SET007 Axioms: SET007+289.ax
%------------------------------------------------------------------------------
% File : SET007+289 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Planes in Affine Spaces
% Version : [Urb08] axioms.
% English :
% Refs : [Mat90] Matuszewski (1990), Formalized Mathematics
% : [Urb07] Urban (2007), MPTP 0.2: Design, Implementation, and In
% : [Urb08] Urban (2006), Email to G. Sutcliffe
% Source : [Urb08]
% Names : aff_4 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 79 ( 0 unt; 0 def)
% Number of atoms : 1104 ( 59 equ)
% Maximal formula atoms : 39 ( 13 avg)
% Number of connectives : 1276 ( 251 ~; 54 |; 557 &)
% ( 12 <=>; 402 =>; 0 <=; 0 <~>)
% Maximal formula depth : 44 ( 15 avg)
% Maximal term depth : 3 ( 1 avg)
% Number of predicates : 19 ( 18 usr; 0 prp; 1-5 aty)
% Number of functors : 9 ( 9 usr; 1 con; 0-3 aty)
% Number of variables : 373 ( 361 !; 12 ?)
% SPC :
% Comments : The individual reference can be found in [Mat90] by looking for
% the name provided by [Urb08].
% : Translated by MPTP from the Mizar Mathematical Library 4.48.930.
% : These set theory axioms are used in encodings of problems in
% various domains, including ALG, CAT, GRP, LAT, SET, and TOP.
%------------------------------------------------------------------------------
fof(t1_aff_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& l1_analoaf(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(A))
=> ~ ( ( r1_aff_1(A,B,C,D)
| r1_aff_1(A,B,D,C) )
& B != C
& ! [F] :
( m1_subset_1(F,u1_struct_0(A))
=> ~ ( r1_aff_1(A,B,E,F)
& r2_analoaf(A,C,E,D,F) ) ) ) ) ) ) ) ) ).
fof(t2_aff_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& l1_analoaf(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
=> ( r2_hidden(B,D)
=> ( ( ~ r2_aff_1(A,B,C,D)
& ~ r2_aff_1(A,C,B,D) )
| r2_hidden(C,D) ) ) ) ) ) ) ).
fof(t3_aff_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& l1_analoaf(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
=> ! [E] :
( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(A)))
=> ~ ( ( r2_aff_1(A,B,C,D)
| r2_aff_1(A,C,B,D) )
& ( r4_aff_1(A,D,E)
| r4_aff_1(A,E,D) )
& ~ ( r2_aff_1(A,B,C,E)
& r2_aff_1(A,C,B,E) ) ) ) ) ) ) ) ).
fof(t4_aff_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& l1_analoaf(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(A))
=> ! [F] :
( m1_subset_1(F,k1_zfmisc_1(u1_struct_0(A)))
=> ~ ( ( r2_aff_1(A,B,C,F)
| r2_aff_1(A,C,B,F) )
& ( r2_analoaf(A,B,C,D,E)
| r2_analoaf(A,D,E,B,C) )
& B != C
& ~ ( r2_aff_1(A,D,E,F)
& r2_aff_1(A,E,D,F) ) ) ) ) ) ) ) ) ).
fof(t5_aff_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& l1_analoaf(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
=> ! [E] :
( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(A)))
=> ~ ( ( r2_aff_1(A,B,C,D)
| r2_aff_1(A,C,B,D) )
& ( r2_aff_1(A,B,C,E)
| r2_aff_1(A,C,B,E) )
& B != C
& ~ ( r4_aff_1(A,D,E)
& r4_aff_1(A,E,D) ) ) ) ) ) ) ) ).
fof(t6_aff_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& l1_analoaf(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(A))
=> ! [F] :
( m1_subset_1(F,k1_zfmisc_1(u1_struct_0(A)))
=> ~ ( ( r2_aff_1(A,B,C,F)
| r2_aff_1(A,C,B,F) )
& ( r2_aff_1(A,D,E,F)
| r2_aff_1(A,E,D,F) )
& ~ ( r2_analoaf(A,B,C,D,E)
& r2_analoaf(A,B,C,E,D) ) ) ) ) ) ) ) ) ).
