SET007 Axioms: SET007+209.ax
%------------------------------------------------------------------------------
% File : SET007+209 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Families of Subsets, Subspaces and Mappings in Topological 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 : tops_2 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 81 ( 13 unt; 0 def)
% Number of atoms : 507 ( 57 equ)
% Maximal formula atoms : 18 ( 6 avg)
% Number of connectives : 454 ( 28 ~; 0 |; 143 &)
% ( 16 <=>; 267 =>; 0 <=; 0 <~>)
% Maximal formula depth : 17 ( 8 avg)
% Maximal term depth : 5 ( 2 avg)
% Number of predicates : 24 ( 22 usr; 1 prp; 0-3 aty)
% Number of functors : 27 ( 27 usr; 1 con; 0-5 aty)
% Number of variables : 227 ( 224 !; 3 ?)
% 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_tops_2,axiom,
! [A] :
( l1_struct_0(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
=> r1_tarski(B,k1_zfmisc_1(k2_pre_topc(A))) ) ) ).
fof(t2_tops_2,axiom,
$true ).
fof(t3_tops_2,axiom,
! [A] :
( l1_struct_0(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
=> ! [C] :
( r1_tarski(C,B)
=> m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A)))) ) ) ) ).
fof(t4_tops_2,axiom,
$true ).
fof(t5_tops_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_struct_0(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
=> ~ ( r1_pre_topc(A,B)
& B = k1_xboole_0 ) ) ) ).
fof(t6_tops_2,axiom,
! [A] :
( l1_struct_0(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
=> r1_tarski(k6_subset_1(u1_struct_0(A),k5_setfam_1(u1_struct_0(A),B),k5_setfam_1(u1_struct_0(A),C)),k5_setfam_1(u1_struct_0(A),k6_subset_1(k1_zfmisc_1(u1_struct_0(A)),B,C))) ) ) ) ).
fof(t7_tops_2,axiom,
$true ).
fof(t8_tops_2,axiom,
$true ).
fof(t9_tops_2,axiom,
$true ).
fof(t10_tops_2,axiom,
! [A,B] :
( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
=> ( ~ ( B != k1_xboole_0
& k7_setfam_1(A,B) = k1_xboole_0 )
& ~ ( k7_setfam_1(A,B) != k1_xboole_0
& B = k1_xboole_0 ) ) ) ).
fof(t11_tops_2,axiom,
! [A,B] :
( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
=> ( B != k1_xboole_0
=> k6_setfam_1(A,k7_setfam_1(A,B)) = k3_subset_1(A,k5_setfam_1(A,B)) ) ) ).
fof(t12_tops_2,axiom,
! [A,B] :
( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
=> ( B != k1_xboole_0
=> k5_setfam_1(A,k7_setfam_1(A,B)) = k3_subset_1(A,k6_setfam_1(A,B)) ) ) ).
fof(t13_tops_2,axiom,
! [A] :
( l1_struct_0(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
=> ( v1_finset_1(k7_setfam_1(u1_struct_0(A),B))
<=> v1_finset_1(B) ) ) ) ).
fof(d1_tops_2,axiom,
! [A] :
( l1_pre_topc(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
=> ( v1_tops_2(B,A)
<=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ( r2_hidden(C,B)
=> v3_pre_topc(C,A) ) ) ) ) ) ).
fof(d2_tops_2,axiom,
! [A] :
( l1_pre_topc(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
=> ( v2_tops_2(B,A)
<=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ( r2_hidden(C,B)
=> v4_pre_topc(C,A) ) ) ) ) ) ).
fof(t14_tops_2,axiom,
$true ).
fof(t15_tops_2,axiom,
$true ).
fof(t16_tops_2,axiom,
! [A] :
( l1_pre_topc(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
=> ( v2_tops_2(B,A)
<=> v1_tops_2(k7_setfam_1(u1_struct_0(A),B),A) ) ) ) ).
fof(t17_tops_2,axiom,
! [A] :
( l1_pre_topc(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
=> ( v1_tops_2(B,A)
<=> v2_tops_2(k7_setfam_1(u1_struct_0(A),B),A) ) ) ) ).
fof(t18_tops_2,axiom,
! [A] :
( l1_pre_topc(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
=> ( ( r1_tarski(B,C)
& v1_tops_2(C,A) )
=> v1_tops_2(B,A) ) ) ) ) ).
