TPTP Problem File: GEO538+1.p
View Solutions
- Solve Problem
%------------------------------------------------------------------------------
% File : GEO538+1 : TPTP v8.2.0. Released v7.0.0.
% Domain : Geometry
% Problem : Hilbert's parallel axiom, case 2
% Version : Especial.
% English :
% Refs : [Urb16] Urban (2016), Email to Geoff Sutcliffe
% : [BW17] Beeson & Wos (2017), Finding Proofs in Tarskian Geomet
% Source : [Urb16]
% Names : Lemma12.11-case2.in.p [BW17]
% Status : Theorem
% Rating : 1.00 v7.0.0
% Syntax : Number of formulae : 233 ( 20 unt; 0 def)
% Number of atoms : 1198 ( 264 equ)
% Maximal formula atoms : 42 ( 5 avg)
% Number of connectives : 1611 ( 646 ~; 855 |; 110 &)
% ( 0 <=>; 0 =>; 0 <=; 0 <~>)
% Maximal formula depth : 31 ( 9 avg)
% Maximal term depth : 3 ( 1 avg)
% Number of predicates : 19 ( 18 usr; 0 prp; 2-8 aty)
% Number of functors : 47 ( 47 usr; 9 con; 0-7 aty)
% Number of variables : 1037 (1037 !; 0 ?)
% SPC : FOF_THM_RFO_SEQ
% Comments :
%------------------------------------------------------------------------------
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,axiom,
! [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(aext1,axiom,
! [Xa,Xb,Xc,Xd] :
( ~ s_col(Xa,Xb,Xc)
| ~ s_col(Xa,Xb,Xd)
| Xa = Xb
| s_col(Xa,Xc,Xd) ) ).
fof(aExtPerp,axiom,
! [Xa,Xb,Xp,Xq,Xr,Xc,Xd] :
( ~ s_col(Xa,Xb,Xp)
| ~ s_col(Xa,Xb,Xq)
| Xp = Xq
| Xa = Xb
| ~ perpAt(Xp,Xq,Xr,Xc,Xd)
| perpAt(Xa,Xb,Xr,Xc,Xd) ) ).
fof(aExtPerp2,axiom,
! [Xa,Xb,Xp,Xq,Xc,Xd] :
( ~ s_col(Xa,Xb,Xp)
| ~ s_col(Xa,Xb,Xq)
| Xp = Xq
| Xa = Xb
| ~ perp(Xp,Xq,Xc,Xd)
| perp(Xa,Xb,Xc,Xd) ) ).
fof(aExtPerp3,axiom,
! [Xa,Xb,Xc,Xd] :
( Xa = Xb
| Xa = Xc
| Xb = Xc
| Xd = Xc
| Xa = Xd
| ~ perp(Xb,Xa,Xa,Xc)
| ~ s_col(Xa,Xc,Xd)
| perp(Xb,Xa,Xa,Xd) ) ).
fof(aExtPerp4,axiom,
! [Xa,Xb,Xp,Xq] :
( ~ perp(Xa,Xb,Xp,Xq)
| perp(Xa,Xb,Xq,Xp) ) ).
fof(aExtPerp5,axiom,
! [Xa,Xb,Xp,Xq,Xc,Xd] :
( ~ s_col(Xa,Xb,Xp)
| ~ s_col(Xa,Xb,Xq)
| Xp = Xq
| Xa = Xb
| perp(Xp,Xq,Xc,Xd)
| ~ perp(Xa,Xb,Xc,Xd) ) ).
fof(aExtPerp6,axiom,
( ~ s_col(xa,xb,xp)
| ~ s_col(xa,xb,xq)
| xp = xq
| xa = xb
| perp(xc,xd,xp,xq)
| ~ perp(xc,xd,xa,xb) ) ).
fof(aExtSameSide1,axiom,
! [Xa,Xb,Xc,Xp,Xq] :
( ~ s_col(Xa,Xb,Xc)
| Xa = Xb
| Xa = Xc
| ~ samesideline(Xp,Xq,Xa,Xb)
| samesideline(Xp,Xq,Xa,Xc) ) ).
fof(aExtSameSide2,axiom,
! [Xa,Xb,Xp,Xq] :
( Xa = Xb
| ~ samesideline(Xp,Xq,Xa,Xb)
| samesideline(Xp,Xq,Xb,Xa) ) ).
fof(aSatz9_2,axiom,
! [Xa,Xp,Xq,Xb] :
( ~ opposite(Xa,Xp,Xq,Xb)
| opposite(Xb,Xp,Xq,Xa) ) ).
fof(aSatz9_3a,axiom,
! [Xa,Xp,Xq,Xc,Xm,Xr,Xb] :
( ~ opposite(Xa,Xp,Xq,Xc)
| ~ s_col(Xp,Xq,Xm)
| ~ s_m(Xa,Xm,Xc)
| ~ s_col(Xp,Xq,Xr)
| ~ sameside(Xa,Xr,Xb)
| opposite(Xb,Xp,Xq,Xc)
| ~ s_t(Xr,Xb,Xa) ) ).