fof(t7_aff_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& l1_analoaf(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(A))
=> ! [F] :
( m1_subset_1(F,k1_zfmisc_1(u1_struct_0(A)))
=> ! [G] :
( m1_subset_1(G,k1_zfmisc_1(u1_struct_0(A)))
=> ( ( r2_hidden(B,F)
& r2_hidden(C,F)
& r2_hidden(D,G) )
=> ( ( ~ r4_aff_1(A,F,G)
& ~ r4_aff_1(A,G,F) )
| B = C
| ( ~ r2_analoaf(A,B,C,D,E)
& ~ r2_analoaf(A,D,E,B,C) )
| r2_hidden(E,G) ) ) ) ) ) ) ) ) ) ).
fof(t8_aff_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& l1_analoaf(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(A))
=> ! [F] :
( m1_subset_1(F,u1_struct_0(A))
=> ! [G] :
( m1_subset_1(G,k1_zfmisc_1(u1_struct_0(A)))
=> ! [H] :
( m1_subset_1(H,k1_zfmisc_1(u1_struct_0(A)))
=> ( ( r2_hidden(B,G)
& r2_hidden(B,H)
& r2_hidden(C,G)
& r2_hidden(D,G)
& r2_hidden(E,H)
& r2_hidden(F,H)
& v1_aff_1(G,A)
& v1_aff_1(H,A)
& B = D )
=> ( B = C
| B = E
| G = H
| ( ~ r2_analoaf(A,C,E,D,F)
& ~ r2_analoaf(A,E,C,F,D) )
| B = F ) ) ) ) ) ) ) ) ) ) ).
fof(t9_aff_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& l1_analoaf(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(A))
=> ! [F] :
( m1_subset_1(F,u1_struct_0(A))
=> ! [G] :
( m1_subset_1(G,k1_zfmisc_1(u1_struct_0(A)))
=> ! [H] :
( m1_subset_1(H,k1_zfmisc_1(u1_struct_0(A)))
=> ( ( r2_hidden(B,G)
& r2_hidden(B,H)
& r2_hidden(C,G)
& r2_hidden(D,G)
& r2_hidden(E,H)
& r2_hidden(F,H)
& v1_aff_1(G,A)
& v1_aff_1(H,A)
& C = D )
=> ( B = C
| B = E
| G = H
| ( ~ r2_analoaf(A,C,E,D,F)
& ~ r2_analoaf(A,E,C,F,D) )
| E = F ) ) ) ) ) ) ) ) ) ) ).
fof(t10_aff_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& l1_analoaf(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(A))
=> ! [F] :
( m1_subset_1(F,k1_zfmisc_1(u1_struct_0(A)))
=> ! [G] :
( m1_subset_1(G,k1_zfmisc_1(u1_struct_0(A)))
=> ( ( r2_hidden(B,F)
& r2_hidden(C,F)
& r2_hidden(D,G)
& r2_hidden(E,G)
& B = C )
=> ( ( ~ r4_aff_1(A,F,G)
& ~ r4_aff_1(A,G,F) )
| F = G
| ( ~ r2_analoaf(A,B,D,C,E)
& ~ r2_analoaf(A,D,B,E,C) )
| D = E ) ) ) ) ) ) ) ) ) ).
fof(t11_aff_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& l1_analoaf(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ? [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
& r2_hidden(B,D)
& r2_hidden(C,D)
& v1_aff_1(D,A) ) ) ) ) ).
fof(t12_aff_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& l1_analoaf(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ~ ( v1_aff_1(B,A)
& ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> r2_hidden(C,B) ) ) ) ) ).
fof(d2_aff_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& l1_analoaf(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ( v1_aff_4(B,A)
<=> ? [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
& ? [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
& v1_aff_1(C,A)
& v1_aff_1(D,A)
& ~ r4_aff_1(A,C,D)
& B = k1_aff_4(A,C,D) ) ) ) ) ) ).
fof(t13_aff_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& l1_analoaf(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ( ~ v1_aff_1(B,A)
=> k1_aff_4(A,B,C) = k1_xboole_0 ) ) ) ) ).
fof(t14_aff_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& l1_analoaf(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ( v1_aff_1(B,A)
=> r1_tarski(C,k1_aff_4(A,B,C)) ) ) ) ) ).
fof(t15_aff_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& l1_analoaf(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ( r4_aff_1(A,B,C)
=> k1_aff_4(A,B,C) = C ) ) ) ) ).
