SET007 Axioms: SET007+565.ax
%------------------------------------------------------------------------------
% File : SET007+565 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : The Definition and Basic Properties of Topological Groups
% 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 : topgrp_1 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 129 ( 1 unt; 0 def)
% Number of atoms : 1078 ( 44 equ)
% Maximal formula atoms : 18 ( 8 avg)
% Number of connectives : 1099 ( 150 ~; 0 |; 661 &)
% ( 17 <=>; 271 =>; 0 <=; 0 <~>)
% Maximal formula depth : 16 ( 9 avg)
% Maximal term depth : 4 ( 1 avg)
% Number of predicates : 43 ( 42 usr; 0 prp; 1-3 aty)
% Number of functors : 41 ( 41 usr; 0 con; 1-5 aty)
% Number of variables : 325 ( 295 !; 30 ?)
% 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_topgrp_1,axiom,
! [A] :
? [B] :
( m1_relset_1(B,A,A)
& v1_relat_1(B)
& v1_funct_1(B)
& v2_funct_1(B)
& v1_funct_2(B,A,A)
& v2_funct_2(B,A,A)
& v3_funct_2(B,A,A) ) ).
fof(fc1_topgrp_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_struct_0(A) )
=> ( v1_relat_1(k2_tops_2(A,A,k7_grcat_1(A)))
& v1_funct_1(k2_tops_2(A,A,k7_grcat_1(A)))
& v2_funct_1(k2_tops_2(A,A,k7_grcat_1(A)))
& ~ v1_xboole_0(k2_tops_2(A,A,k7_grcat_1(A)))
& v1_funct_2(k2_tops_2(A,A,k7_grcat_1(A)),u1_struct_0(A),u1_struct_0(A))
& v1_partfun1(k2_tops_2(A,A,k7_grcat_1(A)),u1_struct_0(A),u1_struct_0(A)) ) ) ).
fof(rc2_topgrp_1,axiom,
! [A] :
( l1_struct_0(A)
=> ? [B] :
( m1_relset_1(B,u1_struct_0(A),u1_struct_0(A))
& v1_relat_1(B)
& v1_funct_1(B)
& v2_funct_1(B)
& v1_funct_2(B,u1_struct_0(A),u1_struct_0(A))
& v2_funct_2(B,u1_struct_0(A),u1_struct_0(A))
& v3_funct_2(B,u1_struct_0(A),u1_struct_0(A)) ) ) ).
fof(fc2_topgrp_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ( v1_relat_1(k4_group_1(A))
& v1_funct_1(k4_group_1(A))
& v2_funct_1(k4_group_1(A))
& ~ v1_xboole_0(k4_group_1(A))
& v1_funct_2(k4_group_1(A),u1_struct_0(A),u1_struct_0(A))
& v2_funct_2(k4_group_1(A),u1_struct_0(A),u1_struct_0(A))
& v3_funct_2(k4_group_1(A),u1_struct_0(A),u1_struct_0(A))
& v1_partfun1(k4_group_1(A),u1_struct_0(A),u1_struct_0(A)) ) ) ).
fof(fc3_topgrp_1,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A)
& m1_subset_1(B,u1_struct_0(A)) )
=> ( v1_relat_1(k1_topgrp_1(A,B))
& v1_funct_1(k1_topgrp_1(A,B))
& v2_funct_1(k1_topgrp_1(A,B))
& ~ v1_xboole_0(k1_topgrp_1(A,B))
& v1_funct_2(k1_topgrp_1(A,B),u1_struct_0(A),u1_struct_0(A))
& v2_funct_2(k1_topgrp_1(A,B),u1_struct_0(A),u1_struct_0(A))
& v3_funct_2(k1_topgrp_1(A,B),u1_struct_0(A),u1_struct_0(A))
& v1_partfun1(k1_topgrp_1(A,B),u1_struct_0(A),u1_struct_0(A)) ) ) ).
fof(fc4_topgrp_1,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A)
& m1_subset_1(B,u1_struct_0(A)) )
=> ( v1_relat_1(k2_topgrp_1(A,B))
& v1_funct_1(k2_topgrp_1(A,B))
& v2_funct_1(k2_topgrp_1(A,B))
& ~ v1_xboole_0(k2_topgrp_1(A,B))
& v1_funct_2(k2_topgrp_1(A,B),u1_struct_0(A),u1_struct_0(A))
& v2_funct_2(k2_topgrp_1(A,B),u1_struct_0(A),u1_struct_0(A))
& v3_funct_2(k2_topgrp_1(A,B),u1_struct_0(A),u1_struct_0(A))
& v1_partfun1(k2_topgrp_1(A,B),u1_struct_0(A),u1_struct_0(A)) ) ) ).