fof(t19_tops_2,axiom,
! [A] :
( l1_pre_topc(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
=> ( ( r1_tarski(B,C)
& v2_tops_2(C,A) )
=> v2_tops_2(B,A) ) ) ) ) ).
fof(t20_tops_2,axiom,
! [A] :
( l1_pre_topc(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
=> ( ( v1_tops_2(B,A)
& v1_tops_2(C,A) )
=> v1_tops_2(k4_subset_1(k1_zfmisc_1(u1_struct_0(A)),B,C),A) ) ) ) ) ).
fof(t21_tops_2,axiom,
! [A] :
( l1_pre_topc(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
=> ( v1_tops_2(B,A)
=> v1_tops_2(k5_subset_1(k1_zfmisc_1(u1_struct_0(A)),B,C),A) ) ) ) ) ).
fof(t22_tops_2,axiom,
! [A] :
( l1_pre_topc(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
=> ( v1_tops_2(B,A)
=> v1_tops_2(k6_subset_1(k1_zfmisc_1(u1_struct_0(A)),B,C),A) ) ) ) ) ).
fof(t23_tops_2,axiom,
! [A] :
( l1_pre_topc(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
=> ( ( v2_tops_2(B,A)
& v2_tops_2(C,A) )
=> v2_tops_2(k4_subset_1(k1_zfmisc_1(u1_struct_0(A)),B,C),A) ) ) ) ) ).
fof(t24_tops_2,axiom,
! [A] :
( l1_pre_topc(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
=> ( v2_tops_2(B,A)
=> v2_tops_2(k5_subset_1(k1_zfmisc_1(u1_struct_0(A)),B,C),A) ) ) ) ) ).
fof(t25_tops_2,axiom,
! [A] :
( l1_pre_topc(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
=> ( v2_tops_2(B,A)
=> v2_tops_2(k6_subset_1(k1_zfmisc_1(u1_struct_0(A)),B,C),A) ) ) ) ) ).
fof(t26_tops_2,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
=> ( v1_tops_2(B,A)
=> v3_pre_topc(k5_setfam_1(u1_struct_0(A),B),A) ) ) ) ).
fof(t27_tops_2,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
=> ( ( v1_tops_2(B,A)
& v1_finset_1(B) )
=> v3_pre_topc(k6_setfam_1(u1_struct_0(A),B),A) ) ) ) ).
fof(t28_tops_2,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
=> ( ( v2_tops_2(B,A)
& v1_finset_1(B) )
=> v4_pre_topc(k5_setfam_1(u1_struct_0(A),B),A) ) ) ) ).
fof(t29_tops_2,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
=> ( v2_tops_2(B,A)
=> v4_pre_topc(k6_setfam_1(u1_struct_0(A),B),A) ) ) ) ).
fof(t30_tops_2,axiom,
$true ).
fof(t31_tops_2,axiom,
! [A] :
( l1_pre_topc(A)
=> ! [B] :
( m1_pre_topc(B,A)
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(B))))
=> m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A)))) ) ) ) ).
fof(t32_tops_2,axiom,
! [A] :
( l1_pre_topc(A)
=> ! [B] :
( m1_pre_topc(B,A)
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(B)))
=> ( v3_pre_topc(C,B)
<=> ? [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
& v3_pre_topc(D,A)
& k3_xboole_0(D,k2_pre_topc(B)) = C ) ) ) ) ) ).
fof(t33_tops_2,axiom,
! [A] :
( l1_pre_topc(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_pre_topc(C,A)
=> ( v3_pre_topc(B,A)
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(C)))
=> ( D = B
=> v3_pre_topc(D,C) ) ) ) ) ) ) ).
fof(t34_tops_2,axiom,
! [A] :
( l1_pre_topc(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_pre_topc(C,A)
=> ( v4_pre_topc(B,A)
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(C)))
=> ( D = B
=> v4_pre_topc(D,C) ) ) ) ) ) ) ).
fof(t35_tops_2,axiom,
! [A] :
( l1_pre_topc(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
=> ! [C] :
( m1_pre_topc(C,A)
=> ( v1_tops_2(B,A)
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(C))))
=> ( D = B
=> v1_tops_2(D,C) ) ) ) ) ) ) ).
