SET007 Axioms: SET007+229.ax
%------------------------------------------------------------------------------
% File : SET007+229 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Parallelity and Lines 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_1 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 85 ( 14 unt; 0 def)
% Number of atoms : 853 ( 56 equ)
% Maximal formula atoms : 20 ( 10 avg)
% Number of connectives : 961 ( 193 ~; 21 |; 391 &)
% ( 14 <=>; 342 =>; 0 <=; 0 <~>)
% Maximal formula depth : 24 ( 12 avg)
% Maximal term depth : 3 ( 1 avg)
% Number of predicates : 16 ( 14 usr; 1 prp; 0-5 aty)
% Number of functors : 4 ( 4 usr; 0 con; 1-3 aty)
% Number of variables : 327 ( 312 !; 15 ?)
% 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(d1_aff_1,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))
=> ( r1_aff_1(A,B,C,D)
<=> r2_analoaf(A,B,C,B,D) ) ) ) ) ) ).
fof(t1_aff_1,axiom,
$true ).
fof(t2_aff_1,axiom,
$true ).
fof(t3_aff_1,axiom,
$true ).
fof(t4_aff_1,axiom,
$true ).
fof(t5_aff_1,axiom,
$true ).
fof(t6_aff_1,axiom,
$true ).
fof(t7_aff_1,axiom,
$true ).
fof(t8_aff_1,axiom,
$true ).
fof(t9_aff_1,axiom,
$true ).
fof(t10_aff_1,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))
=> B = C ) ) ) ).
fof(t11_aff_1,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))
=> ( r2_analoaf(A,B,C,C,B)
& r2_analoaf(A,B,C,B,C) ) ) ) ) ).
fof(t12_aff_1,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))
=> ( r2_analoaf(A,B,C,D,D)
& r2_analoaf(A,D,D,B,C) ) ) ) ) ) ).
fof(t13_aff_1,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))
=> ( r2_analoaf(A,B,C,D,E)
=> ( r2_analoaf(A,B,C,E,D)
& r2_analoaf(A,C,B,D,E)
& r2_analoaf(A,C,B,E,D)
& r2_analoaf(A,D,E,B,C)
& r2_analoaf(A,D,E,C,B)
& r2_analoaf(A,E,D,B,C)
& r2_analoaf(A,E,D,C,B) ) ) ) ) ) ) ) ).
fof(t14_aff_1,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))
=> ~ ( B != C
& ~ ( ~ ( r2_analoaf(A,B,C,D,E)
& r2_analoaf(A,B,C,F,G) )
& ~ ( r2_analoaf(A,B,C,D,E)
& r2_analoaf(A,F,G,B,C) )
& ~ ( r2_analoaf(A,D,E,B,C)
& r2_analoaf(A,F,G,B,C) )
& ~ ( r2_analoaf(A,D,E,B,C)
& r2_analoaf(A,B,C,F,G) ) )
& ~ r2_analoaf(A,D,E,F,G) ) ) ) ) ) ) ) ) ).
fof(t15_aff_1,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))
=> ( r1_aff_1(A,B,C,D)
=> ( r1_aff_1(A,B,D,C)
& r1_aff_1(A,C,B,D)
& r1_aff_1(A,C,D,B)
& r1_aff_1(A,D,B,C)
& r1_aff_1(A,D,C,B) ) ) ) ) ) ) ).
fof(t16_aff_1,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))
=> ( r1_aff_1(A,B,B,C)
& r1_aff_1(A,B,C,C)
& r1_aff_1(A,B,C,B) ) ) ) ) ).
fof(t17_aff_1,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)
& r1_aff_1(A,B,C,E)
& r1_aff_1(A,B,C,F) )
=> ( B = C
| r1_aff_1(A,D,E,F) ) ) ) ) ) ) ) ) ).
fof(t18_aff_1,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)
& r2_analoaf(A,B,C,D,E) )
=> ( B = C
| r1_aff_1(A,B,C,E) ) ) ) ) ) ) ) ).
fof(t19_aff_1,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,C,E) )
=> r2_analoaf(A,B,C,D,E) ) ) ) ) ) ) ).
fof(t20_aff_1,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,D,E,B)
& r1_aff_1(A,D,E,C)
& r1_aff_1(A,B,C,F) )
=> ( B = C
| r1_aff_1(A,D,E,F) ) ) ) ) ) ) ) ) ).
fof(t21_aff_1,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))
=> r1_aff_1(A,B,C,D) ) ) ) ) ).
fof(t22_aff_1,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))
=> ~ ( B != C
& ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> r1_aff_1(A,B,C,D) ) ) ) ) ) ).
