## TPTP Problem File: GEO511+1.p

View Solutions - Solve Problem

```%------------------------------------------------------------------------------
% File     : GEO511+1 : TPTP v8.1.2. Released v7.0.0.
% Domain   : Geometry
% Problem  : Segment with midpoint
% Version  : Especial.
% English  : Given segment ab and perpendiculars ap and qb, and point t on
%            line ab between p and q, with ap <= qb, then segment ab has a
%            midpoint.

% Refs     : [Urb16] Urban (2016), Email to Geoff Sutcliffe
%          : [BW17]  Beeson & Wos (2017), Finding Proofs in Tarskian Geomet
% Source   : [Urb16]
% Names    : Satz8.24.in.p [BW17]

% Status   : Theorem
% Rating   : 0.72 v7.5.0, 0.84 v7.4.0, 0.80 v7.3.0, 0.90 v7.1.0, 0.83 v7.0.0
% Syntax   : Number of formulae    :  162 (  20 unt;   0 def)
%            Number of atoms       :  697 ( 134 equ)
%            Maximal formula atoms :   30 (   4 avg)
%            Number of connectives :  921 ( 386   ~; 462   |;  73   &)
%                                         (   0 <=>;   0  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   22 (   9 avg)
%            Maximal term depth    :    3 (   1 avg)
%            Number of predicates  :   15 (  14 usr;   0 prp; 2-8 aty)
%            Number of functors    :   26 (  26 usr;   4 con; 0-6 aty)
%            Number of variables   :  650 ( 650   !;   0   ?)
% SPC      : FOF_THM_RFO_SEQ

%------------------------------------------------------------------------------
include('Axioms/GEO011+0.ax').
%------------------------------------------------------------------------------
fof(aSatz2_1,axiom,
! [Xa,Xb] : s_e(Xa,Xb,Xa,Xb) ).

fof(aSatz2_2,axiom,
! [Xa,Xb,Xc,Xd] :
( ~ s_e(Xa,Xb,Xc,Xd)
| s_e(Xc,Xd,Xa,Xb) ) ).

fof(aSatz2_3,axiom,
! [Xa,Xb,Xc,Xd,Xe,Xf] :
( ~ s_e(Xa,Xb,Xc,Xd)
| ~ s_e(Xc,Xd,Xe,Xf)
| s_e(Xa,Xb,Xe,Xf) ) ).

fof(aSatz2_4,axiom,
! [Xa,Xb,Xc,Xd] :
( ~ s_e(Xa,Xb,Xc,Xd)
| s_e(Xb,Xa,Xc,Xd) ) ).

fof(aSatz2_5,axiom,
! [Xa,Xb,Xc,Xd] :
( ~ s_e(Xa,Xb,Xc,Xd)
| s_e(Xa,Xb,Xd,Xc) ) ).

fof(aSatz2_8,axiom,
! [Xa,Xb] : s_e(Xa,Xa,Xb,Xb) ).

fof(aSatz2_11,axiom,
! [Xa,Xb,Xc,Xa1,Xb1,Xc1] :
( ~ s_t(Xa,Xb,Xc)
| ~ s_t(Xa1,Xb1,Xc1)
| ~ s_e(Xa,Xb,Xa1,Xb1)
| ~ s_e(Xb,Xc,Xb1,Xc1)
| s_e(Xa,Xc,Xa1,Xc1) ) ).

fof(aSatz2_12,axiom,
! [Xq,Xa,Xd,Xb,Xc] :
( Xq = Xa
| ~ s_t(Xq,Xa,Xd)
| ~ s_e(Xa,Xd,Xb,Xc)
| Xd = ext(Xq,Xa,Xb,Xc) ) ).

fof(aSatz2_13,axiom,
! [Xb,Xc,Xa] :
( ~ s_e(Xb,Xc,Xa,Xa)
| Xb = Xc ) ).

fof(aSatz2_14,axiom,
! [Xa,Xb,Xc,Xd] :
( ~ s_e(Xa,Xb,Xc,Xd)
| s_e(Xb,Xa,Xd,Xc) ) ).

fof(aSatz2_15,axiom,
! [Xa,Xb,Xc,XA,XB,XC] :
( ~ s_t(Xa,Xb,Xc)
| ~ s_t(XA,XB,XC)
| ~ s_e(Xa,Xb,XB,XC)
| ~ s_e(Xb,Xc,XA,XB)
| s_e(Xa,Xc,XA,XC) ) ).

fof(aSatz3_1,axiom,
! [Xa,Xb] : s_t(Xa,Xb,Xb) ).

fof(aSatz3_2,axiom,
! [Xa,Xb,Xc] :
( ~ s_t(Xa,Xb,Xc)
| s_t(Xc,Xb,Xa) ) ).

fof(aSatz3_3,axiom,
! [Xa1,Xb1] : s_t(Xa1,Xa1,Xb1) ).

fof(aSatz3_4,axiom,
! [Xa,Xb,Xc] :
( ~ s_t(Xa,Xb,Xc)
| ~ s_t(Xb,Xa,Xc)
| Xa = Xb ) ).

fof(aSatz3_5a,axiom,
! [Xa,Xb,Xd,Xc] :
( ~ s_t(Xa,Xb,Xd)
| ~ s_t(Xb,Xc,Xd)
| s_t(Xa,Xb,Xc) ) ).

fof(aSatz3_6a,axiom,
! [Xa,Xb,Xc,Xd] :
( ~ s_t(Xa,Xb,Xc)
| ~ s_t(Xa,Xc,Xd)
| s_t(Xb,Xc,Xd) ) ).

fof(aSatz3_7a,axiom,
! [Xa,Xb,Xc,Xd] :
( ~ s_t(Xa,Xb,Xc)
| ~ s_t(Xb,Xc,Xd)
| Xb = Xc
| s_t(Xa,Xc,Xd) ) ).

fof(aSatz3_5b,axiom,
! [Xa,Xb,Xd,Xc] :
( ~ s_t(Xa,Xb,Xd)
| ~ s_t(Xb,Xc,Xd)
| s_t(Xa,Xc,Xd) ) ).

fof(aSatz3_6b,axiom,
! [Xa,Xb,Xc,Xd] :
( ~ s_t(Xa,Xb,Xc)
| ~ s_t(Xa,Xc,Xd)
| s_t(Xa,Xb,Xd) ) ).

fof(aSatz3_7b,axiom,
! [Xa,Xb,Xc,Xd] :
( ~ s_t(Xa,Xb,Xc)
| ~ s_t(Xb,Xc,Xd)
| Xb = Xc
| s_t(Xa,Xb,Xd) ) ).

fof(aSatz3_13a,axiom,
alpha != beta ).

fof(aSatz3_13b,axiom,
beta != gamma ).

fof(aSatz3_13c,axiom,
alpha != gamma ).

