is not between y and z and z is not
 between x and y.




$\mbox{include}(\mbox{'Axioms/GEO001-0.ax'})$

$a\ne c\qquad\mbox{cnf}(\mbox{a\_not\_c},\mbox{hypothesis})$

$a\ne d\qquad\mbox{cnf}(\mbox{a\_not\_d},\mbox{hypothesis})$

$c\ne d\qquad\mbox{cnf}(\mbox{c\_not\_d},\mbox{hypothesis})$

$\mbox{between}(a,c,d)\qquad\mbox{cnf}(\mbox{c\_between\_a\_and\_d},\mbox{hypothesis})$

$\mbox{between}(c,a,d)\mbox{ or }\mbox{between}(a,d,c)\qquad\mbox{cnf}(\mbox{prove\_not\_between\_others},\mbox{negated\_conjecture})$

\medskip\textbf{GEO136+1.p} Underlying curve and one pair of points sufficient for ordering

The underlying curve and one pair of points are sufficient for
 the ordering of the points on the oriented curve.




$\mbox{include}(\mbox{'Axioms/GEO004+0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO004+1.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO004+2.ax'})$

$\forall o,p,q{:}\;( \mbox{ordered\_by}(o,p,q) \ \Rightarrow \ \forall r,s{:}\;( ( \mbox{ordered\_by}(o,r,s)\iff ( \mbox{between}(o,r,p,q)\mbox{ and }( \mbox{between}(o,r,s,q)\mbox{ or }\mbox{between}(o,r,q,s)\mbox{ or }q = s ) ) )\mbox{ or }( \mbox{between}(o,p,r,s)\mbox{ and }( \mbox{between}(o,p,r,q)\mbox{ or }\mbox{between}(o,p,q,r)\mbox{ or }q = r ) )\mbox{ or }( p = r\mbox{ and }( \mbox{between}(o,p,s,q)\mbox{ or }\mbox{between}(o,p,q,s)\mbox{ or }q = s ) ) ) )\qquad\mbox{fof}(\mbox{theorem\_4}_{19},\mbox{conjecture})$

\medskip\textbf{GEO152-1.p} Ordered meeting places

If three objects meet in pairs such that the meeting place of
 x and z precedes that of x and y on the trajectory of x and the
 meeting place of x and y precedes that of y and z on t(y), then
 the meet-ing place of y and z does not precede that of x and z
 on t(z)




$\mbox{include}(\mbox{'Axioms/GEO004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO004-2.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO004-3.ax'})$

$\mbox{connect}(\mbox{sk}_{30},\mbox{sk}_{32},\mbox{sk}_{27})\qquad\mbox{cnf}(\mbox{t18}_{156},\mbox{negated\_conjecture})$

$\mbox{connect}(\mbox{sk}_{30},\mbox{sk}_{31},\mbox{sk}_{28})\qquad\mbox{cnf}(\mbox{t18}_{157},\mbox{negated\_conjecture})$

$\mbox{connect}(\mbox{sk}_{31},\mbox{sk}_{32},\mbox{sk}_{29})\qquad\mbox{cnf}(\mbox{t18}_{158},\mbox{negated\_conjecture})$

$\mbox{ordered\_by}(\mbox{trajectory\_of}(\mbox{sk}_{30}),\mbox{sk}_{27},\mbox{sk}_{28})\qquad\mbox{cnf}(\mbox{t18}_{159},\mbox{negated\_conjecture})$

$\mbox{ordered\_by}(\mbox{trajectory\_of}(\mbox{sk}_{31}),\mbox{sk}_{28},\mbox{sk}_{29})\qquad\mbox{cnf}(\mbox{t18}_{160},\mbox{negated\_conjecture})$

$\mbox{ordered\_by}(\mbox{trajectory\_of}(\mbox{sk}_{32}),\mbox{sk}_{29},\mbox{sk}_{27})\qquad\mbox{cnf}(\mbox{t18}_{161},\mbox{negated\_conjecture})$

\medskip\textbf{GEO189+2.p} Collary to symmetry of incidence

If X, Y, and Z are pairwise distinct, and Z is incident with
 the line connecting X and Y, then Y is incident with the line
 connecting X and Z.




$\mbox{include}(\mbox{'Axioms/GEO008+0.ax'})$

$\forall x,y,z{:}\;( ( \mbox{distinct\_points}(x,y)\mbox{ and }\mbox{distinct\_points}(x,z)\mbox{ and }\mbox{distinct\_points}(y,z)\mbox{ and }\neg\,\mbox{apart\_point\_and\_line}(z,\mbox{line\_connecting}(x,y)) ) \ \Rightarrow \ \neg\,\mbox{apart\_point\_and\_line}(y,\mbox{line\_connecting}(x,z)) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO007-1.p} Betweenness for 4 points on a line

For all pairs of distinct points y and z, if w and x are
 on the line yz to the left of y (i.e. not between y and z),
 then either w is betweeen x and y or x is between w and y.




