SET007 Axioms: SET007+206.ax
%------------------------------------------------------------------------------
% File : SET007+206 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Topological Spaces and Continuous Functions
% 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 : pre_topc [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 96 ( 30 unt; 0 def)
% Number of atoms : 337 ( 37 equ)
% Maximal formula atoms : 11 ( 3 avg)
% Number of connectives : 262 ( 21 ~; 0 |; 88 &)
% ( 14 <=>; 139 =>; 0 <=; 0 <~>)
% Maximal formula depth : 16 ( 5 avg)
% Maximal term depth : 4 ( 1 avg)
% Number of predicates : 26 ( 24 usr; 1 prp; 0-3 aty)
% Number of functors : 20 ( 20 usr; 1 con; 0-4 aty)
% Number of variables : 155 ( 143 !; 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(rc1_pre_topc,axiom,
? [A] :
( l1_pre_topc(A)
& v1_pre_topc(A) ) ).
fof(rc2_pre_topc,axiom,
? [A] :
( l1_pre_topc(A)
& ~ v3_struct_0(A)
& v1_pre_topc(A)
& v2_pre_topc(A) ) ).
fof(fc1_pre_topc,axiom,
! [A] :
( l1_struct_0(A)
=> ( v1_xboole_0(k1_pre_topc(A))
& v1_membered(k1_pre_topc(A))
& v2_membered(k1_pre_topc(A))
& v3_membered(k1_pre_topc(A))
& v4_membered(k1_pre_topc(A))
& v5_membered(k1_pre_topc(A)) ) ) ).
fof(fc2_pre_topc,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_struct_0(A) )
=> ~ v1_xboole_0(k2_pre_topc(A)) ) ).
fof(rc3_pre_topc,axiom,
! [A] :
( l1_pre_topc(A)
=> ? [B] :
( m1_pre_topc(B,A)
& v1_pre_topc(B) ) ) ).
fof(rc4_pre_topc,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_pre_topc(A) )
=> ? [B] :
( m1_pre_topc(B,A)
& ~ v3_struct_0(B)
& v1_pre_topc(B) ) ) ).
fof(cc1_pre_topc,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_pre_topc(B,A)
=> v2_pre_topc(B) ) ) ).
fof(fc3_pre_topc,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& l1_pre_topc(A)
& ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ( ~ v3_struct_0(k3_pre_topc(A,B))
& v1_pre_topc(k3_pre_topc(A,B)) ) ) ).
fof(rc5_pre_topc,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> ? [B] :
( m1_pre_topc(B,A)
& v1_pre_topc(B)
& v2_pre_topc(B) ) ) ).
fof(fc4_pre_topc,axiom,
! [A,B] :
( ( v2_pre_topc(A)
& l1_pre_topc(A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ( v1_pre_topc(k3_pre_topc(A,B))
& v2_pre_topc(k3_pre_topc(A,B)) ) ) ).
fof(fc5_pre_topc,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> v4_pre_topc(k2_pre_topc(A),A) ) ).
fof(rc6_pre_topc,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> ? [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
& v4_pre_topc(B,A) ) ) ).
fof(rc7_pre_topc,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ? [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
& ~ v1_xboole_0(B)
& v4_pre_topc(B,A) ) ) ).
fof(d1_pre_topc,axiom,
! [A] :
( l1_pre_topc(A)
=> ( v2_pre_topc(A)
<=> ( r2_hidden(u1_struct_0(A),u1_pre_topc(A))
& ! [B] :
( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
=> ( r1_tarski(B,u1_pre_topc(A))
=> r2_hidden(k5_setfam_1(u1_struct_0(A),B),u1_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)))
=> ( ( r2_hidden(B,u1_pre_topc(A))
& r2_hidden(C,u1_pre_topc(A)) )
=> r2_hidden(k5_subset_1(u1_struct_0(A),B,C),u1_pre_topc(A)) ) ) ) ) ) ) ).
fof(t1_pre_topc,axiom,
$true ).
fof(t2_pre_topc,axiom,
$true ).
fof(t3_pre_topc,axiom,
$true ).
fof(t4_pre_topc,axiom,
$true ).
fof(t5_pre_topc,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> r2_hidden(k1_xboole_0,u1_pre_topc(A)) ) ).
fof(d2_pre_topc,axiom,
! [A] :
( l1_struct_0(A)
=> k1_pre_topc(A) = k1_xboole_0 ) ).
fof(d3_pre_topc,axiom,
! [A] :
( l1_struct_0(A)
=> k2_pre_topc(A) = u1_struct_0(A) ) ).
fof(t6_pre_topc,axiom,
$true ).
fof(t7_pre_topc,axiom,
$true ).