fof(aSatz3_14a,axiom,
! [Xa,Xb] : s_t(Xa,Xb,ext(Xa,Xb,alpha,gamma)) ).

fof(aSatz3_14b,axiom,
! [Xb,Xa] : Xb != ext(Xa,Xb,alpha,gamma) ).

fof(aSatz3_17,axiom,
! [Xa,Xb,Xc,Xa1,Xb1,Xp] :
( ( ~ s_t(Xa,Xb,Xc)
| ~ s_t(Xa1,Xb1,Xc)
| ~ s_t(Xa,Xp,Xa1)
| s_t(Xp,crossbar(Xa,Xb,Xc,Xa1,Xb1,Xp),Xc) )
& ( ~ s_t(Xa,Xb,Xc)
| ~ s_t(Xa1,Xb1,Xc)
| ~ s_t(Xa,Xp,Xa1)
| s_t(Xb,crossbar(Xa,Xb,Xc,Xa1,Xb1,Xp),Xb1) ) ) ).

fof(aSatz4_2,axiom,
! [Xa,Xb,Xc,Xd,Xa1,Xb1,Xc1,Xd1] :
( ~ s_ifs(Xa,Xb,Xc,Xd,Xa1,Xb1,Xc1,Xd1)
| s_e(Xb,Xd,Xb1,Xd1) ) ).

fof(aSatz4_3,axiom,
! [Xa,Xb,Xc,Xa1,Xb1,Xc1] :
( ~ s_t(Xa,Xb,Xc)
| ~ s_t(Xa1,Xb1,Xc1)
| ~ s_e(Xa,Xc,Xa1,Xc1)
| ~ s_e(Xb,Xc,Xb1,Xc1)
| s_e(Xa,Xb,Xa1,Xb1) ) ).

fof(aSatz4_5,axiom,
! [Xa,Xb,Xc,Xa1,Xc1] :
( ( ~ s_t(Xa,Xb,Xc)
| ~ s_e(Xa,Xc,Xa1,Xc1)
| s_t(Xa1,insert(Xa,Xb,Xa1,Xc1),Xc1) )
& ( ~ s_t(Xa,Xb,Xc)
| ~ s_e(Xa,Xc,Xa1,Xc1)
| s_e3(Xa,Xb,Xc,Xa1,insert(Xa,Xb,Xa1,Xc1),Xc1) ) ) ).

fof(aSatz4_6,axiom,
! [Xa,Xb,Xc,Xa1,Xb1,Xc1] :
( ~ s_t(Xa,Xb,Xc)
| ~ s_e3(Xa,Xb,Xc,Xa1,Xb1,Xc1)
| s_t(Xa1,Xb1,Xc1) ) ).

fof(aSatz4_11a,axiom,
! [Xa,Xb,Xc] :
( ~ s_col(Xa,Xb,Xc)
| s_col(Xb,Xc,Xa) ) ).

fof(aSatz4_11b,axiom,
! [Xa,Xb,Xc] :
( ~ s_col(Xa,Xb,Xc)
| s_col(Xc,Xa,Xb) ) ).

fof(aSatz4_11c,axiom,
! [Xa,Xb,Xc] :
( ~ s_col(Xa,Xb,Xc)
| s_col(Xc,Xb,Xa) ) ).

fof(aSatz4_11d,axiom,
! [Xa,Xb,Xc] :
( ~ s_col(Xa,Xb,Xc)
| s_col(Xb,Xa,Xc) ) ).

fof(aSatz4_11e,axiom,
! [Xa,Xb,Xc] :
( ~ s_col(Xa,Xb,Xc)
| s_col(Xa,Xc,Xb) ) ).

fof(aSatz4_12,axiom,
! [Xa,Xb] : s_col(Xa,Xa,Xb) ).

fof(aSatz4_12b,axiom,
! [Xa,Xb] : s_col(Xa,Xb,Xa) ).

fof(aSatz4_13,axiom,
! [Xa,Xb,Xc,Xa1,Xb1,Xc1] :
( ~ s_col(Xa,Xb,Xc)
| ~ s_e3(Xa,Xb,Xc,Xa1,Xb1,Xc1)
| s_col(Xa1,Xb1,Xc1) ) ).

fof(aSatz4_14,axiom,
! [Xa,Xb,Xc,Xa1,Xb1] :
( ~ s_col(Xa,Xb,Xc)
| ~ s_e(Xa,Xb,Xa1,Xb1)
| s_e3(Xa,Xb,Xc,Xa1,Xb1,insert5(Xa,Xb,Xc,Xa1,Xb1)) ) ).

fof(aSatz4_16,axiom,
! [Xa,Xb,Xc,Xd,Xa1,Xb1,Xc1,Xd1] :
( ~ s_fs(Xa,Xb,Xc,Xd,Xa1,Xb1,Xc1,Xd1)
| Xa = Xb
| s_e(Xc,Xd,Xc1,Xd1) ) ).

fof(aSatz4_17,axiom,
! [Xa,Xb,Xc,Xp,Xq] :
( Xa = Xb
| ~ s_col(Xa,Xb,Xc)
| ~ s_e(Xa,Xp,Xa,Xq)
| ~ s_e(Xb,Xp,Xb,Xq)
| s_e(Xc,Xp,Xc,Xq) ) ).

fof(aSatz4_18,axiom,
! [Xa,Xb,Xc,Xc1] :
( Xa = Xb
| ~ s_col(Xa,Xb,Xc)
| ~ s_e(Xa,Xc,Xa,Xc1)
| ~ s_e(Xb,Xc,Xb,Xc1)
| Xc = Xc1 ) ).

fof(aSatz4_19,axiom,
! [Xa,Xc,Xb,Xc1] :
( ~ s_t(Xa,Xc,Xb)
| ~ s_e(Xa,Xc,Xa,Xc1)
| ~ s_e(Xb,Xc,Xb,Xc1)
| Xc = Xc1 ) ).

fof(aSatz5_1,axiom,
! [Xa,Xb,Xc,Xd] :
( Xa = Xb
| ~ s_t(Xa,Xb,Xc)
| ~ s_t(Xa,Xb,Xd)
| s_t(Xa,Xc,Xd)
| s_t(Xa,Xd,Xc) ) ).

fof(aSatz5_2,axiom,
! [Xa,Xb,Xc,Xd] :
( Xa = Xb
| ~ s_t(Xa,Xb,Xc)
| ~ s_t(Xa,Xb,Xd)
| s_t(Xb,Xc,Xd)
| s_t(Xb,Xd,Xc) ) ).

fof(aSatz5_3,axiom,
! [Xa,Xb,Xd,Xc] :
( ~ s_t(Xa,Xb,Xd)
| ~ s_t(Xa,Xc,Xd)
| s_t(Xa,Xb,Xc)
| s_t(Xa,Xc,Xb) ) ).