fof(fc5_topgrp_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_pre_topc(A) )
=> ( v1_relat_1(k2_tops_2(A,A,k7_grcat_1(A)))
& v1_funct_1(k2_tops_2(A,A,k7_grcat_1(A)))
& v2_funct_1(k2_tops_2(A,A,k7_grcat_1(A)))
& ~ v1_xboole_0(k2_tops_2(A,A,k7_grcat_1(A)))
& v1_funct_2(k2_tops_2(A,A,k7_grcat_1(A)),u1_struct_0(A),u1_struct_0(A))
& v5_pre_topc(k2_tops_2(A,A,k7_grcat_1(A)),A,A)
& v1_partfun1(k2_tops_2(A,A,k7_grcat_1(A)),u1_struct_0(A),u1_struct_0(A)) ) ) ).
fof(cc1_topgrp_1,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A)
& m1_subset_1(B,u1_struct_0(A)) )
=> ! [C] :
( m1_connsp_2(C,A,B)
=> ~ v1_xboole_0(C) ) ) ).
fof(rc3_topgrp_1,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A)
& m1_subset_1(B,u1_struct_0(A)) )
=> ? [C] :
( m1_connsp_2(C,A,B)
& ~ v1_xboole_0(C)
& v3_pre_topc(C,A) ) ) ).
fof(rc4_topgrp_1,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)
& v1_tops_1(B,A) ) ) ).
fof(cc2_topgrp_1,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& l1_pre_topc(A)
& ~ v3_struct_0(B)
& l1_pre_topc(B) )
=> ! [C] :
( m1_relset_1(C,u1_struct_0(A),u1_struct_0(B))
=> ( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
& v3_tops_2(C,A,B) )
=> ( v1_funct_1(C)
& v2_funct_1(C)
& ~ v1_xboole_0(C)
& v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
& v2_funct_2(C,u1_struct_0(A),u1_struct_0(B))
& v3_funct_2(C,u1_struct_0(A),u1_struct_0(B))
& v5_pre_topc(C,A,B)
& v1_partfun1(C,u1_struct_0(A),u1_struct_0(B))
& v1_t_0topsp(C,A,B) ) ) ) ) ).
fof(rc5_topgrp_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_pre_topc(A) )
=> ? [B] :
( m1_relset_1(B,u1_struct_0(A),u1_struct_0(A))
& v1_relat_1(B)
& v1_funct_1(B)
& v2_funct_1(B)
& ~ v1_xboole_0(B)
& v1_funct_2(B,u1_struct_0(A),u1_struct_0(A))
& v2_funct_2(B,u1_struct_0(A),u1_struct_0(A))
& v3_funct_2(B,u1_struct_0(A),u1_struct_0(A))
& v5_pre_topc(B,A,A)
& v3_tops_2(B,A,A)
& v1_partfun1(B,u1_struct_0(A),u1_struct_0(A))
& v1_t_0topsp(B,A,A) ) ) ).
fof(fc6_topgrp_1,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& l1_pre_topc(A)
& v1_funct_1(B)
& v1_funct_2(B,u1_struct_0(A),u1_struct_0(A))
& v3_tops_2(B,A,A)
& m1_relset_1(B,u1_struct_0(A),u1_struct_0(A)) )
=> ( v1_relat_1(k2_tops_2(A,A,B))
& v1_funct_1(k2_tops_2(A,A,B))
& v2_funct_1(k2_tops_2(A,A,B))
& ~ v1_xboole_0(k2_tops_2(A,A,B))
& v1_funct_2(k2_tops_2(A,A,B),u1_struct_0(A),u1_struct_0(A))
& v2_funct_2(k2_tops_2(A,A,B),u1_struct_0(A),u1_struct_0(A))
& v3_funct_2(k2_tops_2(A,A,B),u1_struct_0(A),u1_struct_0(A))
& v5_pre_topc(k2_tops_2(A,A,B),A,A)
& v3_tops_2(k2_tops_2(A,A,B),A,A)
& v1_partfun1(k2_tops_2(A,A,B),u1_struct_0(A),u1_struct_0(A))
& v1_t_0topsp(k2_tops_2(A,A,B),A,A) ) ) ).
fof(cc3_topgrp_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_topgrp_1(B,A)
=> ( v1_relat_1(B)
& v3_tops_2(B,A,A) ) ) ) ).
fof(fc7_topgrp_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_pre_topc(A) )
=> ( ~ v3_struct_0(k4_topgrp_1(A))
& v1_group_1(k4_topgrp_1(A)) ) ) ).
fof(fc8_topgrp_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_pre_topc(A) )
=> ( ~ v3_struct_0(k4_topgrp_1(A))
& v1_group_1(k4_topgrp_1(A))
& v2_group_1(k4_topgrp_1(A))
& v3_group_1(k4_topgrp_1(A))
& v4_group_1(k4_topgrp_1(A)) ) ) ).
fof(cc4_topgrp_1,axiom,
! [A] :
( l1_pre_topc(A)
=> ( ( ~ v3_struct_0(A)
& v3_realset2(A) )
=> ( ~ v3_struct_0(A)
& v1_topgrp_1(A) ) ) ) ).