fof(t16_aff_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& l1_analoaf(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
=> ( r4_aff_1(A,B,C)
=> k1_aff_4(A,B,D) = k1_aff_4(A,C,D) ) ) ) ) ) ).
fof(t17_aff_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& l1_analoaf(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(A))
=> ! [F] :
( m1_subset_1(F,u1_struct_0(A))
=> ! [G] :
( m1_subset_1(G,k1_zfmisc_1(u1_struct_0(A)))
=> ! [H] :
( m1_subset_1(H,k1_zfmisc_1(u1_struct_0(A)))
=> ! [I] :
( m1_subset_1(I,k1_zfmisc_1(u1_struct_0(A)))
=> ! [J] :
( m1_subset_1(J,k1_zfmisc_1(u1_struct_0(A)))
=> ~ ( r2_hidden(B,G)
& r2_hidden(C,G)
& r2_hidden(D,G)
& r2_hidden(B,H)
& r2_hidden(E,H)
& r2_hidden(F,H)
& ~ r2_hidden(B,I)
& ~ r2_hidden(B,J)
& G != H
& r2_hidden(C,I)
& r2_hidden(E,I)
& r2_hidden(D,J)
& r2_hidden(F,J)
& v1_aff_1(G,A)
& v1_aff_1(H,A)
& v1_aff_1(I,A)
& v1_aff_1(J,A)
& ~ r4_aff_1(A,I,J)
& ! [K] :
( m1_subset_1(K,u1_struct_0(A))
=> ~ ( r2_hidden(K,I)
& r2_hidden(K,J) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
fof(t18_aff_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& l1_analoaf(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(A))
=> ! [F] :
( m1_subset_1(F,k1_zfmisc_1(u1_struct_0(A)))
=> ! [G] :
( m1_subset_1(G,k1_zfmisc_1(u1_struct_0(A)))
=> ! [H] :
( m1_subset_1(H,k1_zfmisc_1(u1_struct_0(A)))
=> ! [I] :
( m1_subset_1(I,k1_zfmisc_1(u1_struct_0(A)))
=> ~ ( r2_hidden(B,F)
& r2_hidden(C,F)
& r2_hidden(D,G)
& r2_hidden(E,G)
& r2_hidden(B,H)
& r2_hidden(D,H)
& r2_hidden(C,I)
& r2_hidden(E,I)
& F != G
& r4_aff_1(A,F,G)
& v1_aff_1(H,A)
& v1_aff_1(I,A)
& ~ r4_aff_1(A,H,I)
& ! [J] :
( m1_subset_1(J,u1_struct_0(A))
=> ~ ( r2_hidden(J,H)
& r2_hidden(J,I) ) ) ) ) ) ) ) ) ) ) ) ) ).
fof(t19_aff_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& l1_analoaf(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
=> ( ( v1_aff_4(D,A)
& r2_hidden(B,D)
& r2_hidden(C,D) )
=> ( B = C
| r1_tarski(k2_aff_1(A,B,C),D) ) ) ) ) ) ) ).
fof(t20_aff_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& l1_analoaf(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
=> ( ( v1_aff_1(B,A)
& v1_aff_1(C,A)
& v1_aff_1(D,A)
& r1_tarski(D,k1_aff_4(A,B,C)) )
=> ( r4_aff_1(A,B,C)
| r4_aff_1(A,B,D)
| k1_aff_4(A,B,D) = k1_aff_4(A,B,C) ) ) ) ) ) ) ).
fof(t21_aff_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& l1_analoaf(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
=> ~ ( v1_aff_1(B,A)
& v1_aff_1(C,A)
& v1_aff_1(D,A)
& r1_tarski(D,k1_aff_4(A,B,C))
& ~ r4_aff_1(A,C,D)
& ! [E] :
( m1_subset_1(E,u1_struct_0(A))
=> ~ ( r2_hidden(E,C)
& r2_hidden(E,D) ) ) ) ) ) ) ) ).
fof(t22_aff_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& l1_analoaf(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
=> ~ ( v1_aff_4(B,A)
& v1_aff_1(C,A)
& v1_aff_1(D,A)
& r1_tarski(C,B)
& r1_tarski(D,B)
& ~ r4_aff_1(A,C,D)
& ! [E] :
( m1_subset_1(E,u1_struct_0(A))
=> ~ ( r2_hidden(E,C)
& r2_hidden(E,D) ) ) ) ) ) ) ) ).