fof(aSatz5_5a,axiom,
! [Xa,Xb,Xc,Xd] :
( ( ~ le(Xa,Xb,Xc,Xd)
| s_t(Xa,Xb,ins(Xc,Xd,Xa,Xb)) )
& ( ~ le(Xa,Xb,Xc,Xd)
| s_e(Xa,ins(Xc,Xd,Xa,Xb),Xc,Xd) )
& ( ~ le(Xa,Xb,Xc,Xd)
| ins(Xc,Xd,Xa,Xb) = ext(Xa,Xb,insert(Xa,Xb,Xc,Xd),Xd) ) ) ).

fof(aSatz5_5b,axiom,
! [Xa,Xb,Xc,Xd,Xe] :
( le(Xa,Xb,Xc,Xd)
| ~ s_t(Xa,Xb,Xe)
| ~ s_e(Xa,Xe,Xc,Xd) ) ).

fof(aSatz5_6,axiom,
! [Xa,Xb,Xc,Xd,Xa1,Xb1,Xc1,Xd1] :
( ~ le(Xa,Xb,Xc,Xd)
| ~ s_e(Xa,Xb,Xa1,Xb1)
| ~ s_e(Xc,Xd,Xc1,Xd1)
| le(Xa1,Xb1,Xc1,Xd1) ) ).

fof(aSatz5_7,axiom,
! [Xa,Xb] : le(Xa,Xb,Xa,Xb) ).

fof(aSatz5_8,axiom,
! [Xa,Xb,Xc,Xd,Xe,Xf] :
( ~ le(Xa,Xb,Xc,Xd)
| ~ le(Xc,Xd,Xe,Xf)
| le(Xa,Xb,Xe,Xf) ) ).

fof(aSatz5_9,axiom,
! [Xa,Xb,Xc,Xd] :
( ~ le(Xa,Xb,Xc,Xd)
| ~ le(Xc,Xd,Xa,Xb)
| s_e(Xa,Xb,Xc,Xd) ) ).

fof(aSatz5_10,axiom,
! [Xa,Xb,Xc,Xd] :
( le(Xa,Xb,Xc,Xd)
| le(Xc,Xd,Xa,Xb) ) ).

fof(aSatz5_11,axiom,
! [Xa,Xc,Xd] : le(Xa,Xa,Xc,Xd) ).

fof(aSatz5_12a1,axiom,
! [Xa,Xb,Xc] :
( ~ s_col(Xa,Xb,Xc)
| ~ s_t(Xa,Xb,Xc)
| le(Xa,Xb,Xa,Xc) ) ).

fof(aSatz5_12a2,axiom,
! [Xa,Xb,Xc] :
( ~ s_col(Xa,Xb,Xc)
| ~ s_t(Xa,Xb,Xc)
| le(Xb,Xc,Xa,Xc) ) ).

fof(aNarbouxLemma1,axiom,
! [Xa,Xb,Xc] :
( ~ s_t(Xa,Xb,Xc)
| ~ s_e(Xa,Xc,Xa,Xb)
| Xc = Xb ) ).

fof(aSatz5_12b,axiom,
! [Xa,Xb,Xc] :
( ~ s_col(Xa,Xb,Xc)
| s_t(Xa,Xb,Xc)
| ~ le(Xa,Xb,Xa,Xc)
| ~ le(Xb,Xc,Xa,Xc) ) ).

fof(aSatz6_2a,axiom,
! [Xa,Xp,Xb,Xc] :
( Xa = Xp
| Xb = Xp
| Xc = Xp
| ~ s_t(Xa,Xp,Xc)
| ~ s_t(Xb,Xp,Xc)
| sameside(Xa,Xp,Xb) ) ).

fof(aSatz6_2b,axiom,
! [Xa,Xp,Xb,Xc] :
( Xa = Xp
| Xb = Xp
| Xc = Xp
| ~ s_t(Xa,Xp,Xc)
| s_t(Xb,Xp,Xc)
| ~ sameside(Xa,Xp,Xb) ) ).

fof(aSatz6_3a,axiom,
! [Xa,Xp,Xb] :
( ( ~ sameside(Xa,Xp,Xb)
| Xa != Xp )
& ( ~ sameside(Xa,Xp,Xb)
| Xb != Xp )
& ( ~ sameside(Xa,Xp,Xb)
| c63(Xa,Xp,Xb) != Xp )
& ( ~ sameside(Xa,Xp,Xb)
| s_t(Xa,Xp,c63(Xa,Xp,Xb)) )
& ( ~ sameside(Xa,Xp,Xb)
| s_t(Xb,Xp,c63(Xa,Xp,Xb)) ) ) ).

fof(aSatz6_3b,axiom,
! [Xa,Xp,Xb,Xc] :
( sameside(Xa,Xp,Xb)
| Xa = Xp
| Xb = Xp
| Xc = Xp
| ~ s_t(Xa,Xp,Xc)
| ~ s_t(Xb,Xp,Xc) ) ).

fof(aSatz6_4a,axiom,
! [Xa,Xp,Xb] :
( ( ~ sameside(Xa,Xp,Xb)
| s_col(Xa,Xp,Xb) )
& ( ~ sameside(Xa,Xp,Xb)
| ~ s_t(Xa,Xp,Xb) ) ) ).

fof(aSatz6_4b,axiom,
! [Xa,Xp,Xb] :
( sameside(Xa,Xp,Xb)
| ~ s_col(Xa,Xp,Xb)
| s_t(Xa,Xp,Xb) ) ).

fof(aSatz6_5,axiom,
! [Xa,Xp] :
( Xa = Xp
| sameside(Xa,Xp,Xa) ) ).

fof(aSatz6_6,axiom,
! [Xa,Xp,Xb] :
( ~ sameside(Xa,Xp,Xb)
| sameside(Xb,Xp,Xa) ) ).

fof(aSatz6_7,axiom,
! [Xa,Xp,Xb,Xc] :
( ~ sameside(Xa,Xp,Xb)
| ~ sameside(Xb,Xp,Xc)
| sameside(Xa,Xp,Xc) ) ).

fof(aSatz6_11a,axiom,
! [Xr,Xa,Xb,Xc] :
( ( Xr = Xa
| Xb = Xc
| sameside(insert(Xb,Xc,Xa,Xr),Xa,Xr) )
& ( Xr = Xa
| Xb = Xc
| s_e(Xa,insert(Xb,Xc,Xa,Xr),Xb,Xc) ) ) ).

fof(aSatz6_11b,axiom,
! [Xr,Xa,Xb,Xc,Xp,Xq] :
( Xr = Xa
| Xb = Xc
| ~ sameside(Xp,Xa,Xr)
| ~ s_e(Xa,Xp,Xb,Xc)
| ~ sameside(Xq,Xa,Xr)
| ~ s_e(Xa,Xq,Xb,Xc)
| Xp = Xq ) ).