fof(t23_aff_1,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,D,E)
& r2_analoaf(A,C,D,C,E) )
=> ( r1_aff_1(A,B,C,D)
| D = E ) ) ) ) ) ) ) ).
fof(d2_aff_1,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)))
=> ( D = k1_aff_1(A,B,C)
<=> ! [E] :
( m1_subset_1(E,u1_struct_0(A))
=> ( r2_hidden(E,D)
<=> r1_aff_1(A,B,C,E) ) ) ) ) ) ) ) ).
fof(t24_aff_1,axiom,
$true ).
fof(t25_aff_1,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))
=> k1_aff_1(A,B,C) = k1_aff_1(A,C,B) ) ) ) ).
fof(t26_aff_1,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))
=> ( r2_hidden(B,k2_aff_1(A,B,C))
& r2_hidden(C,k2_aff_1(A,B,C)) ) ) ) ) ).
fof(t27_aff_1,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))
=> ( ( r2_hidden(B,k2_aff_1(A,C,D))
& r2_hidden(E,k2_aff_1(A,C,D)) )
=> ( B = E
| r1_tarski(k2_aff_1(A,B,E),k2_aff_1(A,C,D)) ) ) ) ) ) ) ) ).
fof(t28_aff_1,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))
=> ( ( r2_hidden(B,k2_aff_1(A,C,D))
& r2_hidden(E,k2_aff_1(A,C,D)) )
=> ( C = D
| r1_tarski(k2_aff_1(A,C,D),k2_aff_1(A,B,E)) ) ) ) ) ) ) ) ).
fof(d3_aff_1,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))
& ? [D] :
( m1_subset_1(D,u1_struct_0(A))
& C != D
& B = k2_aff_1(A,C,D) ) ) ) ) ) ).
fof(t29_aff_1,axiom,
$true ).
fof(t30_aff_1,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_1(D,A)
& v1_aff_1(E,A)
& r2_hidden(B,D)
& r2_hidden(C,D)
& r2_hidden(B,E)
& r2_hidden(C,E)
& B != C
& D != E ) ) ) ) ) ) ).
fof(t31_aff_1,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))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ~ ( r2_hidden(C,B)
& r2_hidden(D,B)
& C != D ) ) ) ) ) ) ).
fof(t32_aff_1,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,u1_struct_0(A))
=> ~ ( B != D
& r2_hidden(D,C) ) ) ) ) ) ) ).
fof(t33_aff_1,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))
=> ( r1_aff_1(A,B,C,D)
<=> ? [E] :
( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(A)))
& v1_aff_1(E,A)
& r2_hidden(B,E)
& r2_hidden(C,E)
& r2_hidden(D,E) ) ) ) ) ) ) ).
fof(d4_aff_1,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_aff_1(A,B,C,D)
<=> ? [E] :
( m1_subset_1(E,u1_struct_0(A))
& ? [F] :
( m1_subset_1(F,u1_struct_0(A))
& E != F
& D = k2_aff_1(A,E,F)
& r2_analoaf(A,B,C,E,F) ) ) ) ) ) ) ) ).
fof(d5_aff_1,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)))
=> ( r3_aff_1(A,B,C)
<=> ? [D] :
( m1_subset_1(D,u1_struct_0(A))
& ? [E] :
( m1_subset_1(E,u1_struct_0(A))
& B = k2_aff_1(A,D,E)
& D != E
& r2_aff_1(A,D,E,C) ) ) ) ) ) ) ).
fof(t34_aff_1,axiom,
$true ).
fof(t35_aff_1,axiom,
$true ).
fof(t36_aff_1,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))
=> ( r2_hidden(B,k2_aff_1(A,C,D))
=> ( C = D
| ( r2_hidden(E,k2_aff_1(A,C,D))
<=> r2_analoaf(A,C,D,B,E) ) ) ) ) ) ) ) ) ).
fof(t37_aff_1,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)
& r2_hidden(B,D) )
=> ( r2_hidden(C,D)
<=> r2_aff_1(A,B,C,D) ) ) ) ) ) ) ).
fof(t38_aff_1,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)))
=> ( ( D = k2_aff_1(A,B,C)
=> ( B = C
| ( v1_aff_1(D,A)
& r2_hidden(B,D)
& r2_hidden(C,D)
& B != C ) ) )
& ( ( v1_aff_1(D,A)
& r2_hidden(B,D)
& r2_hidden(C,D) )
=> ( B = C
| ( B != C
& D = k2_aff_1(A,B,C) ) ) ) ) ) ) ) ) ).