$\mbox{include}(\mbox{'Axioms/GEO001-0.ax'})$

$a\ne c\qquad\mbox{cnf}(\mbox{a\_not\_c},\mbox{hypothesis})$

$\mbox{between}(d,a,c)\qquad\mbox{cnf}(\mbox{a\_between\_a\_and\_c},\mbox{hypothesis})$

$\mbox{between}(e,a,c)\qquad\mbox{cnf}(\mbox{a\_between\_a\_and\_e},\mbox{hypothesis})$

$\neg\,\mbox{between}(d,e,a)\qquad\mbox{cnf}(\mbox{e\_not\_between\_d\_and\_a},\mbox{hypothesis})$

$\neg\,\mbox{between}(e,d,a)\qquad\mbox{cnf}(\mbox{prove\_d\_between\_e\_and\_a},\mbox{negated\_conjecture})$

\medskip\textbf{GEO131-1.p} Betweenness and precedence for three points, corollary






$\mbox{include}(\mbox{'Axioms/GEO004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO004-2.ax'})$

$\mbox{between}(\mbox{sk}_{28},\mbox{sk}_{25},\mbox{sk}_{26},\mbox{sk}_{27})\qquad\mbox{cnf}(\mbox{corollary\_4\_16}_{133},\mbox{negated\_conjecture})$

$\mbox{ordered\_by}(\mbox{sk}_{28},\mbox{sk}_{25},\mbox{sk}_{26})\mbox{ or }\mbox{ordered\_by}(\mbox{sk}_{28},\mbox{sk}_{25},\mbox{sk}_{27})\qquad\mbox{cnf}(\mbox{corollary\_4\_16}_{134},\mbox{negated\_conjecture})$

$\mbox{ordered\_by}(\mbox{sk}_{28},\mbox{sk}_{25},\mbox{sk}_{26}) \ \Rightarrow \ \mbox{ordered\_by}(\mbox{sk}_{28},\mbox{sk}_{25},\mbox{sk}_{26})\qquad\mbox{cnf}(\mbox{corollary\_4\_16}_{135},\mbox{negated\_conjecture})$

$\mbox{ordered\_by}(\mbox{sk}_{28},\mbox{sk}_{25},\mbox{sk}_{27}) \ \Rightarrow \ \mbox{ordered\_by}(\mbox{sk}_{28},\mbox{sk}_{25},\mbox{sk}_{27})\qquad\mbox{cnf}(\mbox{corollary\_4\_16}_{136},\mbox{negated\_conjecture})$

$\mbox{ordered\_by}(\mbox{sk}_{28},\mbox{sk}_{25},\mbox{sk}_{27}) \ \Rightarrow \ \neg\,\mbox{ordered\_by}(\mbox{sk}_{28},\mbox{sk}_{25},\mbox{sk}_{26})\qquad\mbox{cnf}(\mbox{corollary\_4\_16}_{137},\mbox{negated\_conjecture})$

\medskip\textbf{GEO157-1.p} Hilbert geometry axioms






$\mbox{include}(\mbox{'Axioms/GEO003-0.ax'})$

\medskip\textbf{GEO128+1.p} Precedence of three points, of whoich two are ordered

If P precedes Q with respect to o, then any point R on o
 precedes Q or is preceded by P.




$\mbox{include}(\mbox{'Axioms/GEO004+0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO004+1.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO004+2.ax'})$

$\forall o,p,q{:}\;( \mbox{ordered\_by}(o,p,q) \ \Rightarrow \ \forall r{:}\;( \mbox{incident\_o}(r,o) \ \Rightarrow \ ( \mbox{ordered\_by}(o,r,q)\mbox{ or }\mbox{ordered\_by}(o,p,r) ) ) )\qquad\mbox{fof}(\mbox{theorem\_4}_{13},\mbox{conjecture})$

\medskip\textbf{GEO234+3.p} Unequally directed lines






$\mbox{include}(\mbox{'Axioms/GEO009+0.ax'})$

$\forall l,m,n{:}\;( \mbox{unequally\_directed\_opposite\_lines}(l,m) \ \Rightarrow \ ( \mbox{unequally\_directed\_lines}(l,n)\mbox{ or }\mbox{unequally\_directed\_opposite\_lines}(m,n) ) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO216+1.p} A line is not orthogonal to itself






$\mbox{include}(\mbox{'Axioms/GEO006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+4.ax'})$

$\forall l{:}\;\neg\,\neg\,\mbox{unorthogonal\_lines}(l,l)\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO001-4.p} Betweenness is symmetric in its outer arguments






$\mbox{between}(x,y,x) \ \Rightarrow \ x{=}y\qquad\mbox{cnf}(\mbox{identity\_for\_betweeness},\mbox{axiom})$

$\mbox{equidistant}(x,y,z,z) \ \Rightarrow \ x{=}y\qquad\mbox{cnf}(\mbox{identity\_for\_equidistance},\mbox{axiom})$

$( \mbox{between}(x,w,v)\mbox{ and }\mbox{between}(y,v,z) ) \ \Rightarrow \ \mbox{between}(x,\mbox{outer\_pasch}(w,x,y,z,v),y)\qquad\mbox{cnf}(\mbox{outer\_pasch}_1,\mbox{axiom})$

$( \mbox{between}(x,w,v)\mbox{ and }\mbox{between}(y,v,z) ) \ \Rightarrow \ \mbox{between}(z,w,\mbox{outer\_pasch}(w,x,y,z,v))\qquad\mbox{cnf}(\mbox{outer\_pasch}_2,\mbox{axiom})$

$\mbox{between}(x,y,\mbox{extension}(x,y,w,v))\qquad\mbox{cnf}(\mbox{segment\_construction}_1,\mbox{axiom})$

$\mbox{equidistant}(y,\mbox{extension}(x,y,w,v),w,v)\qquad\mbox{cnf}(\mbox{segment\_construction}_2,\mbox{axiom})$

$\mbox{between}(a,b,c)\qquad\mbox{cnf}(\mbox{b\_between\_a\_and\_c},\mbox{hypothesis})$

$( x{=}y\mbox{ and }\mbox{between}(w,z,x) ) \ \Rightarrow \ \mbox{between}(w,z,y)\qquad\mbox{cnf}(\mbox{between\_substitution}_3,\mbox{axiom})$