fof(aSatz9_3,axiom,
! [Xa,Xp,Xq,Xc,Xm,Xr,Xb] :
( ~ opposite(Xa,Xp,Xq,Xc)
| ~ s_col(Xp,Xq,Xm)
| ~ s_m(Xa,Xm,Xc)
| ~ s_col(Xp,Xq,Xr)
| ~ sameside(Xa,Xr,Xb)
| opposite(Xb,Xp,Xq,Xc) ) ).
fof(aSideReflect,axiom,
! [Xa,Xb,Xc,Xm] :
( ~ sameside(Xa,Xb,Xc)
| sameside(s(Xm,Xa),s(Xm,Xb),s(Xm,Xc)) ) ).
fof(aSatz9_4a,axiom,
! [Xa,Xp,Xq,Xc,Xcs,Xr,Xm,U] :
( ~ opposite(Xa,Xp,Xq,Xc)
| Xp = Xq
| ~ s_col(Xp,Xq,Xcs)
| ~ s_col(Xp,Xq,Xr)
| ~ perp(Xp,Xq,Xa,Xr)
| ~ perp(Xp,Xq,Xc,Xcs)
| ~ s_m(Xr,Xm,Xcs)
| ~ sameside(U,Xr,Xa)
| ~ le(Xcs,Xc,Xr,Xa)
| Xr = Xcs
| sameside(s(Xm,U),Xcs,Xc) ) ).
fof(aSatz9_4a2,axiom,
! [Xa,Xp,Xq,Xc,Xcs,Xr,Xm,U] :
( ~ opposite(Xa,Xp,Xq,Xc)
| Xp = Xq
| ~ s_col(Xp,Xq,Xcs)
| ~ s_col(Xp,Xq,Xr)
| ~ perp(Xp,Xq,Xa,Xr)
| ~ perp(Xp,Xq,Xc,Xcs)
| ~ s_m(Xr,Xm,Xcs)
| sameside(U,Xr,Xa)
| ~ le(Xcs,Xc,Xr,Xa)
| Xr = Xcs
| ~ sameside(s(Xm,U),Xcs,Xc) ) ).
fof(aSatz9_4b,axiom,
! [Xa,Xp,Xq,Xc,Xcs,Xr,Xm,U] :
( ~ opposite(Xa,Xp,Xq,Xc)
| Xp = Xq
| ~ s_col(Xp,Xq,Xcs)
| ~ s_col(Xp,Xq,Xr)
| ~ perp(Xp,Xq,Xa,Xr)
| ~ perp(Xp,Xq,Xc,Xcs)
| ~ s_m(Xr,Xm,Xcs)
| ~ sameside(U,Xr,Xa)
| Xr != Xcs
| sameside(s(Xm,U),Xcs,Xc) ) ).
fof(aSatz9_4,axiom,
! [Xa,Xp,Xq,Xc,Xcs,Xr,Xm,U] :
( ~ opposite(Xa,Xp,Xq,Xc)
| Xp = Xq
| ~ s_col(Xp,Xq,Xcs)
| ~ s_col(Xp,Xq,Xr)
| ~ perp(Xp,Xq,Xa,Xr)
| ~ perp(Xp,Xq,Xc,Xcs)
| ~ s_m(Xr,Xm,Xcs)
| ~ sameside(U,Xr,Xa)
| sameside(s(Xm,U),Xcs,Xc) ) ).
fof(aSatz9_4c,axiom,
! [Xa,Xp,Xq,Xc,Xcs,Xr,Xm,U,V] :
( ~ opposite(Xa,Xp,Xq,Xc)
| Xp = Xq
| ~ s_col(Xp,Xq,Xcs)
| ~ s_col(Xp,Xq,Xr)
| ~ perp(Xp,Xq,Xa,Xr)
| ~ perp(Xp,Xq,Xc,Xcs)
| ~ s_m(Xr,Xm,Xcs)
| ~ sameside(U,Xr,Xa)
| ~ sameside(V,Xcs,Xc)
| opposite(U,Xp,Xq,V) ) ).
fof(aSatz9_5,axiom,
! [Xa,Xp,Xq,Xc,Xr,Xb] :
( ~ opposite(Xa,Xp,Xq,Xc)
| ~ s_col(Xp,Xq,Xr)
| ~ sameside(Xa,Xr,Xb)
| opposite(Xb,Xp,Xq,Xc) ) ).
fof(aSatz9_6,axiom,
! [Xa,Xc,Xp,Xb,Xq] :
( ( ~ s_t(Xa,Xc,Xp)
| ~ s_t(Xb,Xq,Xc)
| s_t(Xa,op(Xq,Xb,Xp,Xc,Xa),Xb) )
& ( ~ s_t(Xa,Xc,Xp)
| ~ s_t(Xb,Xq,Xc)
| s_t(Xp,Xq,op(Xq,Xb,Xp,Xc,Xa)) ) ) ).