fof(rc6_topgrp_1,axiom,
? [A] :
( l1_pre_topc(A)
& ~ v3_struct_0(A)
& v3_realset2(A)
& v1_pre_topc(A)
& v2_pre_topc(A)
& v1_topgrp_1(A) ) ).
fof(rc7_topgrp_1,axiom,
? [A] :
( l1_topgrp_1(A)
& v2_topgrp_1(A) ) ).
fof(fc9_topgrp_1,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& v1_funct_1(B)
& v1_funct_2(B,k2_zfmisc_1(A,A),A)
& m1_relset_1(B,k2_zfmisc_1(A,A),A)
& m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(A))) )
=> ( ~ v3_struct_0(g1_topgrp_1(A,B,C))
& v2_topgrp_1(g1_topgrp_1(A,B,C)) ) ) ).
fof(fc10_topgrp_1,axiom,
! [A,B,C] :
( ( v1_funct_1(B)
& v1_funct_2(B,k2_zfmisc_1(k1_tarski(A),k1_tarski(A)),k1_tarski(A))
& m1_relset_1(B,k2_zfmisc_1(k1_tarski(A),k1_tarski(A)),k1_tarski(A))
& m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(k1_tarski(A)))) )
=> ( ~ v3_struct_0(g1_topgrp_1(k1_tarski(A),B,C))
& v3_realset2(g1_topgrp_1(k1_tarski(A),B,C))
& v1_topgrp_1(g1_topgrp_1(k1_tarski(A),B,C))
& v2_topgrp_1(g1_topgrp_1(k1_tarski(A),B,C)) ) ) ).
fof(cc5_topgrp_1,axiom,
! [A] :
( l1_group_1(A)
=> ( ( ~ v3_struct_0(A)
& v3_realset2(A) )
=> ( ~ v3_struct_0(A)
& v2_group_1(A)
& v3_group_1(A)
& v4_group_1(A)
& v7_group_1(A) ) ) ) ).
fof(fc11_topgrp_1,axiom,
! [A] :
( ~ v3_struct_0(k2_pcomps_1(k1_tarski(A)))
& v3_realset2(k2_pcomps_1(k1_tarski(A)))
& v1_pre_topc(k2_pcomps_1(k1_tarski(A)))
& v2_pre_topc(k2_pcomps_1(k1_tarski(A)))
& v1_topgrp_1(k2_pcomps_1(k1_tarski(A))) ) ).
fof(rc8_topgrp_1,axiom,
? [A] :
( l1_topgrp_1(A)
& ~ v3_struct_0(A)
& v2_topgrp_1(A) ) ).
fof(rc9_topgrp_1,axiom,
? [A] :
( l1_topgrp_1(A)
& ~ v3_struct_0(A)
& v2_group_1(A)
& v3_group_1(A)
& v4_group_1(A)
& v7_group_1(A)
& v3_realset2(A)
& v2_pre_topc(A)
& v1_topgrp_1(A)
& v2_topgrp_1(A) ) ).
fof(rc10_topgrp_1,axiom,
? [A] :
( l1_topgrp_1(A)
& ~ v3_struct_0(A)
& v2_group_1(A)
& v3_group_1(A)
& v4_group_1(A)
& v7_group_1(A)
& v3_realset2(A)
& v2_pre_topc(A)
& v1_topgrp_1(A)
& v2_topgrp_1(A)
& v3_topgrp_1(A)
& v4_topgrp_1(A) ) ).
fof(fc12_topgrp_1,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& v4_topgrp_1(A)
& l1_topgrp_1(A)
& m1_subset_1(B,u1_struct_0(A)) )
=> ( v1_relat_1(k1_topgrp_1(A,B))
& v1_funct_1(k1_topgrp_1(A,B))
& ~ v1_xboole_0(k1_topgrp_1(A,B))
& v1_funct_2(k1_topgrp_1(A,B),u1_struct_0(A),u1_struct_0(A))
& v5_pre_topc(k1_topgrp_1(A,B),A,A)
& v1_partfun1(k1_topgrp_1(A,B),u1_struct_0(A),u1_struct_0(A)) ) ) ).
fof(fc13_topgrp_1,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& v4_topgrp_1(A)
& l1_topgrp_1(A)
& m1_subset_1(B,u1_struct_0(A)) )
=> ( v1_relat_1(k2_topgrp_1(A,B))
& v1_funct_1(k2_topgrp_1(A,B))
& ~ v1_xboole_0(k2_topgrp_1(A,B))
& v1_funct_2(k2_topgrp_1(A,B),u1_struct_0(A),u1_struct_0(A))
& v5_pre_topc(k2_topgrp_1(A,B),A,A)
& v1_partfun1(k2_topgrp_1(A,B),u1_struct_0(A),u1_struct_0(A)) ) ) ).