fof(t23_aff_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& l1_analoaf(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
=> ! [E] :
( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(A)))
=> ( ( v1_aff_4(C,A)
& r2_hidden(B,C)
& r1_tarski(D,C)
& r2_hidden(B,E) )
=> ( ( ~ r4_aff_1(A,D,E)
& ~ r4_aff_1(A,E,D) )
| r1_tarski(E,C) ) ) ) ) ) ) ) ).
fof(t24_aff_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& l1_analoaf(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
=> ! [E] :
( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(A)))
=> ( ( v1_aff_4(D,A)
& v1_aff_4(E,A)
& r2_hidden(B,D)
& r2_hidden(C,D)
& r2_hidden(B,E)
& r2_hidden(C,E) )
=> ( D = E
| B = C
| v1_aff_1(k5_subset_1(u1_struct_0(A),D,E),A) ) ) ) ) ) ) ) ).
fof(t25_aff_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& l1_analoaf(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ! [E] :
( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(A)))
=> ! [F] :
( m1_subset_1(F,k1_zfmisc_1(u1_struct_0(A)))
=> ( ( v1_aff_4(E,A)
& v1_aff_4(F,A)
& r2_hidden(B,E)
& r2_hidden(C,E)
& r2_hidden(D,E)
& r2_hidden(B,F)
& r2_hidden(C,F)
& r2_hidden(D,F) )
=> ( r1_aff_1(A,B,C,D)
| E = F ) ) ) ) ) ) ) ) ).
fof(t26_aff_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& l1_analoaf(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
=> ! [E] :
( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(A)))
=> ( ( v1_aff_4(B,A)
& v1_aff_4(C,A)
& v1_aff_1(D,A)
& v1_aff_1(E,A)
& r1_tarski(D,B)
& r1_tarski(E,B)
& r1_tarski(D,C)
& r1_tarski(E,C) )
=> ( D = E
| B = C ) ) ) ) ) ) ) ).
fof(d3_aff_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& l1_analoaf(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ( v1_aff_1(C,A)
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
=> ( D = k2_aff_4(A,B,C)
<=> ( r2_hidden(B,D)
& r4_aff_1(A,C,D) ) ) ) ) ) ) ) ).
fof(t27_aff_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& l1_analoaf(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ( v1_aff_1(C,A)
=> v1_aff_1(k2_aff_4(A,B,C),A) ) ) ) ) ).
fof(t28_aff_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& l1_analoaf(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
=> ( ( v1_aff_4(C,A)
& v1_aff_1(D,A)
& r2_hidden(B,C)
& r1_tarski(D,C) )
=> r1_tarski(k2_aff_4(A,B,D),C) ) ) ) ) ) ).
fof(t29_aff_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& l1_analoaf(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(A))
=> ! [F] :
( m1_subset_1(F,k1_zfmisc_1(u1_struct_0(A)))
=> ( ( v1_aff_4(F,A)
& r2_hidden(B,F)
& r2_hidden(C,F)
& r2_hidden(D,F)
& r2_analoaf(A,B,C,D,E) )
=> ( B = C
| r2_hidden(E,F) ) ) ) ) ) ) ) ) ).
fof(t30_aff_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& l1_analoaf(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ( v1_aff_1(C,A)
=> ( r2_hidden(B,C)
<=> k2_aff_4(A,B,C) = C ) ) ) ) ) ).
fof(t31_aff_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& l1_analoaf(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
=> ( v1_aff_1(D,A)
=> k2_aff_4(A,B,D) = k2_aff_4(A,B,k2_aff_4(A,C,D)) ) ) ) ) ) ).
fof(t32_aff_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& l1_analoaf(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
=> ( r4_aff_1(A,C,D)
=> k2_aff_4(A,B,C) = k2_aff_4(A,B,D) ) ) ) ) ) ).
fof(d4_aff_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& l1_analoaf(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ( r1_aff_4(A,B,C)
<=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ! [E] :
( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(A)))
=> ( ( r2_hidden(D,C)
& v1_aff_1(E,A)
& r1_tarski(E,B) )
=> r1_tarski(k2_aff_4(A,D,E),C) ) ) ) ) ) ) ) ).
fof(t33_aff_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& l1_analoaf(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ( r1_tarski(B,C)
=> ( ( ~ ( v1_aff_1(B,A)
& v1_aff_1(C,A) )
& ~ ( v1_aff_4(B,A)
& v1_aff_4(C,A) ) )
| B = C ) ) ) ) ) ).