fof(aSatz9_8,axiom,
! [Xa,Xp,Xq,Xc,Xb] :
( ~ opposite(Xa,Xp,Xq,Xc)
| opposite(Xb,Xp,Xq,Xc)
| ~ samesideline(Xa,Xb,Xp,Xq) ) ).
fof(aSatz9_9,axiom,
! [Xa,Xp,Xq,Xb] :
( ~ opposite(Xa,Xp,Xq,Xb)
| ~ samesideline(Xa,Xb,Xp,Xq) ) ).
fof(aSatz9_13,axiom,
! [Xp,Xq,Xa,Xb,Xc] :
( Xp = Xq
| ~ samesideline(Xa,Xb,Xp,Xq)
| ~ samesideline(Xb,Xc,Xp,Xq)
| samesideline(Xa,Xc,Xp,Xq) ) ).
fof(aLemma9_13a,axiom,
! [Xp,Xq,Xa,Xb] :
( ~ samesideline(Xp,Xq,Xa,Xb)
| samesideline(Xp,Xq,Xb,Xa) ) ).
fof(aLemma9_13b,axiom,
! [Xa,Xb,U,V,W] :
( Xa = Xb
| ~ s_m(U,V,W)
| ~ s_col(Xa,Xb,U)
| ~ s_col(Xa,Xb,W)
| s_col(Xa,Xb,V) ) ).
fof(aLemma9_13c,axiom,
! [Xd,Xe,Xd1,Xa,Xb] :
( Xd = Xe
| ~ s_t(Xe,Xd,Xd1)
| ~ samesideline(Xa,Xb,Xd1,Xe)
| samesideline(Xa,Xb,Xd,Xe) ) ).
fof(aLemma9_13d,axiom,
! [Xa,Xb,Xp,Xq] :
( Xa = Xb
| ~ samesideline(Xp,Xq,Xa,Xb)
| samesideline(Xq,Xp,Xa,Xb) ) ).
fof(aLemma9_13e,axiom,
! [Xa,Xp,Xq,Xb,Xc] :
( ~ opposite(Xa,Xp,Xq,Xb)
| ~ samesideline(Xb,Xc,Xp,Xq)
| opposite(Xa,Xp,Xq,Xc) ) ).
fof(aLemma9_13g,axiom,
! [Xp,Xq,Xc,Xa,Xb] :
( Xp = Xq
| ~ perpAt(Xp,Xq,Xc,Xa,Xc)
| ~ perpAt(Xp,Xq,Xc,Xb,Xc)
| Xa = Xc
| s_col(Xc,Xa,Xb) ) ).
fof(aLemma9_13f_case1,axiom,
! [Xp,Xq,Xr,Xcs] :
( Xp = Xq
| s_col(Xp,Xq,Xr)
| s_col(Xp,Xq,Xcs)
| samesideline(Xr,Xcs,Xp,Xq)
| opposite(Xr,Xp,Xq,Xcs)
| Xp != foot(Xp,Xq,Xr) ) ).
fof(aLemma9_13f_case2,axiom,
! [Xp,Xq,Xr,Xcs] :
( Xp = Xq
| s_col(Xp,Xq,Xr)
| s_col(Xp,Xq,Xcs)
| samesideline(Xr,Xcs,Xp,Xq)
| opposite(Xr,Xp,Xq,Xcs)
| Xp = foot(Xp,Xq,Xr) ) ).
fof(aLemma9_13f,axiom,
! [Xp,Xq,Xr,Xcs] :
( Xp = Xq
| s_col(Xp,Xq,Xr)
| s_col(Xp,Xq,Xcs)
| samesideline(Xr,Xcs,Xp,Xq)
| opposite(Xr,Xp,Xq,Xcs) ) ).
fof(aSatz9_16,axiom,
! [Xa,Xb,Xp,Xq,Xr] :
( ~ samesideline(Xa,Xb,Xp,Xq)
| ~ sameside(Xa,Xq,Xr)
| samesideline(Xr,Xb,Xp,Xq) ) ).
fof(aSatz9_19b,axiom,
! [Xc,Xd,Xp,Xa,Xb] :
( Xc = Xd
| ~ s_col(Xc,Xd,Xp)
| ~ s_col(Xa,Xb,Xp)
| ~ sameside(Xa,Xp,Xb)
| s_col(Xc,Xd,Xa)
| samesideline(Xa,Xb,Xc,Xd) ) ).
fof(aSatz10_2a,axiom,
! [Xa,Xb,Xp] :
( ( Xa = Xb
| s_col(Xa,Xb,midpoint(Xp,reflect(Xa,Xb,Xp))) )
& ( Xa = Xb
| Xp = reflect(Xa,Xb,Xp)
| perp(Xa,Xb,Xp,reflect(Xa,Xb,Xp)) ) ) ).