$\neg\,\mbox{between}(c,b,a)\qquad\mbox{cnf}(\mbox{prove\_b\_between\_c\_and\_a},\mbox{negated\_conjecture})$

\medskip\textbf{GEO042-3.p} First inner transitivity property of betweenness






$\mbox{include}(\mbox{'Axioms/GEO002-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO002-2.ax'})$

$\mbox{equidistant}(u,v,u,v)\qquad\mbox{cnf}(d_1,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(w,x,u,v)\qquad\mbox{cnf}(d_2,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(v,u,w,x)\qquad\mbox{cnf}(d_3,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(u,v,x,w)\qquad\mbox{cnf}(\mbox{d4}_1,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(v,u,x,w)\qquad\mbox{cnf}(\mbox{d4}_2,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(w,x,v,u)\qquad\mbox{cnf}(\mbox{d4}_3,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(x,w,u,v)\qquad\mbox{cnf}(\mbox{d4}_4,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(x,w,v,u)\qquad\mbox{cnf}(\mbox{d4}_5,\mbox{axiom})$

$( \mbox{equidistant}(u,v,w,x)\mbox{ and }\mbox{equidistant}(w,x,y,z) ) \ \Rightarrow \ \mbox{equidistant}(u,v,y,z)\qquad\mbox{cnf}(d_5,\mbox{axiom})$

$v = \mbox{extension}(u,v,w,w)\qquad\mbox{cnf}(e_1,\mbox{axiom})$

$y = \mbox{extension}(u,v,w,x) \ \Rightarrow \ \mbox{between}(u,v,y)\qquad\mbox{cnf}(b_0,\mbox{axiom})$

$\mbox{between}(u,v,\mbox{reflection}(u,v))\qquad\mbox{cnf}(\mbox{r2}_1,\mbox{axiom})$

$\mbox{equidistant}(v,\mbox{reflection}(u,v),u,v)\qquad\mbox{cnf}(\mbox{r2}_2,\mbox{axiom})$

$u = v \ \Rightarrow \ v = \mbox{reflection}(u,v)\qquad\mbox{cnf}(\mbox{r3}_1,\mbox{axiom})$

$u = \mbox{reflection}(u,u)\qquad\mbox{cnf}(\mbox{r3}_2,\mbox{axiom})$

$v = \mbox{reflection}(u,v) \ \Rightarrow \ u = v\qquad\mbox{cnf}(r_4,\mbox{axiom})$

$\mbox{equidistant}(u,u,v,v)\qquad\mbox{cnf}(d_7,\mbox{axiom})$

$( \mbox{equidistant}(u,v,u_1,v_1)\mbox{ and }\mbox{equidistant}(v,w,v_1,w_1)\mbox{ and }\mbox{between}(u,v,w)\mbox{ and }\mbox{between}(u_1,v_1,w_1) ) \ \Rightarrow \ \mbox{equidistant}(u,w,u_1,w_1)\qquad\mbox{cnf}(d_8,\mbox{axiom})$

$( \mbox{between}(u,v,w)\mbox{ and }\mbox{between}(u,v,x)\mbox{ and }\mbox{equidistant}(v,w,v,x) ) \ \Rightarrow \ ( u = v\mbox{ or }w = x )\qquad\mbox{cnf}(d_9,\mbox{axiom})$

$\mbox{between}(u,v,w) \ \Rightarrow \ ( u = v\mbox{ or }w = \mbox{extension}(u,v,v,w) )\qquad\mbox{cnf}(\mbox{d10}_1,\mbox{axiom})$

$\mbox{equidistant}(w,x,y,z) \ \Rightarrow \ ( \mbox{extension}(u,v,w,x) = \mbox{extension}(u,v,y,z)\mbox{ or }u = v )\qquad\mbox{cnf}(\mbox{d10}_2,\mbox{axiom})$

$\mbox{extension}(u,v,u,v) = \mbox{extension}(u,v,v,u)\mbox{ or }u = v\qquad\mbox{cnf}(\mbox{d10}_3,\mbox{axiom})$

$\mbox{equidistant}(v,u,v,\mbox{reflection}(\mbox{reflection}(u,v),v))\qquad\mbox{cnf}(r_5,\mbox{axiom})$

$u = \mbox{reflection}(\mbox{reflection}(u,v),v)\qquad\mbox{cnf}(r_6,\mbox{axiom})$

$\mbox{between}(u,v,v)\qquad\mbox{cnf}(t_3,\mbox{axiom})$

$( \mbox{between}(u,w,x)\mbox{ and }u = x ) \ \Rightarrow \ \mbox{between}(v,w,x)\qquad\mbox{cnf}(b_1,\mbox{axiom})$

$\mbox{between}(u,v,w) \ \Rightarrow \ \mbox{between}(w,v,u)\qquad\mbox{cnf}(t_1,\mbox{axiom})$

$\mbox{between}(u,u,v)\qquad\mbox{cnf}(t_2,\mbox{axiom})$

$( \mbox{between}(u,v,w)\mbox{ and }\mbox{between}(v,u,w) ) \ \Rightarrow \ u = v\qquad\mbox{cnf}(b_2,\mbox{axiom})$

$( \mbox{between}(u,v,w)\mbox{ and }\mbox{between}(u,w,v) ) \ \Rightarrow \ v = w\qquad\mbox{cnf}(b_3,\mbox{axiom})$