fof(t36_tops_2,axiom,
! [A] :
( l1_pre_topc(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
=> ! [C] :
( m1_pre_topc(C,A)
=> ( v2_tops_2(B,A)
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(C))))
=> ( D = B
=> v2_tops_2(D,C) ) ) ) ) ) ) ).
fof(t37_tops_2,axiom,
$true ).
fof(t38_tops_2,axiom,
! [A] :
( l1_pre_topc(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)))
=> m1_subset_1(k5_subset_1(u1_struct_0(A),B,C),k1_zfmisc_1(u1_struct_0(k3_pre_topc(A,C)))) ) ) ) ).
fof(d3_tops_2,axiom,
! [A] :
( l1_pre_topc(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(k3_pre_topc(A,B)))))
=> ( D = k1_tops_2(A,B,C)
<=> ! [E] :
( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(k3_pre_topc(A,B))))
=> ( r2_hidden(E,D)
<=> ? [F] :
( m1_subset_1(F,k1_zfmisc_1(u1_struct_0(A)))
& r2_hidden(F,C)
& k5_subset_1(u1_struct_0(A),F,B) = E ) ) ) ) ) ) ) ) ).
fof(t39_tops_2,axiom,
$true ).
fof(t40_tops_2,axiom,
! [A] :
( l1_pre_topc(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
=> ( r1_tarski(C,D)
=> r1_tarski(k1_tops_2(A,B,C),k1_tops_2(A,B,D)) ) ) ) ) ) ).
fof(t41_tops_2,axiom,
! [A] :
( l1_pre_topc(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(k1_zfmisc_1(u1_struct_0(A))))
=> ( r2_hidden(B,D)
=> r2_hidden(k5_subset_1(u1_struct_0(A),B,C),k1_tops_2(A,C,D)) ) ) ) ) ) ).
fof(t42_tops_2,axiom,
! [A] :
( l1_pre_topc(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(k1_zfmisc_1(u1_struct_0(A))))
=> ( r1_tarski(B,k5_setfam_1(u1_struct_0(A),D))
=> r1_tarski(k5_subset_1(u1_struct_0(A),B,C),k5_setfam_1(u1_struct_0(k3_pre_topc(A,C)),k1_tops_2(A,C,D))) ) ) ) ) ) ).
fof(t43_tops_2,axiom,
! [A] :
( l1_pre_topc(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
=> ( r1_tarski(B,k5_setfam_1(u1_struct_0(A),C))
=> B = k5_setfam_1(u1_struct_0(k3_pre_topc(A,B)),k1_tops_2(A,B,C)) ) ) ) ) ).
fof(t44_tops_2,axiom,
! [A] :
( l1_pre_topc(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
=> r1_tarski(k5_setfam_1(u1_struct_0(k3_pre_topc(A,B)),k1_tops_2(A,B,C)),k5_setfam_1(u1_struct_0(A),C)) ) ) ) ).
fof(t45_tops_2,axiom,
! [A] :
( l1_pre_topc(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
=> ( r1_tarski(B,k5_setfam_1(u1_struct_0(k3_pre_topc(A,B)),k1_tops_2(A,B,C)))
=> r1_tarski(B,k5_setfam_1(u1_struct_0(A),C)) ) ) ) ) ).
fof(t46_tops_2,axiom,
! [A] :
( l1_pre_topc(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
=> ( v1_finset_1(C)
=> v1_finset_1(k1_tops_2(A,B,C)) ) ) ) ) ).
fof(t47_tops_2,axiom,
! [A] :
( l1_pre_topc(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
=> ( v1_tops_2(C,A)
=> v1_tops_2(k1_tops_2(A,B,C),k3_pre_topc(A,B)) ) ) ) ) ).
fof(t48_tops_2,axiom,
! [A] :
( l1_pre_topc(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
=> ( v2_tops_2(C,A)
=> v2_tops_2(k1_tops_2(A,B,C),k3_pre_topc(A,B)) ) ) ) ) ).
fof(t49_tops_2,axiom,
! [A] :
( l1_pre_topc(A)
=> ! [B] :
( m1_pre_topc(B,A)
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(B))))
=> ~ ( v1_tops_2(C,B)
& ! [D] :
( m1_subset_1(D,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
=> ~ ( v1_tops_2(D,A)
& ! [E] :
( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(A)))
=> ( E = k2_pre_topc(B)
=> C = k1_tops_2(A,E,D) ) ) ) ) ) ) ) ) ).