fof(aSatz10_2b,axiom,
! [Xa,Xb,Xp,Xp1] :
( ( Xa = Xb
| ~ s_col(Xa,Xb,midpoint(Xp,Xp1))
| ~ perp(Xa,Xb,Xp,Xp1)
| Xp1 = reflect(Xa,Xb,Xp) )
& ( Xa = Xb
| ~ s_col(Xa,Xb,midpoint(Xp,Xp1))
| Xp != Xp1
| Xp1 = reflect(Xa,Xb,Xp) ) ) ).
fof(aSatz10_4,axiom,
! [Xa,Xb,Xp,Xp1] :
( Xa = Xb
| reflect(Xa,Xb,Xp) != Xp1
| reflect(Xa,Xb,Xp1) = Xp ) ).
fof(aSatz10_5,axiom,
! [Xa,Xb,Xp] :
( Xa = Xb
| reflect(Xa,Xb,reflect(Xa,Xb,Xp)) = Xp ) ).
fof(aSatz10_6,axiom,
! [Xa,Xb,Xp,Xp1] :
( Xa = Xb
| reflect(Xa,Xb,Xp) != Xp1
| Xp = reflect(Xa,Xb,Xp1) ) ).
fof(aSatz10_7,axiom,
! [Xa,Xb,Xp,Xq] :
( Xa = Xb
| reflect(Xa,Xb,Xp) != reflect(Xa,Xb,Xq)
| Xp = Xq ) ).
fof(aSatz10_8a,axiom,
! [Xa,Xb,Xp] :
( Xa = Xb
| reflect(Xa,Xb,Xp) != Xp
| s_col(Xa,Xb,Xp) ) ).
fof(aSatz10_8b,axiom,
! [Xa,Xb,Xp] :
( Xa = Xb
| ~ s_col(Xa,Xb,Xp)
| reflect(Xa,Xb,Xp) = Xp ) ).
fof(aSatz10_10a,axiom,
! [Xa,Xb,Xp,Xq] :
( Xa = Xb
| ~ s_col(Xa,Xb,Xp)
| s_e(Xp,Xq,reflect(Xa,Xb,Xp),reflect(Xa,Xb,Xq)) ) ).
fof(aSatz10_10,axiom,
! [Xa,Xb,Xp,Xq] :
( Xa = Xb
| s_e(Xp,Xq,reflect(Xa,Xb,Xp),reflect(Xa,Xb,Xq)) ) ).
fof(aSatz10_12b,axiom,
! [Xa,Xb,Xc,Xa1] :
( ~ s_r(Xa,Xb,Xc)
| ~ s_r(Xa1,Xb,Xc)
| ~ s_e(Xa,Xb,Xa1,Xb)
| s_e(Xa,Xc,Xa1,Xc) ) ).
fof(aSatz10_12c,axiom,
! [Xa,Xb,Xc,Xa1,Xc1] :
( ~ s_r(Xa,Xb,Xc)
| ~ s_r(Xa1,Xb,Xc1)
| ~ s_e(Xa,Xb,Xa1,Xb)
| ~ s_e(Xb,Xc,Xb,Xc1)
| s_e(Xa,Xc,Xa1,Xc1)
| Xb != midpoint(Xc,Xc1)
| Xc = Xc1 ) ).
fof(aSatz10_12a,axiom,
! [Xa,Xb,Xc,Xa1,Xc1] :
( ~ s_r(Xa,Xb,Xc)
| ~ s_r(Xa1,Xb,Xc1)
| ~ s_e(Xa,Xb,Xa1,Xb)
| ~ s_e(Xb,Xc,Xb,Xc1)
| s_e(Xa,Xc,Xa1,Xc1) ) ).
fof(aSatz10_12,axiom,
! [Xa,Xb,Xc,Xa1,Xb1,Xc1] :
( ~ s_r(Xa,Xb,Xc)
| ~ s_r(Xa1,Xb1,Xc1)
| ~ s_e(Xa,Xb,Xa1,Xb1)
| ~ s_e(Xb,Xc,Xb1,Xc1)
| s_e(Xa,Xc,Xa1,Xc1) ) ).
fof(aSatz10_14,axiom,
! [Xa,Xb,Xp,Xp1] :
( Xa = Xb
| reflect(Xa,Xb,Xp) != Xp1
| s_col(Xa,Xb,Xp)
| opposite(Xp,Xa,Xb,Xp1) ) ).
fof(aSatz10_15,axiom,
! [Xb,Xc,Xa,Xq] :
( ( Xb = Xc
| ~ s_col(Xb,Xc,Xa)
| s_col(Xb,Xc,Xq)
| perp(Xb,Xc,erectsameside(Xb,Xc,Xa,Xq),Xa) )
& ( Xb = Xc
| ~ s_col(Xb,Xc,Xa)
| s_col(Xb,Xc,Xq)
| samesideline(erectsameside(Xb,Xc,Xa,Xq),Xq,Xb,Xc) ) ) ).