fof(t39_aff_1,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)))
=> ( ( v1_aff_1(E,A)
& r2_hidden(B,E)
& r2_hidden(C,E)
& r1_aff_1(A,B,C,D) )
=> ( B = C
| r2_hidden(D,E) ) ) ) ) ) ) ) ).
fof(t40_aff_1,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,u1_struct_0(A))
& ? [D] :
( m1_subset_1(D,u1_struct_0(A))
& r2_aff_1(A,C,D,B) ) )
=> v1_aff_1(B,A) ) ) ) ).
fof(t41_aff_1,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_hidden(B,F)
& r2_hidden(C,F)
& v1_aff_1(F,A) )
=> ( B = C
| ( r2_aff_1(A,D,E,F)
<=> r2_analoaf(A,D,E,B,C) ) ) ) ) ) ) ) ) ) ).
fof(t42_aff_1,axiom,
$true ).
fof(t43_aff_1,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))
=> ( B != C
=> r2_aff_1(A,B,C,k2_aff_1(A,B,C)) ) ) ) ) ).
fof(t44_aff_1,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)
=> ( r2_aff_1(A,B,C,D)
<=> ? [E] :
( m1_subset_1(E,u1_struct_0(A))
& ? [F] :
( m1_subset_1(F,u1_struct_0(A))
& E != F
& r2_hidden(E,D)
& r2_hidden(F,D)
& r2_analoaf(A,B,C,E,F) ) ) ) ) ) ) ) ) ).
fof(t45_aff_1,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_1(F,A)
& r2_aff_1(A,B,C,F)
& r2_aff_1(A,D,E,F) )
=> r2_analoaf(A,B,C,D,E) ) ) ) ) ) ) ) ).
fof(t46_aff_1,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_analoaf(A,B,C,D,E) )
=> ( B = C
| r2_aff_1(A,D,E,F) ) ) ) ) ) ) ) ) ).
fof(t47_aff_1,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_aff_1(A,B,B,C) ) ) ) ) ).
fof(t48_aff_1,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_aff_1(A,B,C,D)
=> r2_aff_1(A,C,B,D) ) ) ) ) ) ).
fof(t49_aff_1,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_aff_1(A,B,C,D)
& ~ r2_hidden(B,D)
& r2_hidden(C,D) ) ) ) ) ) ).
fof(t50_aff_1,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)))
=> ( r3_aff_1(A,B,C)
=> ( v1_aff_1(B,A)
& v1_aff_1(C,A) ) ) ) ) ) ).
fof(t51_aff_1,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)))
=> ( r3_aff_1(A,B,C)
<=> ? [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))
& D != E
& F != G
& r2_analoaf(A,D,E,F,G)
& B = k2_aff_1(A,D,E)
& C = k2_aff_1(A,F,G) ) ) ) ) ) ) ) ) ).
fof(t52_aff_1,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)))
=> ( ( v1_aff_1(F,A)
& v1_aff_1(G,A)
& r2_hidden(B,F)
& r2_hidden(C,F)
& r2_hidden(D,G)
& r2_hidden(E,G) )
=> ( B = C
| D = E
| ( r3_aff_1(A,F,G)
<=> r2_analoaf(A,B,C,D,E) ) ) ) ) ) ) ) ) ) ) ).
fof(t53_aff_1,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)
& r3_aff_1(A,F,G) )
=> r2_analoaf(A,B,C,D,E) ) ) ) ) ) ) ) ) ).
fof(t54_aff_1,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_hidden(B,D)
& r2_hidden(C,D)
& r3_aff_1(A,D,E) )
=> r2_aff_1(A,B,C,E) ) ) ) ) ) ) ).
fof(t55_aff_1,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)
=> r3_aff_1(A,B,B) ) ) ) ).
fof(t56_aff_1,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)))
=> ( r3_aff_1(A,B,C)
=> r3_aff_1(A,C,B) ) ) ) ) ).
fof(t57_aff_1,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)
& r4_aff_1(A,D,E) )
=> r2_aff_1(A,B,C,E) ) ) ) ) ) ) ).
fof(t58_aff_1,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)
& r4_aff_1(A,C,D) )
& ~ ( r4_aff_1(A,B,C)
& r4_aff_1(A,D,C) )
& ~ ( r4_aff_1(A,C,B)
& r4_aff_1(A,C,D) )
& ~ ( r4_aff_1(A,C,B)
& r4_aff_1(A,D,C) ) )
=> r4_aff_1(A,B,D) ) ) ) ) ) ).
fof(t59_aff_1,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)
& r2_hidden(B,C)
& r2_hidden(B,D) )
=> C = D ) ) ) ) ) ).