fof(t34_aff_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& l1_analoaf(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ~ ( v1_aff_4(B,A)
& ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(A))
=> ~ ( r2_hidden(C,B)
& r2_hidden(D,B)
& r2_hidden(E,B)
& ~ r1_aff_1(A,C,D,E) ) ) ) ) ) ) ) ).
fof(t35_aff_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& l1_analoaf(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ~ ( v1_aff_1(B,A)
& v1_aff_4(C,A)
& ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ~ ( r2_hidden(D,C)
& ~ r2_hidden(D,B) ) ) ) ) ) ) ).
fof(t36_aff_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& l1_analoaf(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ~ ( v1_aff_1(C,A)
& ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
=> ~ ( r2_hidden(B,D)
& r1_tarski(C,D)
& v1_aff_4(D,A) ) ) ) ) ) ) ).
fof(t37_aff_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& l1_analoaf(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ? [E] :
( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(A)))
& r2_hidden(B,E)
& r2_hidden(C,E)
& r2_hidden(D,E)
& v1_aff_4(E,A) ) ) ) ) ) ).
fof(t38_aff_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& l1_analoaf(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
=> ~ ( r2_hidden(B,C)
& r2_hidden(B,D)
& v1_aff_1(C,A)
& v1_aff_1(D,A)
& ! [E] :
( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(A)))
=> ~ ( r1_tarski(C,E)
& r1_tarski(D,E)
& v1_aff_4(E,A) ) ) ) ) ) ) ) ).
fof(t39_aff_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& l1_analoaf(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ~ ( r4_aff_1(A,B,C)
& ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
=> ~ ( r1_tarski(B,D)
& r1_tarski(C,D)
& v1_aff_4(D,A) ) ) ) ) ) ) ).
fof(t40_aff_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& l1_analoaf(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ( ( v1_aff_1(B,A)
& v1_aff_1(C,A) )
=> ( r4_aff_1(A,B,C)
<=> r1_aff_4(A,B,C) ) ) ) ) ) ).
fof(t41_aff_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& l1_analoaf(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ( ( v1_aff_1(B,A)
& v1_aff_4(C,A) )
=> ( r1_aff_4(A,B,C)
<=> ? [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
& r1_tarski(D,C)
& ( r4_aff_1(A,B,D)
| r4_aff_1(A,D,B) ) ) ) ) ) ) ) ).
fof(t42_aff_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& l1_analoaf(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ( ( v1_aff_1(B,A)
& v1_aff_4(C,A)
& r1_tarski(B,C) )
=> r1_aff_4(A,B,C) ) ) ) ) ).
fof(t43_aff_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& l1_analoaf(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
=> ( ( v1_aff_1(C,A)
& v1_aff_4(D,A)
& r2_hidden(B,C)
& r2_hidden(B,D)
& r1_aff_4(A,C,D) )
=> r1_tarski(C,D) ) ) ) ) ) ).
fof(d5_aff_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& l1_analoaf(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
=> ( r2_aff_4(A,B,C,D)
<=> ? [E] :
( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(A)))
& r1_tarski(B,E)
& r1_tarski(C,E)
& r1_tarski(D,E)
& v1_aff_4(E,A) ) ) ) ) ) ) ).
fof(t44_aff_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& l1_analoaf(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
=> ( r2_aff_4(A,B,C,D)
=> ( r2_aff_4(A,B,D,C)
& r2_aff_4(A,C,B,D)
& r2_aff_4(A,C,D,B)
& r2_aff_4(A,D,B,C)
& r2_aff_4(A,D,C,B) ) ) ) ) ) ) ).
fof(t45_aff_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& l1_analoaf(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
=> ! [E] :
( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(A)))
=> ( ( v1_aff_1(B,A)
& v1_aff_1(C,A)
& v1_aff_1(D,A)
& v1_aff_1(E,A)
& r2_aff_4(A,D,E,C)
& r2_aff_4(A,D,E,B) )
=> ( D = E
| r2_aff_4(A,D,C,B) ) ) ) ) ) ) ) ).