fof(aSatz6_13a,axiom,
! [Xa,Xp,Xb] :
( ~ sameside(Xa,Xp,Xb)
| ~ le(Xp,Xa,Xp,Xb)
| s_t(Xp,Xa,Xb) ) ).

fof(aSatz6_13b,axiom,
! [Xa,Xp,Xb] :
( ~ sameside(Xa,Xp,Xb)
| le(Xp,Xa,Xp,Xb)
| ~ s_t(Xp,Xa,Xb) ) ).

fof(aSatz6_15a,axiom,
! [Xp,Xq,Xr,Xa] :
( Xp = Xq
| Xp = Xr
| ~ s_t(Xq,Xp,Xr)
| ~ s_col(Xa,Xp,Xq)
| Xa = Xp
| sameside(Xa,Xp,Xq)
| sameside(Xa,Xp,Xr) ) ).

fof(aSatz6_15b,axiom,
! [Xp,Xq,Xr,Xa] :
( Xp = Xq
| Xp = Xr
| ~ s_t(Xq,Xp,Xr)
| ~ sameside(Xa,Xp,Xq)
| s_col(Xa,Xp,Xq) ) ).

fof(aSatz6_15c,axiom,
! [Xp,Xq,Xr,Xa] :
( Xp = Xq
| Xp = Xr
| ~ s_t(Xq,Xp,Xr)
| ~ sameside(Xa,Xp,Xr)
| s_col(Xa,Xp,Xq) ) ).

fof(aSatz6_15d,axiom,
! [Xp,Xq,Xr,Xa] :
( Xp = Xq
| Xp = Xr
| ~ s_t(Xq,Xp,Xr)
| Xa != Xp
| s_col(Xa,Xp,Xq) ) ).

fof(aSatz6_16a,axiom,
! [Xa,Xb,Xc,Xd] :
( Xa = Xb
| ~ s_t(Xc,Xa,Xb)
| ~ s_t(Xd,Xa,Xb)
| s_t(Xd,Xc,Xb)
| s_t(Xc,Xd,Xb) ) ).

fof(aSatz6_16b,axiom,
! [Xp,Xq,Xcs,Xr] :
( Xp = Xq
| Xcs = Xp
| ~ s_col(Xp,Xq,Xcs)
| ~ s_col(Xp,Xq,Xr)
| s_col(Xp,Xcs,Xr) ) ).

fof(aSatz6_17a,axiom,
! [Xp,Xq] :
( Xp = Xq
| s_col(Xp,Xq,Xp) ) ).

fof(aSatz6_17b,axiom,
! [Xp,Xq,Xr] :
( Xp = Xq
| ~ s_col(Xp,Xq,Xr)
| s_col(Xq,Xp,Xr) ) ).

fof(aSatz6_18,axiom,
! [Xa,Xb,Xp,Xq,Xr] :
( Xa = Xb
| Xp = Xq
| ~ s_col(Xp,Xq,Xa)
| ~ s_col(Xp,Xq,Xb)
| ~ s_col(Xp,Xq,Xr)
| s_col(Xa,Xb,Xr) ) ).

fof(aSatz6_21,axiom,
! [Xa,Xb,Xp,Xq,Xc,Xd,Xe] :
( Xa = Xb
| Xp = Xq
| ~ s_col(Xa,Xb,Xc)
| ~ s_col(Xp,Xq,Xc)
| ~ s_col(Xa,Xb,Xd)
| ~ s_col(Xp,Xq,Xd)
| Xc = Xd
| ~ s_col(Xa,Xb,Xe)
| s_col(Xp,Xq,Xe) ) ).

fof(aSatz6_25,axiom,
! [Xa,Xb] :
( Xa = Xb
| ~ s_col(Xa,Xb,pointOffLine(Xa,Xb)) ) ).

fof(aSatz6_28,axiom,
! [Xa,Xb,Xc,Xa1,Xb1,Xc1] :
( ~ sameside(Xa,Xb,Xc)
| ~ sameside(Xa1,Xb1,Xc1)
| ~ s_e(Xb,Xa,Xb1,Xa1)
| ~ s_e(Xb,Xc,Xb1,Xc1)
| s_e(Xa,Xc,Xa1,Xc1) ) ).

fof(aSatz7_2,axiom,
! [Xa,Xm,Xb] :
( ~ s_m(Xa,Xm,Xb)
| s_m(Xb,Xm,Xa) ) ).

fof(aSatz7_3a,axiom,
! [Xa,Xm] :
( ~ s_m(Xa,Xm,Xa)
| Xm = Xa ) ).

fof(aSatz7_3b,axiom,
! [Xa,Xm] :
( s_m(Xa,Xm,Xa)
| Xm != Xa ) ).

fof(aSatz7_4a,axiom,
! [Xp,Xa] : s_m(Xp,Xa,s(Xa,Xp)) ).

fof(aSatz7_4b,axiom,
! [Xp,Xa,Xr,Xq] :
( ~ s_m(Xp,Xa,Xr)
| ~ s_m(Xp,Xa,Xq)
| Xr = Xq ) ).

fof(aSatz7_6,axiom,
! [Xp,Xa,Xq] :
( ~ s_m(Xp,Xa,Xq)
| Xq = s(Xa,Xp) ) ).

fof(aSatz7_7,axiom,
! [Xa,Xp] : s(Xa,s(Xa,Xp)) = Xp ).

fof(aSatz7_8,axiom,
! [Xa,Xp,Xr,Xq] :
( s(Xa,Xp) != Xr
| s(Xa,Xq) != Xr
| Xp = Xq ) ).

fof(aSatz7_9,axiom,
! [Xa,Xp,Xq] :
( s(Xa,Xp) != s(Xa,Xq)
| Xp = Xq ) ).

fof(aSatz7_10a,axiom,
! [Xa,Xp] :
( s(Xa,Xp) != Xp
| Xp = Xa ) ).

fof(aSatz7_10b,axiom,
! [Xa,Xp] :
( s(Xa,Xp) = Xp
| Xp != Xa ) ).

fof(aSatz7_13,axiom,
! [Xp,Xq,Xa] : s_e(Xp,Xq,s(Xa,Xp),s(Xa,Xq)) ).

fof(aSatz7_15a,axiom,
! [Xp,Xq,Xr,Xa] :
( ~ s_t(Xp,Xq,Xr)
| s_t(s(Xa,Xp),s(Xa,Xq),s(Xa,Xr)) ) ).

fof(aSatz7_15b,axiom,
! [Xp,Xq,Xr,Xa] :
( s_t(Xp,Xq,Xr)
| ~ s_t(s(Xa,Xp),s(Xa,Xq),s(Xa,Xr)) ) ).