fof(aSatz10_16a,axiom,
! [Xa,Xb,Xc,Xa1,Xb1,Xp] :
( ( Xa = Xb
| s_col(Xa,Xb,Xc)
| s_col(Xa1,Xb1,Xp)
| ~ s_e(Xa,Xb,Xa1,Xb1)
| s_e3(Xa,Xb,Xc,Xa1,Xb1,triangle(Xa,Xb,Xc,Xa1,Xb1,Xp)) )
& ( Xa = Xb
| s_col(Xa,Xb,Xc)
| s_col(Xa1,Xb1,Xp)
| ~ s_e(Xa,Xb,Xa1,Xb1)
| samesideline(triangle(Xa,Xb,Xc,Xa1,Xb1,Xp),Xp,Xa1,Xb1) ) ) ).
fof(aSatz10_16b,axiom,
! [Xa,Xb,Xc,Xa1,Xb1,Xp,Xc1,Xc2] :
( Xa = Xb
| s_col(Xa,Xb,Xc)
| s_col(Xa1,Xb1,Xp)
| ~ s_e(Xa,Xb,Xa1,Xb1)
| ~ s_e3(Xa,Xb,Xc,Xa1,Xb1,Xc1)
| ~ samesideline(Xc1,Xp,Xa1,Xb1)
| ~ s_e3(Xa,Xb,Xc,Xa1,Xb1,Xc2)
| ~ samesideline(Xc2,Xp,Xa1,Xb1)
| Xc1 = Xc2 ) ).
fof(aSatz11_3a,axiom,
! [Xa,Xb,Xc,Xd,Xe,Xf] :
( ( ~ congruent(Xa,Xb,Xc,Xd,Xe,Xf)
| sameside(ext(Xb,Xa,Xe,Xd),Xb,Xa) )
& ( ~ congruent(Xa,Xb,Xc,Xd,Xe,Xf)
| sameside(ext(Xb,Xc,Xe,Xf),Xb,Xc) )
& ( ~ congruent(Xa,Xb,Xc,Xd,Xe,Xf)
| sameside(ext(Xe,Xd,Xb,Xa),Xe,Xd) )
& ( ~ congruent(Xa,Xb,Xc,Xd,Xe,Xf)
| sameside(ext(Xe,Xf,Xb,Xc),Xe,Xf) )
& ( ~ congruent(Xa,Xb,Xc,Xd,Xe,Xf)
| s_e3(ext(Xb,Xa,Xe,Xd),Xb,ext(Xb,Xc,Xe,Xf),ext(Xe,Xd,Xb,Xa),Xe,ext(Xe,Xf,Xb,Xc)) ) ) ).
fof(aSatz11_3d,axiom,
! [Xa1,Xb,Xa,Xc1,Xc,Xd1,Xe,Xd,Xf1,Xf] :
( ~ sameside(Xa1,Xb,Xa)
| ~ sameside(Xc1,Xb,Xc)
| ~ sameside(Xd1,Xe,Xd)
| ~ sameside(Xf1,Xe,Xf)
| ~ s_e3(Xa1,Xb,Xc1,Xd1,Xe,Xf1)
| congruent(Xa,Xb,Xc,Xd,Xe,Xf) ) ).
fof(aSatz11_4b,axiom,
! [Xa,Xb,Xc,Xd,Xe,Xf] :
( ( Xa = Xb
| Xc = Xb
| Xd = Xe
| Xf = Xe
| sameside(f113a(Xa,Xb,Xc,Xd,Xe,Xf),Xb,Xa)
| congruent(Xa,Xb,Xc,Xd,Xe,Xf) )
& ( Xa = Xb
| Xc = Xb
| Xd = Xe
| Xf = Xe
| sameside(f113c(Xa,Xb,Xc,Xd,Xe,Xf),Xb,Xc)
| congruent(Xa,Xb,Xc,Xd,Xe,Xf) )
& ( Xa = Xb
| Xc = Xb
| Xd = Xe
| Xf = Xe
| sameside(f113d(Xa,Xb,Xc,Xd,Xe,Xf),Xe,Xd)
| congruent(Xa,Xb,Xc,Xd,Xe,Xf) )
& ( Xa = Xb
| Xc = Xb
| Xd = Xe
| Xf = Xe
| sameside(f113f(Xa,Xb,Xc,Xd,Xe,Xf),Xe,Xf)
| congruent(Xa,Xb,Xc,Xd,Xe,Xf) )
& ( Xa = Xb
| Xc = Xb
| Xd = Xe
| Xf = Xe
| s_e(Xb,f113a(Xa,Xb,Xc,Xd,Xe,Xf),Xe,f113d(Xa,Xb,Xc,Xd,Xe,Xf))
| congruent(Xa,Xb,Xc,Xd,Xe,Xf) )
& ( Xa = Xb
| Xc = Xb
| Xd = Xe
| Xf = Xe
| s_e(Xb,f113c(Xa,Xb,Xc,Xd,Xe,Xf),Xe,f113f(Xa,Xb,Xc,Xd,Xe,Xf))
| congruent(Xa,Xb,Xc,Xd,Xe,Xf) )
& ( Xa = Xb
| Xc = Xb
| Xd = Xe
| Xf = Xe
| ~ s_e(f113a(Xa,Xb,Xc,Xd,Xe,Xf),f113c(Xa,Xb,Xc,Xd,Xe,Xf),f113d(Xa,Xb,Xc,Xd,Xe,Xf),f113f(Xa,Xb,Xc,Xd,Xe,Xf))
| congruent(Xa,Xb,Xc,Xd,Xe,Xf) ) ) ).