fof(t46_aff_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& l1_analoaf(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
=> ! [E] :
( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(A)))
=> ( ( v1_aff_1(B,A)
& v1_aff_1(C,A)
& v1_aff_4(D,A)
& r1_tarski(B,D)
& r1_tarski(C,D) )
=> ( B = C
| ( r2_aff_4(A,B,C,E)
<=> r1_tarski(E,D) ) ) ) ) ) ) ) ) ).
fof(t47_aff_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& l1_analoaf(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
=> ( ( r2_hidden(B,C)
& r2_hidden(B,D)
& v1_aff_1(C,A)
& v1_aff_1(D,A) )
=> ( r2_aff_4(A,C,D,D)
& r2_aff_4(A,D,C,D)
& r2_aff_4(A,D,D,C) ) ) ) ) ) ) ).
fof(t48_aff_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& l1_analoaf(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ~ ( ~ ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& v2_diraf(A)
& l1_analoaf(A) )
& v1_aff_4(B,A)
& ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> r2_hidden(C,B) ) ) ) ) ).
fof(t49_aff_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& l1_analoaf(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(A))
=> ! [F] :
( m1_subset_1(F,u1_struct_0(A))
=> ! [G] :
( m1_subset_1(G,u1_struct_0(A))
=> ! [H] :
( m1_subset_1(H,u1_struct_0(A))
=> ! [I] :
( m1_subset_1(I,k1_zfmisc_1(u1_struct_0(A)))
=> ! [J] :
( m1_subset_1(J,k1_zfmisc_1(u1_struct_0(A)))
=> ! [K] :
( m1_subset_1(K,k1_zfmisc_1(u1_struct_0(A)))
=> ( ( r2_hidden(B,I)
& r2_hidden(B,J)
& r2_hidden(B,K)
& r2_hidden(C,I)
& r2_hidden(F,I)
& r2_hidden(D,J)
& r2_hidden(G,J)
& r2_hidden(E,K)
& r2_hidden(H,K)
& v1_aff_1(I,A)
& v1_aff_1(J,A)
& v1_aff_1(K,A)
& r2_analoaf(A,C,D,F,G)
& r2_analoaf(A,C,E,F,H) )
=> ( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& v2_diraf(A)
& l1_analoaf(A) )
| B = C
| B = D
| B = E
| I = J
| I = K
| r2_analoaf(A,D,E,G,H) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
fof(t50_aff_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& l1_analoaf(A) )
=> ( ~ ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& v2_diraf(A)
& l1_analoaf(A) )
=> v4_aff_2(A) ) ) ).
fof(t51_aff_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& l1_analoaf(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(A))
=> ! [F] :
( m1_subset_1(F,u1_struct_0(A))
=> ! [G] :
( m1_subset_1(G,u1_struct_0(A))
=> ! [H] :
( m1_subset_1(H,k1_zfmisc_1(u1_struct_0(A)))
=> ! [I] :
( m1_subset_1(I,k1_zfmisc_1(u1_struct_0(A)))
=> ! [J] :
( m1_subset_1(J,k1_zfmisc_1(u1_struct_0(A)))
=> ( ( r4_aff_1(A,H,I)
& r4_aff_1(A,H,J)
& r2_hidden(B,H)
& r2_hidden(C,H)
& r2_hidden(D,I)
& r2_hidden(E,I)
& r2_hidden(F,J)
& r2_hidden(G,J)
& v1_aff_1(H,A)
& v1_aff_1(I,A)
& v1_aff_1(J,A)
& r2_analoaf(A,B,D,C,E)
& r2_analoaf(A,B,F,C,G) )
=> ( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& v2_diraf(A)
& l1_analoaf(A) )
| H = I
| H = J
| r2_analoaf(A,D,F,E,G) ) ) ) ) ) ) ) ) ) ) ) ) ).
fof(t52_aff_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& l1_analoaf(A) )
=> ( ~ ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& v2_diraf(A)
& l1_analoaf(A) )
=> v11_aff_2(A) ) ) ).
fof(t53_aff_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& l1_analoaf(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(A))
=> ! [F] :
( m1_subset_1(F,u1_struct_0(A))
=> ~ ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& v2_diraf(A)
& l1_analoaf(A)
& ~ r1_aff_1(A,B,C,D)
& ! [G] :
( m1_subset_1(G,u1_struct_0(A))
=> ~ ( r2_analoaf(A,B,D,E,G)
& r2_analoaf(A,C,D,F,G) ) ) ) ) ) ) ) ) ) ).