fof(aSatz7_16a,axiom,
! [Xp,Xq,Xr,Xcs,Xa] :
( ~ s_e(Xp,Xq,Xr,Xcs)
| s_e(s(Xa,Xp),s(Xa,Xq),s(Xa,Xr),s(Xa,Xcs)) ) ).

fof(aSatz7_16b,axiom,
! [Xp,Xq,Xr,Xcs,Xa] :
( s_e(Xp,Xq,Xr,Xcs)
| ~ s_e(s(Xa,Xp),s(Xa,Xq),s(Xa,Xr),s(Xa,Xcs)) ) ).

fof(aSatz7_17,axiom,
! [Xp,Xa,Xq,Xb] :
( ~ s_m(Xp,Xa,Xq)
| ~ s_m(Xp,Xb,Xq)
| Xa = Xb ) ).

fof(aSatz7_18,axiom,
! [Xa,Xp,Xb] :
( s(Xa,Xp) != s(Xb,Xp)
| Xa = Xb ) ).

fof(aSatz7_19,axiom,
! [Xa,Xb,Xp] :
( s(Xa,s(Xb,Xp)) != s(Xb,s(Xa,Xp))
| Xa = Xb ) ).

fof(aSatz7_20,axiom,
! [Xa,Xm,Xb] :
( ~ s_col(Xa,Xm,Xb)
| ~ s_e(Xm,Xa,Xm,Xb)
| Xa = Xb
| s_m(Xa,Xm,Xb) ) ).

fof(aSatz7_21,axiom,
! [Xa,Xb,Xc,Xd,Xp] :
( s_col(Xa,Xb,Xc)
| Xb = Xd
| ~ s_e(Xa,Xb,Xc,Xd)
| ~ s_e(Xb,Xc,Xd,Xa)
| ~ s_col(Xa,Xp,Xc)
| ~ s_col(Xb,Xp,Xd)
| s_m(Xa,Xp,Xc) ) ).

fof(aSatz7_22a,axiom,
! [Xa1,Xm1,Xb1,Xc,Xb2,Xm2,Xa2] :
( ~ s_kf(Xa1,Xm1,Xb1,Xc,Xb2,Xm2,Xa2)
| s_t(Xm1,Xc,Xm2)
| ~ le(Xc,Xa1,Xc,Xa2) ) ).

fof(aSatz7_22b,axiom,
! [Xa1,Xm1,Xb1,Xc,Xb2,Xm2,Xa2] :
( ~ s_kf(Xa1,Xm1,Xb1,Xc,Xb2,Xm2,Xa2)
| s_t(Xm1,Xc,Xm2) ) ).

fof(aSatz7_22,axiom,
! [Xa1,Xc,Xa2,Xb1,Xb2,Xm1,Xm2] :
( ~ s_t(Xa1,Xc,Xa2)
| ~ s_t(Xb1,Xc,Xb2)
| ~ s_e(Xc,Xa1,Xc,Xb1)
| ~ s_e(Xc,Xa2,Xc,Xb2)
| ~ s_m(Xa1,Xm1,Xb1)
| ~ s_m(Xa2,Xm2,Xb2)
| s_t(Xm1,Xc,Xm2) ) ).

fof(aSatz7_25,axiom,
! [Xc,Xa,Xb] :
( ~ s_e(Xc,Xa,Xc,Xb)
| s_m(Xa,isomidpoint(Xa,Xb,Xc),Xb) ) ).

fof(aSatz8_2,axiom,
! [Xa,Xb,Xc] :
( ~ s_r(Xa,Xb,Xc)
| s_r(Xc,Xb,Xa) ) ).

fof(aSatz8_3,axiom,
! [Xa,Xb,Xc,Xa1] :
( ~ s_r(Xa,Xb,Xc)
| Xa = Xb
| ~ s_col(Xb,Xa,Xa1)
| s_r(Xa1,Xb,Xc) ) ).

fof(aSatz8_4,axiom,
! [Xa,Xb,Xc] :
( ~ s_r(Xa,Xb,Xc)
| s_r(Xa,Xb,s(Xb,Xc)) ) ).

fof(aSatz8_5,axiom,
! [Xa,Xb] : s_r(Xa,Xb,Xb) ).

fof(aSatz8_6,axiom,
! [Xa,Xb,Xc,Xa1] :
( ~ s_r(Xa,Xb,Xc)
| ~ s_r(Xa1,Xb,Xc)
| ~ s_t(Xa,Xc,Xa1)
| Xb = Xc ) ).

fof(aSatz8_7,axiom,
! [Xa,Xb,Xc] :
( ~ s_r(Xa,Xb,Xc)
| ~ s_r(Xa,Xc,Xb)
| Xb = Xc ) ).

fof(aSatz8_8,axiom,
! [Xa,Xb] :
( ~ s_r(Xa,Xb,Xa)
| Xa = Xb ) ).

fof(aSatz8_9,axiom,
! [Xa,Xb,Xc] :
( ~ s_r(Xa,Xb,Xc)
| ~ s_col(Xa,Xb,Xc)
| Xa = Xb
| Xc = Xb ) ).

fof(aSatz8_10,axiom,
! [Xa,Xb,Xc,Xa1,Xb1,Xc1] :
( ~ s_r(Xa,Xb,Xc)
| ~ s_e3(Xa,Xb,Xc,Xa1,Xb1,Xc1)
| s_r(Xa1,Xb1,Xc1) ) ).

fof(aSatz8_12a,axiom,
! [Xa,Xb,Xc,Xp,Xq] :
( ~ perpAt(Xa,Xb,Xc,Xp,Xq)
| perpAt(Xp,Xq,Xc,Xa,Xb) ) ).

fof(aSatz8_12b,axiom,
! [Xa,Xb,Xp,Xq] :
( ~ perp(Xa,Xb,Xp,Xq)
| perp(Xp,Xq,Xa,Xb) ) ).

fof(aSatz8_13a,axiom,
! [Xa,Xb,Xc,Xp,Xq] :
( ( ~ perpAt(Xa,Xb,Xc,Xp,Xq)
| Xa != Xb )
& ( ~ perpAt(Xa,Xb,Xc,Xp,Xq)
| Xp != Xq )
& ( ~ perpAt(Xa,Xb,Xc,Xp,Xq)
| s_col(Xp,Xq,Xc) )
& ( ~ perpAt(Xa,Xb,Xc,Xp,Xq)
| s_col(Xa,Xb,Xc) )
& ( ~ perpAt(Xa,Xb,Xc,Xp,Xq)
| s_col(Xa,Xb,f813(Xa,Xb,Xp,Xq,Xc)) )
& ( ~ perpAt(Xa,Xb,Xc,Xp,Xq)
| s_col(Xp,Xq,g813(Xa,Xb,Xp,Xq,Xc)) )
& ( ~ perpAt(Xa,Xb,Xc,Xp,Xq)
| Xc != f813(Xa,Xb,Xp,Xq,Xc) )
& ( ~ perpAt(Xa,Xb,Xc,Xp,Xq)
| Xc != g813(Xa,Xb,Xp,Xq,Xc) )
& ( ~ perpAt(Xa,Xb,Xc,Xp,Xq)
| s_r(f813(Xa,Xb,Xp,Xq,Xc),Xc,g813(Xa,Xb,Xp,Xq,Xc)) ) ) ).