$( \mbox{between}(u,v,w)\mbox{ and }\mbox{between}(v,u,w) ) \ \Rightarrow \ ( u = v\mbox{ or }v = w )\qquad\mbox{cnf}(\mbox{t6}_1,\mbox{axiom})$

$( \mbox{between}(u,v,w)\mbox{ and }\mbox{between}(u,w,v) ) \ \Rightarrow \ ( u = v\mbox{ or }v = w )\qquad\mbox{cnf}(\mbox{t6}_2,\mbox{axiom})$

$\mbox{between}(u,v,x)\qquad\mbox{cnf}(\mbox{v\_between\_u\_and\_x},\mbox{hypothesis})$

$\mbox{between}(v,w,x)\qquad\mbox{cnf}(\mbox{w\_between\_v\_and\_x},\mbox{hypothesis})$

$\neg\,\mbox{between}(u,v,w)\qquad\mbox{cnf}(\mbox{prove\_v\_between\_u\_and\_w},\mbox{negated\_conjecture})$

\medskip\textbf{GEO134-1.p} Betweenness and precedence property 3






$\mbox{include}(\mbox{'Axioms/GEO004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO004-2.ax'})$

$\mbox{ordered\_by}(\mbox{sk}_{25},\mbox{sk}_{26},\mbox{sk}_{27})\qquad\mbox{cnf}(\mbox{theorem\_4\_17\_3}_{133},\mbox{negated\_conjecture})$

$\mbox{sk}_{26}\ne \mbox{sk}_{28}\qquad\mbox{cnf}(\mbox{theorem\_4\_17\_3}_{134},\mbox{negated\_conjecture})$

$\neg\,\mbox{between}(\mbox{sk}_{25},\mbox{sk}_{28},\mbox{sk}_{26},\mbox{sk}_{27})\qquad\mbox{cnf}(\mbox{theorem\_4\_17\_3}_{135},\mbox{negated\_conjecture})$

$\mbox{ordered\_by}(\mbox{sk}_{25},\mbox{sk}_{28},\mbox{sk}_{29})\mbox{ or }\mbox{between}(\mbox{sk}_{25},\mbox{sk}_{26},\mbox{sk}_{28},\mbox{sk}_{29})\qquad\mbox{cnf}(\mbox{theorem\_4\_17\_3}_{136},\mbox{negated\_conjecture})$

$\mbox{ordered\_by}(\mbox{sk}_{25},\mbox{sk}_{28},\mbox{sk}_{29}) \ \Rightarrow \ \mbox{ordered\_by}(\mbox{sk}_{25},\mbox{sk}_{28},\mbox{sk}_{29})\qquad\mbox{cnf}(\mbox{theorem\_4\_17\_3}_{137},\mbox{negated\_conjecture})$

$\mbox{between}(\mbox{sk}_{25},\mbox{sk}_{26},\mbox{sk}_{28},\mbox{sk}_{29}) \ \Rightarrow \ \mbox{between}(\mbox{sk}_{25},\mbox{sk}_{26},\mbox{sk}_{28},\mbox{sk}_{29})\qquad\mbox{cnf}(\mbox{theorem\_4\_17\_3}_{138},\mbox{negated\_conjecture})$

$\mbox{between}(\mbox{sk}_{25},\mbox{sk}_{26},\mbox{sk}_{28},\mbox{sk}_{29}) \ \Rightarrow \ \neg\,\mbox{ordered\_by}(\mbox{sk}_{25},\mbox{sk}_{28},\mbox{sk}_{29})\qquad\mbox{cnf}(\mbox{theorem\_4\_17\_3}_{139},\mbox{negated\_conjecture})$

\medskip\textbf{GEO110-1.p} Betweenness for closed curves






$\mbox{include}(\mbox{'Axioms/GEO004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO004-1.ax'})$

$\mbox{closed}(\mbox{sk}_{15})\qquad\mbox{cnf}(\mbox{theorem\_3\_7}_{77},\mbox{negated\_conjecture})$

$\mbox{sk}_{16}\ne \mbox{sk}_{17}\qquad\mbox{cnf}(\mbox{theorem\_3\_7}_{78},\mbox{negated\_conjecture})$

$\mbox{sk}_{17}\ne \mbox{sk}_{18}\qquad\mbox{cnf}(\mbox{theorem\_3\_7}_{79},\mbox{negated\_conjecture})$

$\mbox{sk}_{16}\ne \mbox{sk}_{18}\qquad\mbox{cnf}(\mbox{theorem\_3\_7}_{80},\mbox{negated\_conjecture})$

$\mbox{incident\_c}(\mbox{sk}_{16},\mbox{sk}_{15})\qquad\mbox{cnf}(\mbox{theorem\_3\_7}_{81},\mbox{negated\_conjecture})$

$\mbox{incident\_c}(\mbox{sk}_{17},\mbox{sk}_{15})\qquad\mbox{cnf}(\mbox{theorem\_3\_7}_{82},\mbox{negated\_conjecture})$

$\mbox{incident\_c}(\mbox{sk}_{18},\mbox{sk}_{15})\qquad\mbox{cnf}(\mbox{theorem\_3\_7}_{83},\mbox{negated\_conjecture})$

$\neg\,\mbox{between\_c}(\mbox{sk}_{15},\mbox{sk}_{16},\mbox{sk}_{17},\mbox{sk}_{18})\qquad\mbox{cnf}(\mbox{theorem\_3\_7}_{84},\mbox{negated\_conjecture})$

\medskip\textbf{GEO128-1.p} Precedence of three points, of whoich two are ordered

If P precedes Q with respect to o, then any point R on o
 precedes Q or is preceded by P.