fof(t54_aff_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& l1_analoaf(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(A))
=> ! [F] :
( m1_subset_1(F,u1_struct_0(A))
=> ~ ( ~ r1_aff_1(A,B,C,D)
& E != F
& r2_analoaf(A,B,C,E,F)
& ! [G] :
( m1_subset_1(G,u1_struct_0(A))
=> ~ ( r2_analoaf(A,B,D,E,G)
& r2_analoaf(A,C,D,F,G) ) ) ) ) ) ) ) ) ) ).
fof(t55_aff_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& l1_analoaf(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ( ( v1_aff_4(B,A)
& v1_aff_4(C,A) )
=> ( r1_aff_4(A,B,C)
<=> ? [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
& ? [E] :
( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(A)))
& ? [F] :
( m1_subset_1(F,k1_zfmisc_1(u1_struct_0(A)))
& ? [G] :
( m1_subset_1(G,k1_zfmisc_1(u1_struct_0(A)))
& ~ r4_aff_1(A,D,E)
& r1_tarski(D,B)
& r1_tarski(E,B)
& r1_tarski(F,C)
& r1_tarski(G,C)
& ( r4_aff_1(A,D,F)
| r4_aff_1(A,F,D) )
& ( r4_aff_1(A,E,G)
| r4_aff_1(A,G,E) ) ) ) ) ) ) ) ) ) ) ).
fof(t56_aff_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& l1_analoaf(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
=> ( ( r4_aff_1(A,B,C)
& r1_aff_4(A,C,D) )
=> r1_aff_4(A,B,D) ) ) ) ) ) ).
fof(t57_aff_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& l1_analoaf(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ( v1_aff_4(B,A)
=> r1_aff_4(A,B,B) ) ) ) ).
fof(t58_aff_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& l1_analoaf(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ( ( v1_aff_4(B,A)
& v1_aff_4(C,A)
& r1_aff_4(A,B,C) )
=> r1_aff_4(A,C,B) ) ) ) ) ).
fof(t59_aff_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& l1_analoaf(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ~ ( v1_aff_4(B,A)
& B = k1_xboole_0 ) ) ) ).
fof(t60_aff_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& l1_analoaf(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
=> ( ( r1_aff_4(A,B,C)
& r1_aff_4(A,C,D) )
=> ( C = k1_xboole_0
| r1_aff_4(A,B,D) ) ) ) ) ) ) ).
fof(t61_aff_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& l1_analoaf(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
=> ( ( v1_aff_4(B,A)
& v1_aff_4(C,A)
& v1_aff_4(D,A) )
=> ( ( ~ ( r1_aff_4(A,B,C)
& r1_aff_4(A,C,D) )
& ~ ( r1_aff_4(A,B,C)
& r1_aff_4(A,D,C) )
& ~ ( r1_aff_4(A,C,B)
& r1_aff_4(A,C,D) )
& ~ ( r1_aff_4(A,C,B)
& r1_aff_4(A,D,C) ) )
| ( r1_aff_4(A,B,D)
& r1_aff_4(A,D,B) ) ) ) ) ) ) ) ).
fof(t62_aff_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& l1_analoaf(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
=> ( ( v1_aff_4(C,A)
& v1_aff_4(D,A)
& r2_hidden(B,C)
& r2_hidden(B,D)
& r1_aff_4(A,C,D) )
=> C = D ) ) ) ) ) ).
fof(t63_aff_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& l1_analoaf(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(A))
=> ! [F] :
( m1_subset_1(F,k1_zfmisc_1(u1_struct_0(A)))
=> ! [G] :
( m1_subset_1(G,k1_zfmisc_1(u1_struct_0(A)))
=> ! [H] :
( m1_subset_1(H,k1_zfmisc_1(u1_struct_0(A)))
=> ( ( v1_aff_4(F,A)
& v1_aff_4(G,A)
& v1_aff_4(H,A)
& r1_aff_4(A,F,G)
& r2_hidden(B,k5_subset_1(u1_struct_0(A),F,H))
& r2_hidden(C,k5_subset_1(u1_struct_0(A),F,H))
& r2_hidden(D,k5_subset_1(u1_struct_0(A),G,H))
& r2_hidden(E,k5_subset_1(u1_struct_0(A),G,H)) )
=> ( F = G
| r2_analoaf(A,B,C,D,E) ) ) ) ) ) ) ) ) ) ) ).