fof(t50_tops_2,axiom,
! [A] :
( l1_pre_topc(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
=> ? [D] :
( v1_relat_1(D)
& v1_funct_1(D)
& k1_relat_1(D) = C
& k2_relat_1(D) = k1_tops_2(A,B,C)
& ! [E] :
( r2_hidden(E,C)
=> ! [F] :
( m1_subset_1(F,k1_zfmisc_1(u1_struct_0(A)))
=> ( F = E
=> k1_funct_1(D,E) = k5_subset_1(u1_struct_0(A),F,B) ) ) ) ) ) ) ) ).
fof(t51_tops_2,axiom,
! [A] :
( l1_struct_0(A)
=> ! [B] :
( ( ~ v3_struct_0(B)
& l1_struct_0(B) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
& m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
=> ( k1_relat_1(C) = k2_pre_topc(A)
& r1_tarski(k2_relat_1(C),k2_pre_topc(B)) ) ) ) ) ).
fof(t52_tops_2,axiom,
! [A] :
( l1_struct_0(A)
=> ! [B] :
( ( ~ v3_struct_0(B)
& l1_struct_0(B) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
& m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
=> k5_pre_topc(A,B,C,k2_pre_topc(B)) = k2_pre_topc(A) ) ) ) ).
fof(t53_tops_2,axiom,
$true ).
fof(t54_tops_2,axiom,
! [A] :
( l1_struct_0(A)
=> ! [B] :
( ( ~ v3_struct_0(B)
& l1_struct_0(B) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
& m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(B))))
=> m1_subset_1(k9_relat_1(k3_funct_3(C),D),k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A)))) ) ) ) ) ).
fof(t55_tops_2,axiom,
! [A] :
( l1_pre_topc(A)
=> ! [B] :
( ( ~ v3_struct_0(B)
& l1_pre_topc(B) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
& m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
=> ( v5_pre_topc(C,A,B)
<=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(B)))
=> ( v3_pre_topc(D,B)
=> v3_pre_topc(k5_pre_topc(A,B,C,D),A) ) ) ) ) ) ) ).
fof(t56_tops_2,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( v2_pre_topc(B)
& l1_pre_topc(B) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
& m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
=> ( v5_pre_topc(C,A,B)
<=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(B)))
=> r1_tarski(k6_pre_topc(A,k5_pre_topc(A,B,C,D)),k5_pre_topc(A,B,C,k6_pre_topc(B,D))) ) ) ) ) ) ).
fof(t57_tops_2,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v2_pre_topc(B)
& l1_pre_topc(B) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
& m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
=> ( v5_pre_topc(C,A,B)
<=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
=> r1_tarski(k4_pre_topc(A,B,C,k6_pre_topc(A,D)),k6_pre_topc(B,k4_pre_topc(A,B,C,D))) ) ) ) ) ) ).
fof(t58_tops_2,axiom,
! [A] :
( l1_pre_topc(A)
=> ! [B] :
( l1_pre_topc(B)
=> ! [C] :
( ( ~ v3_struct_0(C)
& l1_pre_topc(C) )
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,u1_struct_0(A),u1_struct_0(C))
& m2_relset_1(D,u1_struct_0(A),u1_struct_0(C)) )
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,u1_struct_0(C),u1_struct_0(B))
& m2_relset_1(E,u1_struct_0(C),u1_struct_0(B)) )
=> ( ( v5_pre_topc(D,A,C)
& v5_pre_topc(E,C,B) )
=> v5_pre_topc(k7_funct_2(u1_struct_0(A),u1_struct_0(C),u1_struct_0(B),D,E),A,B) ) ) ) ) ) ) ).
fof(t59_tops_2,axiom,
! [A] :
( l1_pre_topc(A)
=> ! [B] :
( ( ~ v3_struct_0(B)
& l1_pre_topc(B) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
& m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(B))))
=> ( ( v5_pre_topc(C,A,B)
& v1_tops_2(D,B) )
=> ! [E] :
( m1_subset_1(E,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
=> ( E = k9_relat_1(k3_funct_3(C),D)
=> v1_tops_2(E,A) ) ) ) ) ) ) ) ).