$\mbox{include}(\mbox{'Axioms/GEO004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO004-2.ax'})$

$\mbox{ordered\_by}(\mbox{sk}_{25},\mbox{sk}_{26},\mbox{sk}_{27})\qquad\mbox{cnf}(\mbox{theorem\_4\_13}_{133},\mbox{negated\_conjecture})$

$\mbox{incident\_o}(\mbox{sk}_{28},\mbox{sk}_{25})\qquad\mbox{cnf}(\mbox{theorem\_4\_13}_{134},\mbox{negated\_conjecture})$

$\neg\,\mbox{ordered\_by}(\mbox{sk}_{25},\mbox{sk}_{28},\mbox{sk}_{27})\qquad\mbox{cnf}(\mbox{theorem\_4\_13}_{135},\mbox{negated\_conjecture})$

$\neg\,\mbox{ordered\_by}(\mbox{sk}_{25},\mbox{sk}_{26},\mbox{sk}_{28})\qquad\mbox{cnf}(\mbox{theorem\_4\_13}_{136},\mbox{negated\_conjecture})$

\medskip\textbf{GEO004-1.p} Every line segment has a midpoint






$\mbox{include}(\mbox{'Axioms/GEO001-0.ax'})$

$a\ne b\qquad\mbox{cnf}(\mbox{a\_not\_b},\mbox{hypothesis})$

$\mbox{equidistant}(a,x,b,x) \ \Rightarrow \ \neg\,\mbox{between}(a,x,b)\qquad\mbox{cnf}(\mbox{prove\_midpoint},\mbox{negated\_conjecture})$

\medskip\textbf{GEO038-2.p} Corollary 1 to the segment contruction axiom






$\mbox{include}(\mbox{'Axioms/GEO002-0.ax'})$

$y = \mbox{extension}(u,v,w,x)\qquad\mbox{cnf}(\mbox{y\_is\_extension},\mbox{hypothesis})$

$\neg\,\mbox{between}(u,v,y)\qquad\mbox{cnf}(\mbox{prove\_corollary},\mbox{negated\_conjecture})$

\medskip\textbf{GEO097+1.p} A subcurves connects any two points on a curve

For any two points on a curve there is a sub-curve that connects
 these two points, that is to say these points are the endpoints
 of the sub-curve.




$\mbox{include}(\mbox{'Axioms/GEO004+0.ax'})$

$\forall p,q,c{:}\;( ( p\ne q\mbox{ and }\mbox{incident\_c}(p,c)\mbox{ and }\mbox{incident\_c}(q,c) ) \ \Rightarrow \ \exists \mbox{cpp}{:}\;( \mbox{part\_of}(\mbox{cpp},c)\mbox{ and }\mbox{end\_point}(p,\mbox{cpp})\mbox{ and }\mbox{end\_point}(q,\mbox{cpp}) ) )\qquad\mbox{fof}(\mbox{theorem\_2}_{15},\mbox{conjecture})$

\medskip\textbf{GEO067-2.p} Any two points are collinear






$\mbox{include}(\mbox{'Axioms/GEO002-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO002-1.ax'})$

$( \mbox{colinear}(x,x,y)\mbox{ and }\mbox{colinear}(x,y,x)\mbox{ and }\mbox{colinear}(y,x,x) ) \ \Rightarrow \ x = y\qquad\mbox{cnf}(\mbox{part}_1,\mbox{negated\_conjecture})$

$( \mbox{colinear}(x,x,y)\mbox{ and }\mbox{colinear}(x,y,x)\mbox{ and }\mbox{colinear}(y,x,x) ) \ \Rightarrow \ \neg\,\mbox{colinear}(x,z,y)\qquad\mbox{cnf}(\mbox{part}_2,\mbox{negated\_conjecture})$

\medskip\textbf{GEO185+2.p} Lemma on symmetry and apartness

If the lines U and V are convergent, then the point that is
 equal to the intersection point is incident to both lines.




$\mbox{include}(\mbox{'Axioms/GEO008+0.ax'})$

$\forall x,y,u,v{:}\;( ( \mbox{distinct\_points}(x,y)\mbox{ and }\mbox{convergent\_lines}(u,v)\mbox{ and }\neg\,\mbox{distinct\_points}(x,\mbox{intersection\_point}(u,v)) ) \ \Rightarrow \ ( \neg\,\mbox{apart\_point\_and\_line}(x,u)\mbox{ and }\neg\,\mbox{apart\_point\_and\_line}(x,v) ) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO002-3.p} For all points x and y, x is between x and y






$\mbox{include}(\mbox{'Axioms/GEO002-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO002-2.ax'})$

$\mbox{equidistant}(u,v,u,v)\qquad\mbox{cnf}(d_1,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(w,x,u,v)\qquad\mbox{cnf}(d_2,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(v,u,w,x)\qquad\mbox{cnf}(d_3,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(u,v,x,w)\qquad\mbox{cnf}(\mbox{d4}_1,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(v,u,x,w)\qquad\mbox{cnf}(\mbox{d4}_2,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(w,x,v,u)\qquad\mbox{cnf}(\mbox{d4}_3,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(x,w,u,v)\qquad\mbox{cnf}(\mbox{d4}_4,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(x,w,v,u)\qquad\mbox{cnf}(\mbox{d4}_5,\mbox{axiom})$

$( \mbox{equidistant}(u,v,w,x)\mbox{ and }\mbox{equidistant}(w,x,y,z) ) \ \Rightarrow \ \mbox{equidistant}(u,v,y,z)\qquad\mbox{cnf}(d_5,\mbox{axiom})$