fof(t8_pre_topc,axiom,
$true ).
fof(t9_pre_topc,axiom,
$true ).
fof(t10_pre_topc,axiom,
$true ).
fof(t11_pre_topc,axiom,
$true ).
fof(t12_pre_topc,axiom,
! [A] :
( l1_struct_0(A)
=> k2_pre_topc(A) = u1_struct_0(A) ) ).
fof(t13_pre_topc,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_struct_0(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> r2_hidden(B,k2_pre_topc(A)) ) ) ).
fof(t14_pre_topc,axiom,
! [A] :
( l1_struct_0(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> r1_tarski(B,k2_pre_topc(A)) ) ) ).
fof(t15_pre_topc,axiom,
! [A] :
( l1_struct_0(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> k5_subset_1(u1_struct_0(A),B,k2_pre_topc(A)) = B ) ) ).
fof(t16_pre_topc,axiom,
! [A] :
( l1_struct_0(A)
=> ! [B] :
( r1_tarski(B,k2_pre_topc(A))
=> m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) ) ) ).
fof(t17_pre_topc,axiom,
! [A] :
( l1_struct_0(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> k3_subset_1(u1_struct_0(A),B) = k6_subset_1(u1_struct_0(A),k2_pre_topc(A),B) ) ) ).
fof(t18_pre_topc,axiom,
! [A] :
( l1_struct_0(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> k4_subset_1(u1_struct_0(A),B,k3_subset_1(u1_struct_0(A),B)) = k2_pre_topc(A) ) ) ).
fof(t19_pre_topc,axiom,
$true ).
fof(t20_pre_topc,axiom,
$true ).
fof(t21_pre_topc,axiom,
$true ).
fof(t22_pre_topc,axiom,
! [A] :
( l1_struct_0(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> k6_subset_1(u1_struct_0(A),k2_pre_topc(A),k6_subset_1(u1_struct_0(A),k2_pre_topc(A),B)) = B ) ) ).
fof(t23_pre_topc,axiom,
! [A] :
( l1_struct_0(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ( ~ ( B != k2_pre_topc(A)
& k6_subset_1(u1_struct_0(A),k2_pre_topc(A),B) = k1_xboole_0 )
& ~ ( k6_subset_1(u1_struct_0(A),k2_pre_topc(A),B) != k1_xboole_0
& B = k2_pre_topc(A) ) ) ) ) ).
fof(t24_pre_topc,axiom,
! [A] :
( l1_struct_0(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)))
=> ( k6_subset_1(u1_struct_0(A),k2_pre_topc(A),B) = C
=> k2_pre_topc(A) = k4_subset_1(u1_struct_0(A),B,C) ) ) ) ) ).
fof(t25_pre_topc,axiom,
! [A] :
( l1_struct_0(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)))
=> ( ( k2_pre_topc(A) = k4_subset_1(u1_struct_0(A),B,C)
& r1_xboole_0(B,C) )
=> C = k6_subset_1(u1_struct_0(A),k2_pre_topc(A),B) ) ) ) ) ).
fof(t26_pre_topc,axiom,
$true ).
fof(t27_pre_topc,axiom,
! [A] :
( l1_struct_0(A)
=> k2_pre_topc(A) = k3_subset_1(u1_struct_0(A),k1_pre_topc(A)) ) ).
fof(d4_pre_topc,axiom,
$true ).
fof(d5_pre_topc,axiom,
! [A] :
( l1_pre_topc(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ( v3_pre_topc(B,A)
<=> r2_hidden(B,u1_pre_topc(A)) ) ) ) ).
fof(d6_pre_topc,axiom,
! [A] :
( l1_pre_topc(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ( v4_pre_topc(B,A)
<=> v3_pre_topc(k6_subset_1(u1_struct_0(A),k2_pre_topc(A),B),A) ) ) ) ).
fof(d7_pre_topc,axiom,
$true ).
fof(d8_pre_topc,axiom,
! [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)
<=> k2_pre_topc(A) = k5_setfam_1(u1_struct_0(A),B) ) ) ) ).
fof(d9_pre_topc,axiom,
! [A] :
( l1_pre_topc(A)
=> ! [B] :
( l1_pre_topc(B)
=> ( m1_pre_topc(B,A)
<=> ( r1_tarski(k2_pre_topc(B),k2_pre_topc(A))
& ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(B)))
=> ( r2_hidden(C,u1_pre_topc(B))
<=> ? [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
& r2_hidden(D,u1_pre_topc(A))
& C = k3_xboole_0(D,k2_pre_topc(B)) ) ) ) ) ) ) ) ).