fof(t60_tops_2,axiom,
! [A] :
( l1_pre_topc(A)
=> ! [B] :
( l1_pre_topc(B)
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
& m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(B))))
=> ( ( v5_pre_topc(C,A,B)
& v2_tops_2(D,B) )
=> ! [E] :
( m1_subset_1(E,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
=> ( E = k9_relat_1(k3_funct_3(C),D)
=> v2_tops_2(E,A) ) ) ) ) ) ) ) ).
fof(d4_tops_2,axiom,
! [A] :
( l1_struct_0(A)
=> ! [B] :
( l1_struct_0(B)
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
& m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
=> ( ( k2_relat_1(C) = k2_pre_topc(B)
& v2_funct_1(C) )
=> k2_tops_2(A,B,C) = k2_funct_1(C) ) ) ) ) ).
fof(t61_tops_2,axiom,
$true ).
fof(t62_tops_2,axiom,
! [A] :
( l1_struct_0(A)
=> ! [B] :
( ( ~ v3_struct_0(B)
& l1_struct_0(B) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
& m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
=> ( ( k2_relat_1(C) = k2_pre_topc(B)
& v2_funct_1(C) )
=> ( k1_relat_1(k2_tops_2(A,B,C)) = k2_pre_topc(B)
& k2_relat_1(k2_tops_2(A,B,C)) = k2_pre_topc(A) ) ) ) ) ) ).
fof(t63_tops_2,axiom,
! [A] :
( l1_struct_0(A)
=> ! [B] :
( l1_struct_0(B)
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
& m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
=> ( ( k2_relat_1(C) = k2_pre_topc(B)
& v2_funct_1(C) )
=> v2_funct_1(k2_tops_2(A,B,C)) ) ) ) ) ).
fof(t64_tops_2,axiom,
! [A] :
( l1_struct_0(A)
=> ! [B] :
( ( ~ v3_struct_0(B)
& l1_struct_0(B) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
& m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
=> ( ( k2_relat_1(C) = k2_pre_topc(B)
& v2_funct_1(C) )
=> k2_tops_2(B,A,k2_tops_2(A,B,C)) = C ) ) ) ) ).
fof(t65_tops_2,axiom,
! [A] :
( l1_struct_0(A)
=> ! [B] :
( l1_struct_0(B)
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
& m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
=> ( ( k2_relat_1(C) = k2_pre_topc(B)
& v2_funct_1(C) )
=> ( k5_relat_1(C,k2_tops_2(A,B,C)) = k6_relat_1(k1_relat_1(C))
& k5_relat_1(k2_tops_2(A,B,C),C) = k6_relat_1(k2_relat_1(C)) ) ) ) ) ) ).
fof(t66_tops_2,axiom,
! [A] :
( l1_struct_0(A)
=> ! [B] :
( ( ~ v3_struct_0(B)
& l1_struct_0(B) )
=> ! [C] :
( ( ~ v3_struct_0(C)
& l1_struct_0(C) )
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,u1_struct_0(A),u1_struct_0(B))
& m2_relset_1(D,u1_struct_0(A),u1_struct_0(B)) )
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,u1_struct_0(B),u1_struct_0(C))
& m2_relset_1(E,u1_struct_0(B),u1_struct_0(C)) )
=> ( ( k1_relat_1(D) = k2_pre_topc(A)
& k2_relat_1(D) = k2_pre_topc(B)
& v2_funct_1(D)
& k1_relat_1(E) = k2_pre_topc(B)
& k2_relat_1(E) = k2_pre_topc(C)
& v2_funct_1(E) )
=> k2_tops_2(A,C,k7_funct_2(u1_struct_0(A),u1_struct_0(B),u1_struct_0(C),D,E)) = k7_funct_2(u1_struct_0(C),u1_struct_0(B),u1_struct_0(A),k2_tops_2(B,C,E),k2_tops_2(A,B,D)) ) ) ) ) ) ) ).
fof(t67_tops_2,axiom,
! [A] :
( l1_struct_0(A)
=> ! [B] :
( l1_struct_0(B)
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
& m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
=> ( ( k2_relat_1(C) = k2_pre_topc(B)
& v2_funct_1(C) )
=> k4_pre_topc(A,B,C,D) = k5_pre_topc(B,A,k2_tops_2(A,B,C),D) ) ) ) ) ) ).