fof(cc6_topgrp_1,axiom,
! [A] :
( l1_topgrp_1(A)
=> ( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& v2_pre_topc(A)
& v4_topgrp_1(A) )
=> ( ~ v3_struct_0(A)
& v2_group_1(A)
& v3_group_1(A)
& v4_group_1(A)
& v2_pre_topc(A)
& v1_topgrp_1(A) ) ) ) ).
fof(fc14_topgrp_1,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& v2_pre_topc(A)
& v4_topgrp_1(A)
& l1_topgrp_1(A)
& v4_pre_topc(B,A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
& m1_subset_1(C,u1_struct_0(A)) )
=> v4_pre_topc(k4_group_2(A,C,B),A) ) ).
fof(fc15_topgrp_1,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& v2_pre_topc(A)
& v4_topgrp_1(A)
& l1_topgrp_1(A)
& v4_pre_topc(B,A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
& m1_subset_1(C,u1_struct_0(A)) )
=> v4_pre_topc(k3_group_2(A,C,B),A) ) ).
fof(fc16_topgrp_1,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& v2_pre_topc(A)
& v3_topgrp_1(A)
& l1_topgrp_1(A)
& v4_pre_topc(B,A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> v4_pre_topc(k1_group_2(A,B),A) ) ).
fof(fc17_topgrp_1,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& v2_pre_topc(A)
& v4_topgrp_1(A)
& l1_topgrp_1(A)
& v3_pre_topc(B,A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
& m1_subset_1(C,u1_struct_0(A)) )
=> v3_pre_topc(k4_group_2(A,C,B),A) ) ).
fof(fc18_topgrp_1,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& v2_pre_topc(A)
& v4_topgrp_1(A)
& l1_topgrp_1(A)
& v3_pre_topc(B,A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
& m1_subset_1(C,u1_struct_0(A)) )
=> v3_pre_topc(k3_group_2(A,C,B),A) ) ).
fof(fc19_topgrp_1,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& v2_pre_topc(A)
& v3_topgrp_1(A)
& l1_topgrp_1(A)
& v3_pre_topc(B,A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> v3_pre_topc(k1_group_2(A,B),A) ) ).
fof(fc20_topgrp_1,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& v2_pre_topc(A)
& v4_topgrp_1(A)
& l1_topgrp_1(A)
& v3_pre_topc(B,A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
=> v3_pre_topc(k2_group_2(A,B,C),A) ) ).
fof(fc21_topgrp_1,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& v2_pre_topc(A)
& v4_topgrp_1(A)
& l1_topgrp_1(A)
& v3_pre_topc(B,A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
=> v3_pre_topc(k2_group_2(A,C,B),A) ) ).
fof(fc22_topgrp_1,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& v2_pre_topc(A)
& v3_topgrp_1(A)
& l1_topgrp_1(A)
& v1_tops_1(B,A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> v1_tops_1(k1_group_2(A,B),A) ) ).
fof(fc23_topgrp_1,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& v2_pre_topc(A)
& v4_topgrp_1(A)
& l1_topgrp_1(A)
& v1_tops_1(B,A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
& m1_subset_1(C,u1_struct_0(A)) )
=> v1_tops_1(k4_group_2(A,C,B),A) ) ).
fof(fc24_topgrp_1,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& v2_pre_topc(A)
& v4_topgrp_1(A)
& l1_topgrp_1(A)
& v1_tops_1(B,A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
& m1_subset_1(C,u1_struct_0(A)) )
=> v1_tops_1(k3_group_2(A,C,B),A) ) ).
fof(cc7_topgrp_1,axiom,
! [A] :
( l1_topgrp_1(A)
=> ( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& v2_pre_topc(A)
& v3_topgrp_1(A)
& v4_topgrp_1(A) )
=> ( ~ v3_struct_0(A)
& v2_group_1(A)
& v3_group_1(A)
& v4_group_1(A)
& v2_pre_topc(A)
& v4_compts_1(A)
& v1_topgrp_1(A)
& v3_topgrp_1(A)
& v4_topgrp_1(A) ) ) ) ).
fof(t1_topgrp_1,axiom,
! [A] :
( l1_struct_0(A)
=> k2_relat_1(k7_grcat_1(A)) = k2_pre_topc(A) ) ).
fof(t2_topgrp_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_struct_0(A) )
=> k2_tops_2(A,A,k7_grcat_1(A)) = k7_grcat_1(A) ) ).
fof(t3_topgrp_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_struct_0(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> k5_pre_topc(A,A,k7_grcat_1(A),B) = B ) ) ).
fof(t4_topgrp_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_group_1(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)))
=> ( ( r1_tarski(B,C)
& r1_tarski(D,E) )
=> r1_tarski(k2_group_2(A,B,D),k2_group_2(A,C,E)) ) ) ) ) ) ) ).
fof(t5_topgrp_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_group_1(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,u1_struct_0(A))
=> ( r1_tarski(B,C)
=> r1_tarski(k4_group_2(A,D,B),k4_group_2(A,D,C)) ) ) ) ) ) ).