fof(aSatz8_13b,axiom,
! [Xa,Xb,Xp,Xq,Xcx,U,V] :
( Xa = Xb
| Xp = Xq
| ~ s_col(Xp,Xq,Xcx)
| ~ s_col(Xa,Xb,Xcx)
| ~ s_col(Xa,Xb,U)
| ~ s_col(Xp,Xq,V)
| Xcx = U
| Xcx = V
| ~ s_r(U,Xcx,V)
| perpAt(Xa,Xb,Xcx,Xp,Xq) ) ).

fof(aSatz8_14a,axiom,
! [Xa,Xb,Xp,Xq] :
( ~ perp(Xa,Xb,Xp,Xq)
| ~ s_col(Xa,Xb,Xp)
| ~ s_col(Xa,Xb,Xq) ) ).

fof(aSatz8_14b,axiom,
! [Xa,Xb,Xp,Xq,Xc] :
( ~ perp(Xa,Xb,Xp,Xq)
| ~ s_col(Xa,Xb,Xc)
| ~ s_col(Xp,Xq,Xc)
| perpAt(Xa,Xb,Xc,Xp,Xq) ) ).

fof(aSatz8_14c,axiom,
! [Xa,Xb,Xc,Xp,Xq] :
( ~ perpAt(Xa,Xb,Xc,Xp,Xq)
| Xc = il(Xa,Xb,Xp,Xq) ) ).

fof(aSatz8_15,axiom,
! [Xa,Xb,Xd,Xc] :
( Xa = Xb
| ~ s_col(Xa,Xb,Xd)
| ~ perp(Xa,Xb,Xc,Xd)
| perpAt(Xa,Xb,Xd,Xc,Xd) ) ).

fof(aSatz8_16a,axiom,
! [Xa,Xb,Xp,Xq,Xc] :
( ( Xa = Xb
| ~ s_col(Xa,Xb,Xp)
| ~ s_col(Xa,Xb,Xq)
| Xq = Xp
| ~ s_col(Xa,Xb,Xc)
| ~ perp(Xa,Xb,Xc,Xp) )
& ( Xa = Xb
| ~ s_col(Xa,Xb,Xp)
| ~ s_col(Xa,Xb,Xq)
| Xq = Xp
| s_r(Xc,Xp,Xq)
| ~ perp(Xa,Xb,Xc,Xp) ) ) ).

fof(aSatz8_16b,axiom,
! [Xa,Xb,Xp,Xq,Xc] :
( Xa = Xb
| ~ s_col(Xa,Xb,Xp)
| ~ s_col(Xa,Xb,Xq)
| Xq = Xp
| perp(Xa,Xb,Xc,Xp)
| s_col(Xa,Xb,Xc)
| ~ s_r(Xc,Xp,Xq) ) ).

fof(aSatz8_18a,axiom,
! [Xa,Xb,Xc,Xp,Xq] :
( s_col(Xa,Xb,Xc)
| ~ s_col(Xa,Xb,Xp)
| ~ s_col(Xa,Xb,Xq)
| ~ perp(Xa,Xb,Xc,Xp)
| ~ perp(Xa,Xb,Xc,Xq)
| Xp = Xq ) ).

fof(aSatz8_18,axiom,
! [Xa,Xb,Xc] :
( ( s_col(Xa,Xb,Xc)
| s_col(Xa,Xb,foot(Xa,Xb,Xc)) )
& ( s_col(Xa,Xb,Xc)
| perp(Xa,Xb,Xc,foot(Xa,Xb,Xc)) ) ) ).

fof(aSatz8_20a,axiom,
! [Xa,Xb,Xc,Xp] :
( ~ s_r(Xa,Xb,Xc)
| ~ s_m(s(Xa,Xc),Xp,s(Xb,Xc))
| s_r(Xb,Xa,Xp) ) ).

fof(aSatz8_20b,axiom,
! [Xa,Xb,Xc,Xp] :
( ~ s_r(Xa,Xb,Xc)
| ~ s_m(s(Xa,Xc),Xp,s(Xb,Xc))
| Xb = Xc
| Xa != Xp ) ).

fof(aperp1,axiom,
! [Xa,Xb,Xp,Xq] :
( ~ perp(Xa,Xb,Xp,Xq)
| perp(Xb,Xa,Xp,Xq) ) ).

fof(aExtCol2,axiom,
! [Xa,Xb,Xc,Xd,Xp] :
( Xa = Xb
| Xc = Xd
| ~ s_col(Xa,Xb,Xc)
| ~ s_col(Xa,Xb,Xd)
| ~ s_col(Xc,Xd,Xp)
| s_col(Xa,Xb,Xp) ) ).

fof(aSatz8_21a,axiom,
! [Xa,Xb,Xc] :
( ( Xa = Xb
| perp(Xa,Xb,erect21a(Xa,Xb,Xc),Xa)
| s_col(Xa,Xb,Xc) )
& ( Xa = Xb
| s_col(Xa,Xb,erectAux21a(Xa,Xb,Xc))
| s_col(Xa,Xb,Xc) )
& ( Xa = Xb
| s_t(Xc,erectAux21a(Xa,Xb,Xc),erect21a(Xa,Xb,Xc))
| s_col(Xa,Xb,Xc) ) ) ).

fof(aSatz8_21,axiom,
! [Xa,Xb,Xc] :
( ( Xa = Xb
| perp(Xa,Xb,erect(Xa,Xb,Xc),Xa) )
& ( Xa = Xb
| s_col(Xa,Xb,erectAux(Xa,Xb,Xc)) )
& ( Xa = Xb
| s_t(Xc,erectAux(Xa,Xb,Xc),erect(Xa,Xb,Xc)) ) ) ).

fof(aSatz8_22b,axiom,
! [Xa,Xp,Xb,Xq,Xt] :
( ~ le(Xa,Xp,Xb,Xq)
| ~ perp(Xa,Xb,Xa,Xp)
| ~ perp(Xa,Xb,Xb,Xq)
| ~ s_t(Xp,Xt,Xq)
| ~ s_col(Xa,Xb,Xt)
| s_m(Xa,midpointAux(Xa,Xb,Xp,Xq,Xt),Xb) ) ).

fof(aSatz8_22,axiom,
! [Xa,Xb] : s_m(Xa,midpoint(Xa,Xb),Xb) ).