$v = \mbox{extension}(u,v,w,w)\qquad\mbox{cnf}(e_1,\mbox{axiom})$

$y = \mbox{extension}(u,v,w,x) \ \Rightarrow \ \mbox{between}(u,v,y)\qquad\mbox{cnf}(b_0,\mbox{axiom})$

$\mbox{between}(u,v,\mbox{reflection}(u,v))\qquad\mbox{cnf}(\mbox{r2}_1,\mbox{axiom})$

$\mbox{equidistant}(v,\mbox{reflection}(u,v),u,v)\qquad\mbox{cnf}(\mbox{r2}_2,\mbox{axiom})$

$u = v \ \Rightarrow \ v = \mbox{reflection}(u,v)\qquad\mbox{cnf}(\mbox{r3}_1,\mbox{axiom})$

$u = \mbox{reflection}(u,u)\qquad\mbox{cnf}(\mbox{r3}_2,\mbox{axiom})$

$v = \mbox{reflection}(u,v) \ \Rightarrow \ u = v\qquad\mbox{cnf}(r_4,\mbox{axiom})$

$\mbox{equidistant}(u,u,v,v)\qquad\mbox{cnf}(d_7,\mbox{axiom})$

$( \mbox{equidistant}(u,v,u_1,v_1)\mbox{ and }\mbox{equidistant}(v,w,v_1,w_1)\mbox{ and }\mbox{between}(u,v,w)\mbox{ and }\mbox{between}(u_1,v_1,w_1) ) \ \Rightarrow \ \mbox{equidistant}(u,w,u_1,w_1)\qquad\mbox{cnf}(d_8,\mbox{axiom})$

$( \mbox{between}(u,v,w)\mbox{ and }\mbox{between}(u,v,x)\mbox{ and }\mbox{equidistant}(v,w,v,x) ) \ \Rightarrow \ ( u = v\mbox{ or }w = x )\qquad\mbox{cnf}(d_9,\mbox{axiom})$

$\mbox{between}(u,v,w) \ \Rightarrow \ ( u = v\mbox{ or }w = \mbox{extension}(u,v,v,w) )\qquad\mbox{cnf}(\mbox{d10}_1,\mbox{axiom})$

$\mbox{equidistant}(w,x,y,z) \ \Rightarrow \ ( \mbox{extension}(u,v,w,x) = \mbox{extension}(u,v,y,z)\mbox{ or }u = v )\qquad\mbox{cnf}(\mbox{d10}_2,\mbox{axiom})$

$\mbox{extension}(u,v,u,v) = \mbox{extension}(u,v,v,u)\mbox{ or }u = v\qquad\mbox{cnf}(\mbox{d10}_3,\mbox{axiom})$

$\mbox{equidistant}(v,u,v,\mbox{reflection}(\mbox{reflection}(u,v),v))\qquad\mbox{cnf}(r_5,\mbox{axiom})$

$u = \mbox{reflection}(\mbox{reflection}(u,v),v)\qquad\mbox{cnf}(r_6,\mbox{axiom})$

$\mbox{between}(u,v,v)\qquad\mbox{cnf}(t_3,\mbox{axiom})$

$( \mbox{between}(u,w,x)\mbox{ and }u = x ) \ \Rightarrow \ \mbox{between}(v,w,x)\qquad\mbox{cnf}(b_1,\mbox{axiom})$

$\mbox{between}(u,v,w) \ \Rightarrow \ \mbox{between}(w,v,u)\qquad\mbox{cnf}(t_1,\mbox{axiom})$

$\neg\,\mbox{between}(a,a,b)\qquad\mbox{cnf}(\mbox{prove\_a\_between\_a\_and\_b},\mbox{negated\_conjecture})$

\medskip\textbf{GEO084+1.p} Sum is monotone, part 2






$\mbox{include}(\mbox{'Axioms/GEO004+0.ax'})$

$\forall c_1,c_2,c_3,p{:}\;( ( \mbox{part\_of}(c_1,c_3)\mbox{ and }\mbox{part\_of}(c_2,c_3)\mbox{ and }p\wedge c_1{=}c_2 ) \ \Rightarrow \ \mbox{part\_of}(c_1 + c_2,c_3) )\qquad\mbox{fof}(\mbox{corollary\_2\_6}_2,\mbox{conjecture})$

\medskip\textbf{GEO087-1.p} If one curve is part of another curve then they cannot meet






$\mbox{include}(\mbox{'Axioms/GEO004-0.ax'})$

$\mbox{part\_of}(\mbox{sk}_{14},\mbox{sk}_{15})\qquad\mbox{cnf}(\mbox{corollary\_2\_9}_{67},\mbox{negated\_conjecture})$

$\mbox{sk}_{16}\wedge \mbox{sk}_{14}{=}\mbox{sk}_{15}\qquad\mbox{cnf}(\mbox{corollary\_2\_9}_{68},\mbox{negated\_conjecture})$

\medskip\textbf{GEO194+1.p} Corollary to symmetry of apartness

If the lines X, Y, and Z are pairwise convergent, and the
 intersection point of X and Y is apart from a line Z, then
 the intersection point of X and Z is apart from Y.




$\mbox{include}(\mbox{'Axioms/GEO006+0.ax'})$

$\forall x,y,z{:}\;( ( \mbox{convergent\_lines}(x,y)\mbox{ and }\mbox{convergent\_lines}(z,y)\mbox{ and }\mbox{convergent\_lines}(x,z) ) \ \Rightarrow \ ( \mbox{apart\_point\_and\_line}(\mbox{intersection\_point}(x,y),z) \ \Rightarrow \ \mbox{apart\_point\_and\_line}(\mbox{intersection\_point}(x,z),y) ) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO200+2.p} Line equals its converse

If the points X and Y are distinct, then the line connecting
 X and Y is equal to the line connecting Y and X.