fof(aSatz11_4,axiom,
! [Xa,Xb,Xc,Xd,Xe,Xf,Xa2,Xc2,Xd2,Xf2] :
( ~ congruent(Xa,Xb,Xc,Xd,Xe,Xf)
| ~ sameside(Xa2,Xb,Xa)
| ~ sameside(Xc2,Xb,Xc)
| ~ sameside(Xd2,Xe,Xd)
| ~ sameside(Xf2,Xe,Xf)
| ~ s_e(Xb,Xa2,Xe,Xd2)
| ~ s_e(Xb,Xc2,Xe,Xf2)
| s_e(Xa2,Xc2,Xd2,Xf2) ) ).
fof(aSatz11_4d,axiom,
! [Xa1,Xb,Xa2,Xc1,Xc2,Xd1,Xe,Xd2,Xf1,Xf2,Xa,Xc,Xd,Xf] :
( ~ sameside(Xa1,Xb,Xa2)
| ~ sameside(Xc1,Xb,Xc2)
| ~ sameside(Xd1,Xe,Xd2)
| ~ sameside(Xf1,Xe,Xf2)
| ~ s_e3(Xa1,Xb,Xc1,Xd1,Xe,Xf1)
| Xa = Xb
| Xc = Xb
| Xd = Xe
| Xf = Xe
| ~ sameside(Xa2,Xb,Xa)
| ~ sameside(Xc2,Xb,Xc)
| ~ sameside(Xd2,Xe,Xd)
| ~ sameside(Xf2,Xe,Xf)
| ~ s_e(Xb,Xa2,Xe,Xd2)
| ~ s_e(Xb,Xc2,Xe,Xf2)
| s_e(Xa2,Xc2,Xd2,Xf2) ) ).
fof(aSatz11_6,axiom,
! [Xa,Xb,Xc] :
( Xa = Xb
| Xc = Xb
| congruent(Xa,Xb,Xc,Xa,Xb,Xc) ) ).
fof(aSatz11_7,axiom,
! [Xa,Xb,Xc,Xd,Xe,Xf] :
( ~ congruent(Xa,Xb,Xc,Xd,Xe,Xf)
| congruent(Xd,Xe,Xf,Xa,Xb,Xc) ) ).
fof(aSatz11_8,axiom,
! [Xa1,Xb1,Xc1,Xa2,Xb2,Xc2,Xa3,Xb3,Xc3] :
( ~ congruent(Xa1,Xb1,Xc1,Xa2,Xb2,Xc2)
| ~ congruent(Xa2,Xb2,Xc2,Xa3,Xb3,Xc3)
| congruent(Xa1,Xb1,Xc1,Xa3,Xb3,Xc3) ) ).
fof(aSatz11_9,axiom,
! [Xa,Xb,Xc] :
( Xa = Xb
| Xc = Xb
| congruent(Xa,Xb,Xc,Xc,Xb,Xa) ) ).
fof(aSatz11_15a,axiom,
! [Xa,Xb,Xc,Xd,Xe,Xp] :
( ( s_col(Xa,Xb,Xc)
| s_col(Xd,Xe,Xp)
| congruent(Xa,Xb,Xc,Xd,Xe,constructAngle(Xa,Xb,Xc,Xd,Xe,Xp)) )
& ( s_col(Xa,Xb,Xc)
| s_col(Xd,Xe,Xp)
| samesideline(constructAngle(Xa,Xb,Xc,Xd,Xe,Xp),Xp,Xd,Xe) ) ) ).
fof(aSatz11_15b,axiom,
! [Xa,Xb,Xc,Xd,Xe,Xp,Xf1,Xf2] :
( s_col(Xa,Xb,Xc)
| s_col(Xd,Xe,Xp)
| ~ congruent(Xa,Xb,Xc,Xd,Xe,Xf1)
| ~ samesideline(Xf1,Xp,Xe,Xd)
| ~ congruent(Xa,Xb,Xc,Xd,Xe,Xf2)
| ~ samesideline(Xf2,Xp,Xe,Xd)
| sameside(Xf1,Xe,Xf2) ) ).