fof(t6_topgrp_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_group_1(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,u1_struct_0(A))
=> ( r1_tarski(B,C)
=> r1_tarski(k3_group_2(A,D,B),k3_group_2(A,D,C)) ) ) ) ) ) ).
fof(t7_topgrp_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(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,k1_group_2(A,B))
<=> r2_hidden(k3_group_1(A,C),B) ) ) ) ) ).
fof(t8_topgrp_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> k1_group_2(A,k1_group_2(A,B)) = B ) ) ).
fof(t9_topgrp_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(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(k1_group_2(A,B),k1_group_2(A,C)) ) ) ) ) ).
fof(t10_topgrp_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> k4_pre_topc(A,A,k4_group_1(A),B) = k1_group_2(A,B) ) ) ).
fof(t11_topgrp_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> k5_pre_topc(A,A,k4_group_1(A),B) = k1_group_2(A,B) ) ) ).
fof(t12_topgrp_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> v2_funct_1(k4_group_1(A)) ) ).
fof(t13_topgrp_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> k2_relat_1(k4_group_1(A)) = u1_struct_0(A) ) ).
fof(t14_topgrp_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> k6_funct_2(u1_struct_0(A),k4_group_1(A)) = k4_group_1(A) ) ).
fof(t15_topgrp_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_group_1(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_funct_2(k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),u1_struct_0(A),u1_group_1(A),k2_zfmisc_1(B,C)) = k2_group_2(A,B,C) ) ) ) ).
fof(d1_topgrp_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_group_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),u1_struct_0(A))
& m2_relset_1(C,u1_struct_0(A),u1_struct_0(A)) )
=> ( C = k1_topgrp_1(A,B)
<=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> k8_funct_2(u1_struct_0(A),u1_struct_0(A),C,D) = k1_group_1(A,B,D) ) ) ) ) ) ).
fof(d2_topgrp_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_group_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),u1_struct_0(A))
& m2_relset_1(C,u1_struct_0(A),u1_struct_0(A)) )
=> ( C = k2_topgrp_1(A,B)
<=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> k8_funct_2(u1_struct_0(A),u1_struct_0(A),C,D) = k1_group_1(A,D,B) ) ) ) ) ) ).
fof(t16_topgrp_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_group_1(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> k4_pre_topc(A,A,k1_topgrp_1(A,C),B) = k3_group_2(A,C,B) ) ) ) ).
fof(t17_topgrp_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_group_1(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> k4_pre_topc(A,A,k2_topgrp_1(A,C),B) = k4_group_2(A,C,B) ) ) ) ).
fof(t18_topgrp_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> k2_tops_2(A,A,k1_topgrp_1(A,B)) = k1_topgrp_1(A,k3_group_1(A,B)) ) ) ).
fof(t19_topgrp_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> k2_tops_2(A,A,k2_topgrp_1(A,B)) = k2_topgrp_1(A,k3_group_1(A,B)) ) ) ).
fof(t20_topgrp_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_pre_topc(A) )
=> v3_tops_2(k7_grcat_1(A),A,A) ) ).
fof(t21_topgrp_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> m1_connsp_2(k2_pre_topc(A),A,B) ) ) ).
fof(t22_topgrp_1,axiom,
! [A] :
( ( ~ v3_struct_0(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)) )
=> ( v1_t_0topsp(C,A,B)
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ! [E] :
( m1_connsp_2(E,A,D)
=> ? [F] :
( v3_pre_topc(F,B)
& m1_connsp_2(F,B,k8_funct_2(u1_struct_0(A),u1_struct_0(B),C,D))
& r1_tarski(F,k4_pre_topc(A,B,C,E)) ) ) ) ) ) ) ) ).
fof(t23_topgrp_1,axiom,
! [A] :
( ( ~ v3_struct_0(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)) )
=> ( ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ! [E] :
( ( v3_pre_topc(E,A)
& m1_connsp_2(E,A,D) )
=> ? [F] :
( m1_connsp_2(F,B,k8_funct_2(u1_struct_0(A),u1_struct_0(B),C,D))
& r1_tarski(F,k4_pre_topc(A,B,C,E)) ) ) )
=> v1_t_0topsp(C,A,B) ) ) ) ) ).
fof(t24_topgrp_1,axiom,
! [A] :
( ( ~ v3_struct_0(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(B)))
=> ( v4_pre_topc(D,B)
<=> v4_pre_topc(k5_pre_topc(A,B,C,D),A) ) ) ) ) ) ) ) ).
fof(t25_topgrp_1,axiom,
! [A] :
( ( ~ v3_struct_0(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)))
=> ( v3_pre_topc(D,A)
<=> v3_pre_topc(k4_pre_topc(A,B,C,D),B) ) ) ) ) ) ) ) ).
fof(t26_topgrp_1,axiom,
! [A] :
( ( ~ v3_struct_0(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(B)))
=> ( v3_pre_topc(D,B)
<=> v3_pre_topc(k5_pre_topc(A,B,C,D),A) ) ) ) ) ) ) ) ).