fof(aSatz8_24a,axiom,
! [Xp,Xa,Xb,Xq,Xt,Xr,Xcx] :
( ~ perp(Xp,Xa,Xa,Xb)
| ~ perp(Xq,Xb,Xa,Xb)
| ~ s_col(Xa,Xb,Xt)
| ~ s_t(Xp,Xt,Xq)
| ~ s_t(Xb,Xr,Xq)
| ~ s_e(Xa,Xp,Xb,Xr)
| Xcx != ip(Xp,Xt,Xq,Xb,Xr)
| s_m(Xa,Xcx,Xb) ) ).

fof(aSatz8_24b,axiom,
! [Xp,Xa,Xb,Xq,Xt,Xr,Xcx] :
( ~ perp(Xp,Xa,Xa,Xb)
| ~ perp(Xq,Xb,Xa,Xb)
| ~ s_col(Xa,Xb,Xt)
| ~ s_t(Xp,Xt,Xq)
| ~ s_t(Xb,Xr,Xq)
| ~ s_e(Xa,Xp,Xb,Xr)
| Xcx != ip(Xp,Xt,Xq,Xb,Xr)
| s_m(Xp,Xcx,Xr) ) ).

fof(aSatz8_24,conjecture,
! [Xp,Xa,Xb,Xq,Xt,Xr] :
( ~ perp(Xp,Xa,Xa,Xb)
| ~ perp(Xq,Xb,Xa,Xb)
| ~ s_col(Xa,Xb,Xt)
| ~ s_t(Xp,Xt,Xq)
| ~ s_t(Xb,Xr,Xq)
| ~ s_e(Xa,Xp,Xb,Xr)
| s_m(Xp,midpoint(Xa,Xb),Xr) ) ).

fof(d_insert,axiom,
! [Xa,Xb,Xa1,Xc1] : insert(Xa,Xb,Xa1,Xc1) = ext(ext(Xc1,Xa1,alpha,gamma),Xa1,Xa,Xb) ).

fof(d_Defn2_10,axiom,
! [Xa,Xb,Xc,Xd,Za,Zb,Zc,Zd] :
( ( ~ s_afs(Xa,Xb,Xc,Xd,Za,Zb,Zc,Zd)
| s_t(Xa,Xb,Xc) )
& ( ~ s_afs(Xa,Xb,Xc,Xd,Za,Zb,Zc,Zd)
| s_t(Za,Zb,Zc) )
& ( ~ s_afs(Xa,Xb,Xc,Xd,Za,Zb,Zc,Zd)
| s_e(Xa,Xb,Za,Zb) )
& ( ~ s_afs(Xa,Xb,Xc,Xd,Za,Zb,Zc,Zd)
| s_e(Xb,Xc,Zb,Zc) )
& ( ~ s_afs(Xa,Xb,Xc,Xd,Za,Zb,Zc,Zd)
| s_e(Xa,Xd,Za,Zd) )
& ( ~ s_afs(Xa,Xb,Xc,Xd,Za,Zb,Zc,Zd)
| s_e(Xb,Xd,Zb,Zd) )
& ( ~ s_t(Xa,Xb,Xc)
| ~ s_t(Za,Zb,Zc)
| ~ s_e(Xa,Xb,Za,Zb)
| ~ s_e(Xb,Xc,Zb,Zc)
| ~ s_e(Xa,Xd,Za,Zd)
| ~ s_e(Xb,Xd,Zb,Zd)
| s_afs(Xa,Xb,Xc,Xd,Za,Zb,Zc,Zd) ) ) ).

fof(d_Defn4_1,axiom,
! [Xa,Xb,Xc,Xd,Za,Zb,Zc,Zd] :
( ( ~ s_ifs(Xa,Xb,Xc,Xd,Za,Zb,Zc,Zd)
| s_t(Xa,Xb,Xc) )
& ( ~ s_ifs(Xa,Xb,Xc,Xd,Za,Zb,Zc,Zd)
| s_t(Za,Zb,Zc) )
& ( ~ s_ifs(Xa,Xb,Xc,Xd,Za,Zb,Zc,Zd)
| s_e(Xa,Xc,Za,Zc) )
& ( ~ s_ifs(Xa,Xb,Xc,Xd,Za,Zb,Zc,Zd)
| s_e(Xb,Xc,Zb,Zc) )
& ( ~ s_ifs(Xa,Xb,Xc,Xd,Za,Zb,Zc,Zd)
| s_e(Xa,Xd,Za,Zd) )
& ( ~ s_ifs(Xa,Xb,Xc,Xd,Za,Zb,Zc,Zd)
| s_e(Xc,Xd,Zc,Zd) )
& ( ~ s_t(Xa,Xb,Xc)
| ~ s_t(Za,Zb,Zc)
| ~ s_e(Xa,Xc,Za,Zc)
| ~ s_e(Xb,Xc,Zb,Zc)
| ~ s_e(Xa,Xd,Za,Zd)
| ~ s_e(Xc,Xd,Zc,Zd)
| s_ifs(Xa,Xb,Xc,Xd,Za,Zb,Zc,Zd) ) ) ).

fof(d_Defn4_4,axiom,
! [Xa1,Xa2,Xa3,Xb1,Xb2,Xb3] :
( ( ~ s_e3(Xa1,Xa2,Xa3,Xb1,Xb2,Xb3)
| s_e(Xa1,Xa2,Xb1,Xb2) )
& ( ~ s_e3(Xa1,Xa2,Xa3,Xb1,Xb2,Xb3)
| s_e(Xa1,Xa3,Xb1,Xb3) )
& ( ~ s_e3(Xa1,Xa2,Xa3,Xb1,Xb2,Xb3)
| s_e(Xa2,Xa3,Xb2,Xb3) )
& ( ~ s_e(Xa1,Xa2,Xb1,Xb2)
| ~ s_e(Xa1,Xa3,Xb1,Xb3)
| ~ s_e(Xa2,Xa3,Xb2,Xb3)
| s_e3(Xa1,Xa2,Xa3,Xb1,Xb2,Xb3) ) ) ).

fof(d_Defn4_10,axiom,
! [Xa,Xb,Xc] :
( ( ~ s_col(Xa,Xb,Xc)
| s_t(Xa,Xb,Xc)
| s_t(Xb,Xc,Xa)
| s_t(Xc,Xa,Xb) )
& ( s_col(Xa,Xb,Xc)
| ~ s_t(Xa,Xb,Xc) )
& ( s_col(Xa,Xb,Xc)
| ~ s_t(Xb,Xc,Xa) )
& ( s_col(Xa,Xb,Xc)
| ~ s_t(Xc,Xa,Xb) ) ) ).