fof(t60_aff_1,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_aff_1(A,C,D,E)
& C != D
& r1_aff_1(A,B,C,D) ) ) ) ) ) ) ).
fof(t61_aff_1,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)))
=> ( ( r2_aff_1(A,B,C,G)
& r2_aff_1(A,D,E,G)
& r1_aff_1(A,F,B,D)
& r1_aff_1(A,F,C,E)
& r2_hidden(F,G)
& B = C )
=> ( r2_hidden(B,G)
| D = E ) ) ) ) ) ) ) ) ) ).
fof(t62_aff_1,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_1(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(t63_aff_1,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)
& r4_aff_1(A,C,D) ) ) ) ) ) ) ).
fof(t64_aff_1,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)))
=> ( ( r4_aff_1(A,C,D)
& r4_aff_1(A,C,E)
& r2_hidden(B,D)
& r2_hidden(B,E) )
=> D = E ) ) ) ) ) ) ).
fof(t65_aff_1,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_1(F,A)
& r2_hidden(B,F)
& r2_hidden(C,F)
& r2_hidden(D,F)
& r2_hidden(E,F) )
=> r2_analoaf(A,B,C,D,E) ) ) ) ) ) ) ) ).
fof(t66_aff_1,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)
& r2_hidden(B,D)
& r2_hidden(C,D) )
=> r2_aff_1(A,B,C,D) ) ) ) ) ) ).
fof(t67_aff_1,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,B,C,E) )
=> ( B = C
| r4_aff_1(A,D,E) ) ) ) ) ) ) ) ).
fof(t68_aff_1,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,E)
& r1_aff_1(A,B,D,F)
& r2_analoaf(A,C,D,E,F)
& E = F )
=> ( r1_aff_1(A,B,C,D)
| ( E = B
& F = B ) ) ) ) ) ) ) ) ) ).
fof(t69_aff_1,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,E)
& r1_aff_1(A,B,D,F)
& r2_analoaf(A,C,D,E,F)
& E = B )
=> ( r1_aff_1(A,B,C,D)
| F = B ) ) ) ) ) ) ) ) ).
fof(t70_aff_1,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))
=> ( ( r1_aff_1(A,B,C,E)
& r1_aff_1(A,B,D,F)
& r1_aff_1(A,B,D,G)
& r2_analoaf(A,C,D,E,F)
& r2_analoaf(A,C,D,E,G) )
=> ( r1_aff_1(A,B,C,D)
| F = G ) ) ) ) ) ) ) ) ) ).
fof(t71_aff_1,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)
& r2_hidden(B,D)
& r2_hidden(C,D) )
=> ( B = C
| D = k2_aff_1(A,B,C) ) ) ) ) ) ) ).
fof(t72_aff_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& v2_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)
& ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ~ ( r2_hidden(D,B)
& r2_hidden(D,C) ) ) ) ) ) ) ).
fof(t73_aff_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& v2_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)
& ~ r2_aff_1(A,B,C,D)
& ! [E] :
( m1_subset_1(E,u1_struct_0(A))
=> ~ ( r2_hidden(E,D)
& r1_aff_1(A,B,C,E) ) ) ) ) ) ) ) ).
fof(t74_aff_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& v2_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))
=> ~ ( ~ r2_analoaf(A,B,C,D,E)
& ! [F] :
( m1_subset_1(F,u1_struct_0(A))
=> ~ ( r1_aff_1(A,B,C,F)
& r1_aff_1(A,D,E,F) ) ) ) ) ) ) ) ) ).
fof(symmetry_r4_aff_1,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))) )
=> ( r4_aff_1(A,B,C)
=> r4_aff_1(A,C,B) ) ) ).
fof(redefinition_r4_aff_1,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))) )
=> ( r4_aff_1(A,B,C)
<=> r3_aff_1(A,B,C) ) ) ).
fof(dt_k1_aff_1,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,u1_struct_0(A)) )
=> m1_subset_1(k1_aff_1(A,B,C),k1_zfmisc_1(u1_struct_0(A))) ) ).
fof(dt_k2_aff_1,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,u1_struct_0(A)) )
=> m1_subset_1(k2_aff_1(A,B,C),k1_zfmisc_1(u1_struct_0(A))) ) ).
fof(commutativity_k2_aff_1,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,u1_struct_0(A)) )
=> k2_aff_1(A,B,C) = k2_aff_1(A,C,B) ) ).
fof(redefinition_k2_aff_1,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,u1_struct_0(A)) )
=> k2_aff_1(A,B,C) = k1_aff_1(A,B,C) ) ).
%------------------------------------------------------------------------------