fof(t27_topgrp_1,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(B)))
=> r1_tarski(k5_pre_topc(A,B,C,k1_tops_1(B,D)),k1_tops_1(A,k5_pre_topc(A,B,C,D))) ) ) ) ) ) ).
fof(t28_topgrp_1,axiom,
! [A] :
( ( ~ v3_struct_0(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)) )
=> ! [D] :
( ( v1_tops_1(D,A)
& m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A))) )
=> ( v3_tops_2(C,A,B)
=> v1_tops_1(k4_pre_topc(A,B,C,D),B) ) ) ) ) ) ).
fof(t29_topgrp_1,axiom,
! [A] :
( ( ~ v3_struct_0(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)) )
=> ! [D] :
( ( v1_tops_1(D,B)
& m1_subset_1(D,k1_zfmisc_1(u1_struct_0(B))) )
=> ( v3_tops_2(C,A,B)
=> v1_tops_1(k5_pre_topc(A,B,C,D),A) ) ) ) ) ) ).
fof(d3_topgrp_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,u1_struct_0(A),u1_struct_0(A))
& m2_relset_1(B,u1_struct_0(A),u1_struct_0(A)) )
=> ( m1_topgrp_1(B,A)
<=> v3_tops_2(B,A,A) ) ) ) ).
fof(t30_topgrp_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_topgrp_1(B,A)
=> m1_topgrp_1(k2_tops_2(A,A,B),A) ) ) ).
fof(t31_topgrp_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_topgrp_1(B,A)
=> ! [C] :
( m1_topgrp_1(C,A)
=> m1_topgrp_1(k7_funct_2(u1_struct_0(A),u1_struct_0(A),u1_struct_0(A),C,B),A) ) ) ) ).
fof(d4_topgrp_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( v1_group_1(B)
& l1_group_1(B) )
=> ( B = k4_topgrp_1(A)
<=> ! [C] :
( ( r2_hidden(C,u1_struct_0(B))
=> m1_topgrp_1(C,A) )
& ( m1_topgrp_1(C,A)
=> r2_hidden(C,u1_struct_0(B)) )
& ! [D] :
( m1_topgrp_1(D,A)
=> ! [E] :
( m1_topgrp_1(E,A)
=> k1_binop_1(u1_group_1(B),D,E) = k7_funct_2(u1_struct_0(A),u1_struct_0(A),u1_struct_0(A),D,E) ) ) ) ) ) ) ).
fof(t32_topgrp_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_topgrp_1(B,A)
=> ! [C] :
( m1_topgrp_1(C,A)
=> ! [D] :
( m1_subset_1(D,u1_struct_0(k4_topgrp_1(A)))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(k4_topgrp_1(A)))
=> ( ( B = D
& C = E )
=> k1_group_1(k4_topgrp_1(A),D,E) = k7_funct_2(u1_struct_0(A),u1_struct_0(A),u1_struct_0(A),B,C) ) ) ) ) ) ) ).
fof(t33_topgrp_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_pre_topc(A) )
=> k3_topgrp_1(A) = k2_group_1(k4_topgrp_1(A)) ) ).
fof(t34_topgrp_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_topgrp_1(B,A)
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k4_topgrp_1(A)))
=> ( B = C
=> k3_group_1(k4_topgrp_1(A),C) = k2_tops_2(A,A,B) ) ) ) ) ).
fof(d5_topgrp_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_pre_topc(A) )
=> ( v1_topgrp_1(A)
<=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ? [D] :
( m1_topgrp_1(D,A)
& k8_funct_2(u1_struct_0(A),u1_struct_0(A),D,B) = C ) ) ) ) ) ).
fof(t35_topgrp_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& v1_topgrp_1(A)
& l1_pre_topc(A) )
=> ( ? [B] :
( m1_subset_1(B,u1_struct_0(A))
& v4_pre_topc(k1_struct_0(A,B),A) )
=> v1_urysohn1(A) ) ) ).
fof(t36_topgrp_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& v1_topgrp_1(A)
& l1_pre_topc(A) )
=> ( ? [B] :
( m1_subset_1(B,u1_struct_0(A))
& ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ~ ( v3_pre_topc(C,A)
& r2_hidden(B,C)
& ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
=> ~ ( r2_hidden(B,D)
& v3_pre_topc(D,A)
& r1_tarski(k6_pre_topc(A,D),C) ) ) ) ) )
=> v4_compts_1(A) ) ) ).
fof(d6_topgrp_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_topgrp_1(A) )
=> ( v3_topgrp_1(A)
<=> v5_pre_topc(k5_topgrp_1(A),A,A) ) ) ).
fof(d7_topgrp_1,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_topgrp_1(A) )
=> ( v4_topgrp_1(A)
<=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,u1_struct_0(k6_borsuk_1(A,A)),u1_struct_0(A))
& m2_relset_1(B,u1_struct_0(k6_borsuk_1(A,A)),u1_struct_0(A)) )
=> ( B = u1_group_1(A)
=> v5_pre_topc(B,k6_borsuk_1(A,A),A) ) ) ) ) ).