fof(aSatz11_49a,axiom,
! [Xa,Xb,Xc,Xa1,Xb1,Xc1] :
( ~ congruent(Xa,Xb,Xc,Xa1,Xb1,Xc1)
| ~ s_e(Xb,Xa,Xb1,Xa1)
| ~ s_e(Xb,Xc,Xb1,Xc1)
| s_e(Xa,Xc,Xa1,Xc1) ) ).
fof(aSatz11_49b,axiom,
! [Xa,Xb,Xc,Xa1,Xb1,Xc1] :
( ~ congruent(Xa,Xb,Xc,Xa1,Xb1,Xc1)
| ~ s_e(Xb,Xa,Xb1,Xa1)
| ~ s_e(Xb,Xc,Xb1,Xc1)
| Xa = Xc
| congruent(Xb,Xa,Xc,Xb1,Xa1,Xc1) ) ).
fof(aExtParallel,axiom,
! [Xa1,Xa2,Xb1,Xb2] :
( ~ parallel(Xa1,Xa2,Xb1,Xb2)
| parallel(Xa1,Xa2,Xb2,Xb1) ) ).
fof(aSatz12_6,axiom,
! [Xa1,Xa2,Xb1,Xb2,Xb3,Xb4] :
( ~ parallel(Xa1,Xa2,Xb1,Xb2)
| ~ s_col(Xb1,Xb2,Xb3)
| ~ s_col(Xb1,Xb2,Xb4)
| samesideline(Xb3,Xb4,Xa1,Xa2) ) ).
fof(aLemma12_11_case1,axiom,
! [Xc2,Xp1,Xp2,Xa,Xt,Xr,Xcs,Xq] :
( ( Xc2 != ext(Xp1,Xp2,Xp1,Xa)
| ~ opposite(Xt,Xr,Xcs,Xc2)
| s_col(Xt,Xq,Xa)
| ~ s_col(Xr,Xcs,Xa)
| ~ s_col(Xp1,Xp2,Xa)
| ~ parallel(Xt,Xq,Xr,Xcs)
| ~ parallel(Xt,Xq,Xp1,Xp2)
| s_t(c12111(Xt,Xq,Xr,Xcs,Xp1,Xp2,Xa),b12111(Xt,Xq,Xr,Xcs,Xp1,Xp2,Xa),Xt) )
& ( Xc2 != ext(Xp1,Xp2,Xp1,Xa)
| ~ opposite(Xt,Xr,Xcs,Xc2)
| s_col(Xt,Xq,Xa)
| ~ s_col(Xr,Xcs,Xa)
| ~ s_col(Xp1,Xp2,Xa)
| ~ parallel(Xt,Xq,Xr,Xcs)
| ~ parallel(Xt,Xq,Xp1,Xp2)
| s_col(Xp1,Xp2,c12111(Xt,Xq,Xr,Xcs,Xp1,Xp2,Xa)) )
& ( Xc2 != ext(Xp1,Xp2,Xp1,Xa)
| ~ opposite(Xt,Xr,Xcs,Xc2)
| s_col(Xt,Xq,Xa)
| ~ s_col(Xr,Xcs,Xa)
| ~ s_col(Xp1,Xp2,Xa)
| ~ parallel(Xt,Xq,Xr,Xcs)
| ~ parallel(Xt,Xq,Xp1,Xp2)
| s_col(Xr,Xcs,b12111(Xt,Xq,Xr,Xcs,Xp1,Xp2,Xa)) )
& ( Xc2 != ext(Xp1,Xp2,Xp1,Xa)
| ~ opposite(Xt,Xr,Xcs,Xc2)
| s_col(Xt,Xq,Xa)
| ~ s_col(Xr,Xcs,Xa)
| ~ s_col(Xp1,Xp2,Xa)
| ~ parallel(Xt,Xq,Xr,Xcs)
| ~ parallel(Xt,Xq,Xp1,Xp2)
| Xa != c12111(Xt,Xq,Xr,Xcs,Xp1,Xp2,Xa) ) ) ).