fof(t64_aff_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& l1_analoaf(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(A))
=> ! [F] :
( m1_subset_1(F,k1_zfmisc_1(u1_struct_0(A)))
=> ! [G] :
( m1_subset_1(G,k1_zfmisc_1(u1_struct_0(A)))
=> ! [H] :
( m1_subset_1(H,k1_zfmisc_1(u1_struct_0(A)))
=> ( ( v1_aff_4(F,A)
& v1_aff_4(G,A)
& v1_aff_4(H,A)
& r1_aff_4(A,F,G)
& r2_hidden(B,k5_subset_1(u1_struct_0(A),F,H))
& r2_hidden(C,k5_subset_1(u1_struct_0(A),F,H))
& r2_hidden(D,k5_subset_1(u1_struct_0(A),G,H))
& r2_hidden(E,k5_subset_1(u1_struct_0(A),G,H)) )
=> ( F = G
| B = C
| D = E
| r4_aff_1(A,k5_subset_1(u1_struct_0(A),F,H),k5_subset_1(u1_struct_0(A),G,H)) ) ) ) ) ) ) ) ) ) ) ).
fof(t65_aff_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& l1_analoaf(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ~ ( v1_aff_4(C,A)
& ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
=> ~ ( r2_hidden(B,D)
& r1_aff_4(A,C,D)
& v1_aff_4(D,A) ) ) ) ) ) ) ).
fof(d6_aff_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& l1_analoaf(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ( v1_aff_4(C,A)
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
=> ( D = k3_aff_4(A,B,C)
<=> ( r2_hidden(B,D)
& r1_aff_4(A,C,D)
& v1_aff_4(D,A) ) ) ) ) ) ) ) ).
fof(t66_aff_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& l1_analoaf(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ( v1_aff_4(C,A)
=> ( r2_hidden(B,C)
<=> k3_aff_4(A,B,C) = C ) ) ) ) ) ).
fof(t67_aff_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& l1_analoaf(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
=> ( v1_aff_4(D,A)
=> k3_aff_4(A,B,D) = k3_aff_4(A,B,k3_aff_4(A,C,D)) ) ) ) ) ) ).
fof(t68_aff_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& l1_analoaf(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
=> ( ( v1_aff_1(C,A)
& v1_aff_4(D,A)
& r1_aff_4(A,C,D) )
=> r1_tarski(k2_aff_4(A,B,C),k3_aff_4(A,B,D)) ) ) ) ) ) ).
fof(t69_aff_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& l1_analoaf(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
=> ( ( v1_aff_4(C,A)
& v1_aff_4(D,A)
& r1_aff_4(A,C,D) )
=> k3_aff_4(A,B,C) = k3_aff_4(A,B,D) ) ) ) ) ) ).
fof(dt_k1_aff_4,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& l1_analoaf(A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
=> m1_subset_1(k1_aff_4(A,B,C),k1_zfmisc_1(u1_struct_0(A))) ) ).
fof(dt_k2_aff_4,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& l1_analoaf(A)
& m1_subset_1(B,u1_struct_0(A))
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
=> m1_subset_1(k2_aff_4(A,B,C),k1_zfmisc_1(u1_struct_0(A))) ) ).
fof(dt_k3_aff_4,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& l1_analoaf(A)
& m1_subset_1(B,u1_struct_0(A))
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
=> m1_subset_1(k3_aff_4(A,B,C),k1_zfmisc_1(u1_struct_0(A))) ) ).
fof(d1_aff_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& l1_analoaf(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> k1_aff_4(A,B,C) = a_3_0_aff_4(A,B,C) ) ) ) ).
fof(fraenkel_a_3_0_aff_4,axiom,
! [A,B,C,D] :
( ( ~ v3_struct_0(B)
& ~ v3_realset2(B)
& v1_diraf(B)
& l1_analoaf(B)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(B)))
& m1_subset_1(D,k1_zfmisc_1(u1_struct_0(B))) )
=> ( r2_hidden(A,a_3_0_aff_4(B,C,D))
<=> ? [E] :
( m1_subset_1(E,u1_struct_0(B))
& A = E
& ? [F] :
( m1_subset_1(F,u1_struct_0(B))
& r2_aff_1(B,E,F,C)
& r2_hidden(F,D) ) ) ) ) ).
%------------------------------------------------------------------------------