fof(t37_topgrp_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& v4_topgrp_1(A)
& l1_topgrp_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_connsp_2(D,A,k1_group_1(A,B,C))
=> ? [E] :
( v3_pre_topc(E,A)
& m1_connsp_2(E,A,B)
& ? [F] :
( v3_pre_topc(F,A)
& m1_connsp_2(F,A,C)
& r1_tarski(k2_group_2(A,E,F),D) ) ) ) ) ) ) ).
fof(t38_topgrp_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_topgrp_1(A) )
=> ( ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_connsp_2(D,A,k1_group_1(A,B,C))
=> ? [E] :
( m1_connsp_2(E,A,B)
& ? [F] :
( m1_connsp_2(F,A,C)
& r1_tarski(k2_group_2(A,E,F),D) ) ) ) ) )
=> v4_topgrp_1(A) ) ) ).
fof(t39_topgrp_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& v2_pre_topc(A)
& v3_topgrp_1(A)
& l1_topgrp_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_connsp_2(C,A,k3_group_1(A,B))
=> ? [D] :
( v3_pre_topc(D,A)
& m1_connsp_2(D,A,B)
& r1_tarski(k1_group_2(A,D),C) ) ) ) ) ).
fof(t40_topgrp_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& v2_pre_topc(A)
& l1_topgrp_1(A) )
=> ( ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_connsp_2(C,A,k3_group_1(A,B))
=> ? [D] :
( m1_connsp_2(D,A,B)
& r1_tarski(k1_group_2(A,D),C) ) ) )
=> v3_topgrp_1(A) ) ) ).
fof(t41_topgrp_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& v2_pre_topc(A)
& v3_topgrp_1(A)
& v4_topgrp_1(A)
& l1_topgrp_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_connsp_2(D,A,k1_group_1(A,B,k3_group_1(A,C)))
=> ? [E] :
( v3_pre_topc(E,A)
& m1_connsp_2(E,A,B)
& ? [F] :
( v3_pre_topc(F,A)
& m1_connsp_2(F,A,C)
& r1_tarski(k2_group_2(A,E,k1_group_2(A,F)),D) ) ) ) ) ) ) ).
fof(t42_topgrp_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& v2_pre_topc(A)
& l1_topgrp_1(A) )
=> ( ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_connsp_2(D,A,k1_group_1(A,B,k3_group_1(A,C)))
=> ? [E] :
( m1_connsp_2(E,A,B)
& ? [F] :
( m1_connsp_2(F,A,C)
& r1_tarski(k2_group_2(A,E,k1_group_2(A,F)),D) ) ) ) ) )
=> ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& v2_pre_topc(A)
& v3_topgrp_1(A)
& v4_topgrp_1(A)
& l1_topgrp_1(A) ) ) ) ).
fof(t43_topgrp_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& v2_pre_topc(A)
& v4_topgrp_1(A)
& l1_topgrp_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> m1_topgrp_1(k1_topgrp_1(A,B),A) ) ) ).
fof(t44_topgrp_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& v2_pre_topc(A)
& v4_topgrp_1(A)
& l1_topgrp_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> m1_topgrp_1(k2_topgrp_1(A,B),A) ) ) ).
fof(t45_topgrp_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& v2_pre_topc(A)
& v3_topgrp_1(A)
& l1_topgrp_1(A) )
=> m1_topgrp_1(k5_topgrp_1(A),A) ) ).
fof(t46_topgrp_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& v2_pre_topc(A)
& v4_topgrp_1(A)
& l1_topgrp_1(A) )
=> ! [B] :
( ( v4_pre_topc(B,A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> v4_pre_topc(k4_group_2(A,C,B),A) ) ) ) ).
fof(t47_topgrp_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& v2_pre_topc(A)
& v4_topgrp_1(A)
& l1_topgrp_1(A) )
=> ! [B] :
( ( v4_pre_topc(B,A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> v4_pre_topc(k3_group_2(A,C,B),A) ) ) ) ).
fof(t48_topgrp_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& v2_pre_topc(A)
& v3_topgrp_1(A)
& l1_topgrp_1(A) )
=> ! [B] :
( ( v4_pre_topc(B,A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> v4_pre_topc(k1_group_2(A,B),A) ) ) ).
fof(t49_topgrp_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& v2_pre_topc(A)
& v4_topgrp_1(A)
& l1_topgrp_1(A) )
=> ! [B] :
( ( v3_pre_topc(B,A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> v3_pre_topc(k4_group_2(A,C,B),A) ) ) ) ).
fof(t50_topgrp_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& v2_pre_topc(A)
& v4_topgrp_1(A)
& l1_topgrp_1(A) )
=> ! [B] :
( ( v3_pre_topc(B,A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> v3_pre_topc(k3_group_2(A,C,B),A) ) ) ) ).