$\mbox{include}(\mbox{'Axioms/GEO008+0.ax'})$

$\forall x,y{:}\;( \mbox{distinct\_points}(x,y) \ \Rightarrow \ \neg\,\mbox{distinct\_lines}(\mbox{line\_connecting}(x,y),\mbox{line\_connecting}(y,x)) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO017-3.p} Corollary 1 to symmetries of equidistance

Show that if the distance from A to B equals the distance
 from C to D, then the distance from A to B equals the
 distance from D to C.




$\mbox{include}(\mbox{'Axioms/GEO002-0.ax'})$

$\mbox{equidistant}(u,v,u,v)\qquad\mbox{cnf}(d_1,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(w,x,u,v)\qquad\mbox{cnf}(d_2,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x) \ \Rightarrow \ \mbox{equidistant}(v,u,w,x)\qquad\mbox{cnf}(d_3,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x)\qquad\mbox{cnf}(\mbox{u\_to\_v\_equals\_w\_to\_x},\mbox{hypothesis})$

$\neg\,\mbox{equidistant}(u,v,x,w)\qquad\mbox{cnf}(\mbox{prove\_symmetry},\mbox{negated\_conjecture})$

\medskip\textbf{GEO215+3.p} Third line not orthogonal to two convergent lines

If two lines L and M are convergent, then a third line N is
 not orthogonal to L or M.




$\mbox{include}(\mbox{'Axioms/GEO006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+1.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+2.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+3.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+4.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+5.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+6.ax'})$

$\forall l,m,n{:}\;( \mbox{convergent\_lines}(l,m) \ \Rightarrow \ ( \mbox{not\_orthogonal\_lines}(l,n)\mbox{ or }\mbox{not\_orthogonal\_lines}(m,n) ) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO100+1.p} Subcurves with common endpoint can be complemented

Every proper sub-curve of an open curve that has a common
 endpoint with the open curve can be complemented by another
 curve so that their sum constitute the open curve.




$\mbox{include}(\mbox{'Axioms/GEO004+0.ax'})$

$\forall c,c_1,p{:}\;( ( \mbox{open}(c)\mbox{ and }\mbox{part\_of}(c_1,c)\mbox{ and }c_1\ne c\mbox{ and }\mbox{end\_point}(p,c_1)\mbox{ and }\mbox{end\_point}(p,c) ) \ \Rightarrow \ \exists q,c_2{:}\;( p\ne q\mbox{ and }q\wedge c_1{=}c_2\mbox{ and }c = c_1 + c_2 ) )\qquad\mbox{fof}(\mbox{theorem\_2}_{18},\mbox{conjecture})$

\medskip\textbf{GEO015-3.p} Equidistance is symmetric between its argument pairs

Show that if the distance from A to B equals the distance
 from C to D, then the distance from C to D equals the
 distance from A to B.




$\mbox{include}(\mbox{'Axioms/GEO002-0.ax'})$

$\mbox{equidistant}(u,v,u,v)\qquad\mbox{cnf}(d_1,\mbox{axiom})$

$\mbox{equidistant}(u,v,w,x)\qquad\mbox{cnf}(\mbox{u\_to\_v\_equals\_w\_to\_x},\mbox{hypothesis})$

$\neg\,\mbox{equidistant}(w,x,u,v)\qquad\mbox{cnf}(\mbox{prove\_symmetry},\mbox{negated\_conjecture})$

\medskip\textbf{GEO070-2.p} Non-collinear points in the bisecting diagonal theorem

Under the hypotheses of the bisecting diagonal theorem, the
 points u, v, w cannot be colinear.




$\mbox{include}(\mbox{'Axioms/GEO002-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO002-1.ax'})$

$\mbox{equidistant}(u,v,w,x)\qquad\mbox{cnf}(\mbox{u\_to\_v\_equals\_w\_to\_x},\mbox{hypothesis})$

$\mbox{equidistant}(v,w,x,u)\qquad\mbox{cnf}(\mbox{v\_to\_w\_equals\_x\_to\_u},\mbox{hypothesis})$

$\mbox{equidistant}(u,w,v,x)\qquad\mbox{cnf}(\mbox{u\_to\_w\_equals\_v\_to\_x},\mbox{hypothesis})$

$\mbox{between}(u,y,w)\qquad\mbox{cnf}(\mbox{y\_between\_u\_and\_w},\mbox{hypothesis})$

$\mbox{between}(v,y,x)\qquad\mbox{cnf}(\mbox{y\_between\_v\_and\_x},\mbox{hypothesis})$

$u\ne v\qquad\mbox{cnf}(\mbox{u\_not\_v},\mbox{hypothesis})$

$x\ne u\qquad\mbox{cnf}(\mbox{x\_not\_u},\mbox{hypothesis})$

$\mbox{colinear}(u,v,w)\qquad\mbox{cnf}(\mbox{prove\_uvw\_not\_colinear},\mbox{negated\_conjecture})$

\medskip\textbf{GEO089-1.p} Inner points of a sub-curve of a curve are inner points






$\mbox{include}(\mbox{'Axioms/GEO004-0.ax'})$

$\mbox{part\_of}(\mbox{sk}_{16},\mbox{sk}_{14})\qquad\mbox{cnf}(\mbox{corollary\_2\_11}_{67},\mbox{negated\_conjecture})$