fof(t68_tops_2,axiom,
! [A] :
( l1_struct_0(A)
=> ! [B] :
( l1_struct_0(B)
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
& m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(B)))
=> ( ( k2_relat_1(C) = k2_pre_topc(B)
& v2_funct_1(C) )
=> k5_pre_topc(A,B,C,D) = k4_pre_topc(B,A,k2_tops_2(A,B,C),D) ) ) ) ) ) ).
fof(d5_tops_2,axiom,
! [A] :
( l1_pre_topc(A)
=> ! [B] :
( l1_pre_topc(B)
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
& m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
=> ( v3_tops_2(C,A,B)
<=> ( k1_relat_1(C) = k2_pre_topc(A)
& k2_relat_1(C) = k2_pre_topc(B)
& v2_funct_1(C)
& v5_pre_topc(C,A,B)
& v5_pre_topc(k2_tops_2(A,B,C),B,A) ) ) ) ) ) ).
fof(t69_tops_2,axiom,
$true ).
fof(t70_tops_2,axiom,
! [A] :
( l1_pre_topc(A)
=> ! [B] :
( ( ~ v3_struct_0(B)
& l1_pre_topc(B) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
& m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
=> ( v3_tops_2(C,A,B)
=> v3_tops_2(k2_tops_2(A,B,C),B,A) ) ) ) ) ).
fof(t71_tops_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& l1_pre_topc(B) )
=> ! [C] :
( ( ~ v3_struct_0(C)
& l1_pre_topc(C) )
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,u1_struct_0(A),u1_struct_0(B))
& m2_relset_1(D,u1_struct_0(A),u1_struct_0(B)) )
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,u1_struct_0(B),u1_struct_0(C))
& m2_relset_1(E,u1_struct_0(B),u1_struct_0(C)) )
=> ( ( v3_tops_2(D,A,B)
& v3_tops_2(E,B,C) )
=> v3_tops_2(k7_funct_2(u1_struct_0(A),u1_struct_0(B),u1_struct_0(C),D,E),A,C) ) ) ) ) ) ) ).
fof(t72_tops_2,axiom,
! [A] :
( l1_pre_topc(A)
=> ! [B] :
( ( ~ v3_struct_0(B)
& l1_pre_topc(B) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
& m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
=> ( v3_tops_2(C,A,B)
<=> ( k1_relat_1(C) = k2_pre_topc(A)
& k2_relat_1(C) = k2_pre_topc(B)
& v2_funct_1(C)
& ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
=> ( v4_pre_topc(D,A)
<=> v4_pre_topc(k4_pre_topc(A,B,C,D),B) ) ) ) ) ) ) ) ).
fof(t73_tops_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( v2_pre_topc(B)
& l1_pre_topc(B) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
& m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
=> ( v3_tops_2(C,A,B)
<=> ( k1_relat_1(C) = k2_pre_topc(A)
& k2_relat_1(C) = k2_pre_topc(B)
& v2_funct_1(C)
& ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(B)))
=> k5_pre_topc(A,B,C,k6_pre_topc(B,D)) = k6_pre_topc(A,k5_pre_topc(A,B,C,D)) ) ) ) ) ) ) ).
fof(t74_tops_2,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v2_pre_topc(B)
& l1_pre_topc(B) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
& m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
=> ( v3_tops_2(C,A,B)
<=> ( k1_relat_1(C) = k2_pre_topc(A)
& k2_relat_1(C) = k2_pre_topc(B)
& v2_funct_1(C)
& ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
=> k4_pre_topc(A,B,C,k6_pre_topc(A,D)) = k6_pre_topc(B,k4_pre_topc(A,B,C,D)) ) ) ) ) ) ) ).
fof(dt_k1_tops_2,axiom,
! [A,B,C] :
( ( l1_pre_topc(A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
& m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A)))) )
=> m1_subset_1(k1_tops_2(A,B,C),k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(k3_pre_topc(A,B))))) ) ).
fof(dt_k2_tops_2,axiom,
! [A,B,C] :
( ( l1_struct_0(A)
& l1_struct_0(B)
& v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
& m1_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
=> ( v1_funct_1(k2_tops_2(A,B,C))
& v1_funct_2(k2_tops_2(A,B,C),u1_struct_0(B),u1_struct_0(A))
& m2_relset_1(k2_tops_2(A,B,C),u1_struct_0(B),u1_struct_0(A)) ) ) ).
%------------------------------------------------------------------------------