fof(aLemma12_11_case2,conjecture,
! [Xc2,Xp1,Xp2,Xa,Xt,Xr,Xcs,Xq] :
( ( Xc2 != ext(Xp1,Xp2,Xp1,Xa)
| opposite(Xt,Xr,Xcs,Xc2)
| s_col(Xt,Xq,Xa)
| ~ s_col(Xr,Xcs,Xa)
| ~ s_col(Xp1,Xp2,Xa)
| ~ parallel(Xt,Xq,Xr,Xcs)
| ~ parallel(Xt,Xq,Xp1,Xp2)
| s_t(c12112(Xt,Xq,Xr,Xcs,Xp1,Xp2,Xa),b12112(Xt,Xq,Xr,Xcs,Xp1,Xp2,Xa),Xt) )
& ( Xc2 != ext(Xp1,Xp2,Xp1,Xa)
| opposite(Xt,Xr,Xcs,Xc2)
| s_col(Xt,Xq,Xa)
| ~ s_col(Xr,Xcs,Xa)
| ~ s_col(Xp1,Xp2,Xa)
| ~ parallel(Xt,Xq,Xr,Xcs)
| ~ parallel(Xt,Xq,Xp1,Xp2)
| s_col(Xp1,Xp2,c12112(Xt,Xq,Xr,Xcs,Xp1,Xp2,Xa)) )
& ( Xc2 != ext(Xp1,Xp2,Xp1,Xa)
| opposite(Xt,Xr,Xcs,Xc2)
| s_col(Xt,Xq,Xa)
| ~ s_col(Xr,Xcs,Xa)
| ~ s_col(Xp1,Xp2,Xa)
| ~ parallel(Xt,Xq,Xr,Xcs)
| ~ parallel(Xt,Xq,Xp1,Xp2)
| s_col(Xr,Xcs,b12112(Xt,Xq,Xr,Xcs,Xp1,Xp2,Xa)) )
& ( Xc2 != ext(Xp1,Xp2,Xp1,Xa)
| opposite(Xt,Xr,Xcs,Xc2)
| s_col(Xt,Xq,Xa)
| ~ s_col(Xr,Xcs,Xa)
| ~ s_col(Xp1,Xp2,Xa)
| ~ parallel(Xt,Xq,Xr,Xcs)
| ~ parallel(Xt,Xq,Xp1,Xp2)
| Xa != c12112(Xt,Xq,Xr,Xcs,Xp1,Xp2,Xa) ) ) ).
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) ) ) ).
fof(d_Defn9_1,axiom,
! [Xp,Xq,Xa,Xb,Xt] :
( ( Xp = Xq
| s_col(Xp,Xq,Xa)
| s_col(Xp,Xq,Xb)
| ~ s_t(Xa,Xt,Xb)
| ~ s_col(Xp,Xq,Xt)
| opposite(Xa,Xp,Xq,Xb) )
& ( ~ opposite(Xa,Xp,Xq,Xb)
| ~ s_col(Xp,Xq,Xa) )
& ( ~ opposite(Xa,Xp,Xq,Xb)
| ~ s_col(Xp,Xq,Xb) )
& ( ~ opposite(Xa,Xp,Xq,Xb)
| s_t(Xa,il(Xa,Xb,Xp,Xq),Xb) )
& ( ~ opposite(Xa,Xp,Xq,Xb)
| s_col(Xp,Xq,il(Xa,Xb,Xp,Xq)) ) ) ).
fof(d_Defn9_7,axiom,
! [Xa,Xb,Xp,Xq,Xu,Xc,Xv] :
( ( ~ samesideline(Xa,Xb,Xp,Xq)
| s_col(Xp,Xq,ss1(Xa,Xb,Xp,Xq)) )
& ( ~ samesideline(Xa,Xb,Xp,Xq)
| s_col(Xp,Xq,ss2(Xa,Xb,Xp,Xq)) )
& ( ~ samesideline(Xa,Xb,Xp,Xq)
| s_t(Xa,ss1(Xa,Xb,Xp,Xq),ss3(Xa,Xb,Xp,Xq)) )
& ( ~ samesideline(Xa,Xb,Xp,Xq)
| s_t(Xb,ss2(Xa,Xb,Xp,Xq),ss3(Xa,Xb,Xp,Xq)) )
& ( ~ samesideline(Xa,Xb,Xp,Xq)
| Xa != ss1(Xa,Xb,Xp,Xq) )
& ( ~ samesideline(Xa,Xb,Xp,Xq)
| Xb != ss2(Xa,Xb,Xp,Xq) )
& ( ~ samesideline(Xa,Xb,Xp,Xq)
| ss3(Xa,Xb,Xp,Xq) != ss1(Xa,Xb,Xp,Xq) )
& ( ~ samesideline(Xa,Xb,Xp,Xq)
| ss3(Xa,Xb,Xp,Xq) != ss2(Xa,Xb,Xp,Xq) )
& ( ~ samesideline(Xa,Xb,Xp,Xq)
| Xp != Xq )
& ( ~ samesideline(Xa,Xb,Xp,Xq)
| ~ s_col(Xp,Xq,Xa) )
& ( ~ samesideline(Xa,Xb,Xp,Xq)
| ~ s_col(Xp,Xq,Xb) )
& ( ~ s_t(Xa,Xu,Xc)
| ~ s_t(Xb,Xv,Xc)
| ~ s_col(Xp,Xq,Xu)
| ~ s_col(Xp,Xq,Xv)
| Xp = Xq
| Xc = Xu
| Xc = Xv
| Xa = Xu
| Xb = Xv
| s_col(Xp,Xq,Xa)
| s_col(Xp,Xq,Xb)
| samesideline(Xa,Xb,Xp,Xq) ) ) ).
%------------------------------------------------------------------------------