fof(d10_pre_topc,axiom,
! [A] :
( l1_pre_topc(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( ( v1_pre_topc(C)
& m1_pre_topc(C,A) )
=> ( C = k3_pre_topc(A,B)
<=> k2_pre_topc(C) = B ) ) ) ) ).
fof(d11_pre_topc,axiom,
$true ).
fof(d12_pre_topc,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)) )
=> ( v5_pre_topc(C,A,B)
<=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(B)))
=> ( v4_pre_topc(D,B)
=> v4_pre_topc(k5_pre_topc(A,B,C,D),A) ) ) ) ) ) ) ).
fof(t28_pre_topc,axiom,
$true ).
fof(t29_pre_topc,axiom,
$true ).
fof(t30_pre_topc,axiom,
$true ).
fof(t31_pre_topc,axiom,
$true ).
fof(t32_pre_topc,axiom,
$true ).
fof(t33_pre_topc,axiom,
$true ).
fof(t34_pre_topc,axiom,
$true ).
fof(t35_pre_topc,axiom,
$true ).
fof(t36_pre_topc,axiom,
$true ).
fof(t37_pre_topc,axiom,
$true ).
fof(t38_pre_topc,axiom,
$true ).
fof(t39_pre_topc,axiom,
! [A] :
( l1_pre_topc(A)
=> ! [B] :
( m1_pre_topc(B,A)
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(B)))
=> m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) ) ) ) ).
fof(t40_pre_topc,axiom,
$true ).
fof(t41_pre_topc,axiom,
! [A] :
( l1_pre_topc(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ~ ( B != k1_pre_topc(A)
& ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ~ r2_hidden(C,B) ) ) ) ) ).
fof(t42_pre_topc,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> v4_pre_topc(k2_pre_topc(A),A) ) ).
fof(t43_pre_topc,axiom,
! [A] :
( l1_pre_topc(A)
=> ! [B] :
( m1_pre_topc(B,A)
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(B)))
=> ( v4_pre_topc(C,B)
<=> ? [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
& v4_pre_topc(D,A)
& k3_xboole_0(D,k2_pre_topc(B)) = C ) ) ) ) ) ).
fof(t44_pre_topc,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))))
=> ( ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ( r2_hidden(C,B)
=> v4_pre_topc(C,A) ) )
=> v4_pre_topc(k6_setfam_1(u1_struct_0(A),B),A) ) ) ) ).
fof(d13_pre_topc,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)))
=> ( C = k6_pre_topc(A,B)
<=> ! [D] :
( r2_hidden(D,u1_struct_0(A))
=> ( r2_hidden(D,C)
<=> ! [E] :
( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(A)))
=> ~ ( v3_pre_topc(E,A)
& r2_hidden(D,E)
& r1_xboole_0(B,E) ) ) ) ) ) ) ) ) ).
fof(t45_pre_topc,axiom,
! [A] :
( l1_pre_topc(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( r2_hidden(C,u1_struct_0(A))
=> ( r2_hidden(C,k6_pre_topc(A,B))
<=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
=> ( ( v4_pre_topc(D,A)
& r1_tarski(B,D) )
=> r2_hidden(C,D) ) ) ) ) ) ) ).
fof(t46_pre_topc,axiom,
! [A] :
( ( v2_pre_topc(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(u1_struct_0(A)))
=> ( r2_hidden(D,C)
<=> ( v4_pre_topc(D,A)
& r1_tarski(B,D) ) ) )
& k6_pre_topc(A,B) = k6_setfam_1(u1_struct_0(A),C) ) ) ) ).
fof(t47_pre_topc,axiom,
! [A] :
( l1_pre_topc(A)
=> ! [B] :
( m1_pre_topc(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(B)))
=> ( C = D
=> k6_pre_topc(B,D) = k3_xboole_0(k6_pre_topc(A,C),k2_pre_topc(B)) ) ) ) ) ) ).
fof(t48_pre_topc,axiom,
! [A] :
( l1_pre_topc(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> r1_tarski(B,k6_pre_topc(A,B)) ) ) ).
fof(t49_pre_topc,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)))
=> ( r1_tarski(B,C)
=> r1_tarski(k6_pre_topc(A,B),k6_pre_topc(A,C)) ) ) ) ) ).
fof(t50_pre_topc,axiom,
! [A] :
( ( v2_pre_topc(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)))
=> k6_pre_topc(A,k4_subset_1(u1_struct_0(A),B,C)) = k4_subset_1(u1_struct_0(A),k6_pre_topc(A,B),k6_pre_topc(A,C)) ) ) ) ).
fof(t51_pre_topc,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)))
=> r1_tarski(k6_pre_topc(A,k5_subset_1(u1_struct_0(A),B,C)),k5_subset_1(u1_struct_0(A),k6_pre_topc(A,B),k6_pre_topc(A,C))) ) ) ) ).