$\mbox{inner\_point}(\mbox{sk}_{15},\mbox{sk}_{16})\qquad\mbox{cnf}(\mbox{corollary\_2\_11}_{68},\mbox{negated\_conjecture})$

$\neg\,\mbox{inner\_point}(\mbox{sk}_{15},\mbox{sk}_{14})\qquad\mbox{cnf}(\mbox{corollary\_2\_11}_{69},\mbox{negated\_conjecture})$

\medskip\textbf{GEO078-4.p} Every plane contains 3 noncollinear points






$\mbox{include}(\mbox{'Axioms/GEO003-0.ax'})$

$\mbox{plane}(\mbox{a\_plane})\qquad\mbox{cnf}(\mbox{there\_is\_a\_plane},\mbox{hypothesis})$

$( \mbox{point}(x_1)\mbox{ and }\mbox{point}(x_2)\mbox{ and }\mbox{point}(x_3)\mbox{ and }\mbox{on}(x_1,\mbox{a\_plane})\mbox{ and }\mbox{on}(x_2,\mbox{a\_plane})\mbox{ and }\mbox{on}(x_3,\mbox{a\_plane}) ) \ \Rightarrow \ ( \mbox{collinear}(x_1,x_2,x_3)\mbox{ or }x_1 = x_2\mbox{ or }x_1 = x_3\mbox{ or }x_2 = x_3 )\qquad\mbox{cnf}(\mbox{prove\_every\_plane\_contains\_3\_noncollinear\_points},\mbox{negated\_conjecture})$

\medskip\textbf{GEO174+1.p} Lemma on symmetry and apartness

If two points are distinct and a line U is apart from at least
 one of these points, then this line is distinct from the line
 connecting these points




$\mbox{include}(\mbox{'Axioms/GEO006+0.ax'})$

$\forall x,y,u,v{:}\;( ( \mbox{distinct\_points}(x,y)\mbox{ and }\mbox{convergent\_lines}(u,v)\mbox{ and }( \mbox{apart\_point\_and\_line}(x,u)\mbox{ or }\mbox{apart\_point\_and\_line}(y,u) ) ) \ \Rightarrow \ \mbox{distinct\_lines}(u,\mbox{line\_connecting}(x,y)) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO352-1.p} Tarski geometry axioms






$\mbox{include}(\mbox{'Axioms/GEO002-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO002-1.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO002-2.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO002-3.ax'})$

\medskip\textbf{GEO190+3.p} Collary to symmetry of incidence

If X, Y, and Z are pairwise distinct, and Z is incident with
 the line connecting X and Y, then Z is incident with the line
 connecting Y and X.




$\mbox{include}(\mbox{'Axioms/GEO006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+1.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+2.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+3.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+4.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+5.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+6.ax'})$

$\forall x,y,z{:}\;( ( \mbox{distinct\_points}(x,y)\mbox{ and }\mbox{distinct\_points}(x,z)\mbox{ and }\mbox{distinct\_points}(y,z)\mbox{ and }\mbox{incident\_point\_and\_line}(z,\mbox{line\_connecting}(x,y)) ) \ \Rightarrow \ \mbox{incident\_point\_and\_line}(z,\mbox{line\_connecting}(y,x)) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{GEO139-1.p} Oppositely oriented curve exists

For every oriented curve there is an oppositely oriented curve
 with the same underlying curve.




$\mbox{include}(\mbox{'Axioms/GEO004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO004-2.ax'})$

$\mbox{underlying\_curve}(\mbox{sk}_{25}) = \mbox{underlying\_curve}(a) \ \Rightarrow \ \mbox{ordered\_by}(\mbox{sk}_{25},\mbox{sk}_{26}(a),\mbox{sk}_{27}(a))\qquad\mbox{cnf}(\mbox{theorem\_4\_22}_{133},\mbox{negated\_conjecture})$

$\mbox{underlying\_curve}(\mbox{sk}_{25}) = \mbox{underlying\_curve}(a) \ \Rightarrow \ \neg\,\mbox{ordered\_by}(a,\mbox{sk}_{27}(a),\mbox{sk}_{26}(a))\qquad\mbox{cnf}(\mbox{theorem\_4\_22}_{134},\mbox{negated\_conjecture})$

\medskip\textbf{GEO059-2.p} Congruence for double reflection






$\mbox{include}(\mbox{'Axioms/GEO002-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO002-2.ax'})$

$\neg\,\mbox{equidistant}(v,u,v,\mbox{reflection}(\mbox{reflection}(u,v),v))\qquad\mbox{cnf}(\mbox{prove\_congruence},\mbox{negated\_conjecture})$

\medskip\textbf{GEO353+1.p} Apartness geometry






$\mbox{include}(\mbox{'Axioms/GEO006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+1.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+2.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+3.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+4.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+5.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+6.ax'})$

\medskip\textbf{GEO210+2.p} Uniqueness of orthogonality

If the point A is incident with line L, and the line L is
 orthogonal to M, then L is equal to the orthogonal to M
 through A.




$\mbox{include}(\mbox{'Axioms/GEO008+0.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+2.ax'})$

$\mbox{include}(\mbox{'Axioms/GEO006+3.ax'})$

$\forall a,l,m{:}\;( ( \neg\,\mbox{apart\_point\_and\_line}(a,l)\mbox{ and }\neg\,\mbox{unorthogonal\_lines}(l,m) ) \ \Rightarrow \ \neg\,\mbox{distinct\_lines}(l,\mbox{orthogonal\_through\_point}(m,a)) )\qquad\mbox{fof}(\mbox{con},\mbox{conjecture})$

\medskip
\end{document}