fof(d_Defn4_15,axiom,
! [Xa,Xb,Xc,Xd,Xa1,Xb1,Xc1,Xd1] :
( ( ~ s_fs(Xa,Xb,Xc,Xd,Xa1,Xb1,Xc1,Xd1)
| s_col(Xa,Xb,Xc) )
& ( ~ s_fs(Xa,Xb,Xc,Xd,Xa1,Xb1,Xc1,Xd1)
| s_e3(Xa,Xb,Xc,Xa1,Xb1,Xc1) )
& ( ~ s_fs(Xa,Xb,Xc,Xd,Xa1,Xb1,Xc1,Xd1)
| s_e(Xa,Xd,Xa1,Xd1) )
& ( ~ s_fs(Xa,Xb,Xc,Xd,Xa1,Xb1,Xc1,Xd1)
| s_e(Xb,Xd,Xb1,Xd1) )
& ( ~ s_col(Xa,Xb,Xc)
| ~ s_e3(Xa,Xb,Xc,Xa1,Xb1,Xc1)
| ~ s_e(Xa,Xd,Xa1,Xd1)
| ~ s_e(Xb,Xd,Xb1,Xd1)
| s_fs(Xa,Xb,Xc,Xd,Xa1,Xb1,Xc1,Xd1) ) ) ).

fof(d_Defn5_4,axiom,
! [Xa,Xb,Xc,Xd,Y] :
( ( ~ le(Xa,Xb,Xc,Xd)
| s_t(Xc,insert(Xa,Xb,Xc,Xd),Xd) )
& ( ~ le(Xa,Xb,Xc,Xd)
| s_e(Xa,Xb,Xc,insert(Xa,Xb,Xc,Xd)) )
& ( ~ s_t(Xc,Y,Xd)
| ~ s_e(Xa,Xb,Xc,Y)
| le(Xa,Xb,Xc,Xd) ) ) ).

fof(d_Defn6_1,axiom,
! [Xa,Xp,Xb] :
( ( ~ sameside(Xa,Xp,Xb)
| Xa != Xp )
& ( ~ sameside(Xa,Xp,Xb)
| Xb != Xp )
& ( ~ sameside(Xa,Xp,Xb)
| s_t(Xp,Xa,Xb)
| s_t(Xp,Xb,Xa) )
& ( ~ s_t(Xp,Xa,Xb)
| Xa = Xp
| xb = Xp
| sameside(Xa,Xp,Xb) )
& ( ~ s_t(Xp,Xb,Xa)
| Xa = Xp
| Xb = Xp
| sameside(Xa,Xp,Xb) ) ) ).

fof(d_Defn7_1,axiom,
! [Xa,Xm,Xb] :
( ( ~ s_m(Xa,Xm,Xb)
| s_t(Xa,Xm,Xb) )
& ( ~ s_m(Xa,Xm,Xb)
| s_e(Xm,Xa,Xm,Xb) )
& ( ~ s_t(Xa,Xm,Xb)
| ~ s_e(Xm,Xa,Xm,Xb)
| s_m(Xa,Xm,Xb) ) ) ).

fof(d_Defn7_23,axiom,
! [Xa1,Xm1,Xb1,Xc,Xb2,Xm2,Xa2] :
( ( ~ s_kf(Xa1,Xm1,Xb1,Xc,Xb2,Xm2,Xa2)
| s_t(Xa1,Xc,Xa2) )
& ( ~ s_kf(Xa1,Xm1,Xb1,Xc,Xb2,Xm2,Xa2)
| s_t(Xb1,Xc,Xb2) )
& ( ~ s_kf(Xa1,Xm1,Xb1,Xc,Xb2,Xm2,Xa2)
| s_e(Xc,Xa1,Xc,Xb1) )
& ( ~ s_kf(Xa1,Xm1,Xb1,Xc,Xb2,Xm2,Xa2)
| s_e(Xc,Xa2,Xc,Xb2) )
& ( ~ s_kf(Xa1,Xm1,Xb1,Xc,Xb2,Xm2,Xa2)
| s_m(Xa1,Xm1,Xb1) )
& ( ~ s_kf(Xa1,Xm1,Xb1,Xc,Xb2,Xm2,Xa2)
| s_m(Xa2,Xm2,Xb2) )
& ( ~ s_t(Xa1,Xc,Xa2)
| ~ s_t(Xb1,Xc,Xb2)
| ~ s_e(Xc,Xa1,Xc,Xb1)
| ~ s_e(Xc,Xa2,Xc,Xb2)
| ~ s_m(Xa1,Xm1,Xb1)
| ~ s_m(Xa2,Xm2,Xb2)
| s_kf(Xa1,Xm1,Xb1,Xc,Xb2,Xm2,Xa2) ) ) ).

fof(d_Defn8_1,axiom,
! [Xa,Xb,Xc] :
( ( ~ s_r(Xa,Xb,Xc)
| s_e(Xa,Xc,Xa,s(Xb,Xc)) )
& ( s_r(Xa,Xb,Xc)
| ~ s_e(Xa,Xc,Xa,s(Xb,Xc)) ) ) ).

fof(d_Defn8_11a,axiom,
! [Y,Z,X,Y1,Z1,U,V] :
( ( ~ perpAt(Y,Z,X,Y1,Z1)
| s_col(Y,Z,X) )
& ( ~ perpAt(Y,Z,X,Y1,Z1)
| s_col(Y1,Z1,X) )
& ( ~ perpAt(Y,Z,X,Y1,Z1)
| Y != Z )
& ( ~ perpAt(Y,Z,X,Y1,Z1)
| Y1 != Z1 )
& ( ~ perpAt(Y,Z,X,Y1,Z1)
| ~ s_col(Y,Z,U)
| ~ s_col(Y1,Z1,V)
| s_r(U,X,V) )
& ( perpAt(Y,Z,X,Y1,Z1)
| Y = Z
| Y1 = Z1
| ~ s_col(Y,Z,X)
| ~ s_col(Y1,Z1,X)
| s_col(Y,Z,f811(Y,Z,Y1,Z1,X)) )
& ( perpAt(Y,Z,X,Y1,Z1)
| Y = Z
| Y1 = Z1
| ~ s_col(Y,Z,X)
| ~ s_col(Y1,Z1,X)
| s_col(Y1,Z1,g811(Y,Z,Y1,Z1,X)) )
& ( perpAt(Y,Z,X,Y1,Z1)
| Y = Z
| Y1 = Z1
| ~ s_col(Y,Z,X)
| ~ s_col(Y1,Z1,X)
| ~ s_r(f811(Y,Z,Y1,Z1,X),X,g811(Y,Z,Y1,Z1,X)) ) ) ).

fof(d_Defn8_11b,axiom,
! [Xp,Xq,Xp1,Xq1,X] :
( ( ~ perp(Xp,Xq,Xp1,Xq1)
| perpAt(Xp,Xq,il(Xp,Xq,Xp1,Xq1),Xp1,Xq1) )
& ( perp(Xp,Xq,Xp1,Xq1)
| ~ perpAt(Xp,Xq,X,Xp1,Xq1) ) ) ).

%------------------------------------------------------------------------------
```