fof(t52_pre_topc,axiom,
! [A] :
( l1_pre_topc(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ( ( v4_pre_topc(B,A)
=> k6_pre_topc(A,B) = B )
& ( ( v2_pre_topc(A)
& k6_pre_topc(A,B) = B )
=> v4_pre_topc(B,A) ) ) ) ) ).
fof(t53_pre_topc,axiom,
! [A] :
( l1_pre_topc(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ( ( v3_pre_topc(B,A)
=> k6_pre_topc(A,k6_subset_1(u1_struct_0(A),k2_pre_topc(A),B)) = k6_subset_1(u1_struct_0(A),k2_pre_topc(A),B) )
& ( ( v2_pre_topc(A)
& k6_pre_topc(A,k6_subset_1(u1_struct_0(A),k2_pre_topc(A),B)) = k6_subset_1(u1_struct_0(A),k2_pre_topc(A),B) )
=> v3_pre_topc(B,A) ) ) ) ) ).
fof(t54_pre_topc,axiom,
! [A] :
( l1_pre_topc(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( r2_hidden(C,k6_pre_topc(A,B))
<=> ( ~ v3_struct_0(A)
& ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
=> ~ ( v3_pre_topc(D,A)
& r2_hidden(C,D)
& r1_xboole_0(B,D) ) ) ) ) ) ) ) ).
fof(dt_m1_pre_topc,axiom,
! [A] :
( l1_pre_topc(A)
=> ! [B] :
( m1_pre_topc(B,A)
=> l1_pre_topc(B) ) ) ).
fof(existence_m1_pre_topc,axiom,
! [A] :
( l1_pre_topc(A)
=> ? [B] : m1_pre_topc(B,A) ) ).
fof(dt_l1_pre_topc,axiom,
! [A] :
( l1_pre_topc(A)
=> l1_struct_0(A) ) ).
fof(existence_l1_pre_topc,axiom,
? [A] : l1_pre_topc(A) ).
fof(abstractness_v1_pre_topc,axiom,
! [A] :
( l1_pre_topc(A)
=> ( v1_pre_topc(A)
=> A = g1_pre_topc(u1_struct_0(A),u1_pre_topc(A)) ) ) ).
fof(dt_k1_pre_topc,axiom,
! [A] :
( l1_struct_0(A)
=> m1_subset_1(k1_pre_topc(A),k1_zfmisc_1(u1_struct_0(A))) ) ).
fof(dt_k2_pre_topc,axiom,
! [A] :
( l1_struct_0(A)
=> m1_subset_1(k2_pre_topc(A),k1_zfmisc_1(u1_struct_0(A))) ) ).
fof(dt_k3_pre_topc,axiom,
! [A,B] :
( ( l1_pre_topc(A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ( v1_pre_topc(k3_pre_topc(A,B))
& m1_pre_topc(k3_pre_topc(A,B),A) ) ) ).
fof(dt_k4_pre_topc,axiom,
! [A,B,C,D] :
( ( 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)) )
=> m1_subset_1(k4_pre_topc(A,B,C,D),k1_zfmisc_1(u1_struct_0(B))) ) ).
fof(redefinition_k4_pre_topc,axiom,
! [A,B,C,D] :
( ( 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)) )
=> k4_pre_topc(A,B,C,D) = k9_relat_1(C,D) ) ).
fof(dt_k5_pre_topc,axiom,
! [A,B,C,D] :
( ( 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)) )
=> m1_subset_1(k5_pre_topc(A,B,C,D),k1_zfmisc_1(u1_struct_0(A))) ) ).
fof(redefinition_k5_pre_topc,axiom,
! [A,B,C,D] :
( ( 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)) )
=> k5_pre_topc(A,B,C,D) = k10_relat_1(C,D) ) ).
fof(dt_k6_pre_topc,axiom,
! [A,B] :
( ( l1_pre_topc(A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> m1_subset_1(k6_pre_topc(A,B),k1_zfmisc_1(u1_struct_0(A))) ) ).
fof(dt_u1_pre_topc,axiom,
! [A] :
( l1_pre_topc(A)
=> m1_subset_1(u1_pre_topc(A),k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A)))) ) ).
fof(dt_g1_pre_topc,axiom,
! [A,B] :
( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
=> ( v1_pre_topc(g1_pre_topc(A,B))
& l1_pre_topc(g1_pre_topc(A,B)) ) ) ).
fof(free_g1_pre_topc,axiom,
! [A,B] :
( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
=> ! [C,D] :
( g1_pre_topc(A,B) = g1_pre_topc(C,D)
=> ( A = C
& B = D ) ) ) ).
%------------------------------------------------------------------------------