fof(t51_topgrp_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& v2_pre_topc(A)
& v3_topgrp_1(A)
& l1_topgrp_1(A) )
=> ! [B] :
( ( v3_pre_topc(B,A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> v3_pre_topc(k1_group_2(A,B),A) ) ) ).
fof(t52_topgrp_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& v2_pre_topc(A)
& v4_topgrp_1(A)
& l1_topgrp_1(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)))
=> ( v3_pre_topc(C,A)
=> v3_pre_topc(k2_group_2(A,C,B),A) ) ) ) ) ).
fof(t53_topgrp_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& v2_pre_topc(A)
& v4_topgrp_1(A)
& l1_topgrp_1(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)))
=> ( v3_pre_topc(C,A)
=> v3_pre_topc(k2_group_2(A,B,C),A) ) ) ) ) ).
fof(t54_topgrp_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& v2_pre_topc(A)
& v3_topgrp_1(A)
& l1_topgrp_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_connsp_2(C,A,B)
=> m1_connsp_2(k1_group_2(A,C),A,k3_group_1(A,B)) ) ) ) ).
fof(t55_topgrp_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& v2_pre_topc(A)
& v3_topgrp_1(A)
& v4_topgrp_1(A)
& l1_topgrp_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_connsp_2(C,A,k1_group_1(A,B,k3_group_1(A,B)))
=> ? [D] :
( v3_pre_topc(D,A)
& m1_connsp_2(D,A,B)
& r1_tarski(k2_group_2(A,D,k1_group_2(A,D)),C) ) ) ) ) ).
fof(t56_topgrp_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& v2_pre_topc(A)
& v3_topgrp_1(A)
& l1_topgrp_1(A) )
=> ! [B] :
( ( v1_tops_1(B,A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> v1_tops_1(k1_group_2(A,B),A) ) ) ).
fof(t57_topgrp_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& v2_pre_topc(A)
& v4_topgrp_1(A)
& l1_topgrp_1(A) )
=> ! [B] :
( ( v1_tops_1(B,A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> v1_tops_1(k3_group_2(A,C,B),A) ) ) ) ).
fof(t58_topgrp_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& v2_pre_topc(A)
& v4_topgrp_1(A)
& l1_topgrp_1(A) )
=> ! [B] :
( ( v1_tops_1(B,A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> v1_tops_1(k4_group_2(A,C,B),A) ) ) ) ).
fof(t60_topgrp_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& v2_pre_topc(A)
& v3_topgrp_1(A)
& v4_topgrp_1(A)
& l1_topgrp_1(A) )
=> v4_compts_1(A) ) ).
fof(dt_m1_topgrp_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_topgrp_1(B,A)
=> ( v1_funct_1(B)
& v1_funct_2(B,u1_struct_0(A),u1_struct_0(A))
& m2_relset_1(B,u1_struct_0(A),u1_struct_0(A)) ) ) ) ).
fof(existence_m1_topgrp_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_pre_topc(A) )
=> ? [B] : m1_topgrp_1(B,A) ) ).
fof(dt_l1_topgrp_1,axiom,
! [A] :
( l1_topgrp_1(A)
=> ( l1_group_1(A)
& l1_pre_topc(A) ) ) ).
fof(existence_l1_topgrp_1,axiom,
? [A] : l1_topgrp_1(A) ).
fof(abstractness_v2_topgrp_1,axiom,
! [A] :
( l1_topgrp_1(A)
=> ( v2_topgrp_1(A)
=> A = g1_topgrp_1(u1_struct_0(A),u1_group_1(A),u1_pre_topc(A)) ) ) ).
fof(dt_k1_topgrp_1,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& l1_group_1(A)
& m1_subset_1(B,u1_struct_0(A)) )
=> ( v1_funct_1(k1_topgrp_1(A,B))
& v1_funct_2(k1_topgrp_1(A,B),u1_struct_0(A),u1_struct_0(A))
& m2_relset_1(k1_topgrp_1(A,B),u1_struct_0(A),u1_struct_0(A)) ) ) ).
fof(dt_k2_topgrp_1,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& l1_group_1(A)
& m1_subset_1(B,u1_struct_0(A)) )
=> ( v1_funct_1(k2_topgrp_1(A,B))
& v1_funct_2(k2_topgrp_1(A,B),u1_struct_0(A),u1_struct_0(A))
& m2_relset_1(k2_topgrp_1(A,B),u1_struct_0(A),u1_struct_0(A)) ) ) ).
fof(dt_k3_topgrp_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_pre_topc(A) )
=> m1_topgrp_1(k3_topgrp_1(A),A) ) ).
fof(redefinition_k3_topgrp_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_pre_topc(A) )
=> k3_topgrp_1(A) = k7_grcat_1(A) ) ).
fof(dt_k4_topgrp_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_pre_topc(A) )
=> ( v1_group_1(k4_topgrp_1(A))
& l1_group_1(k4_topgrp_1(A)) ) ) ).
fof(dt_k5_topgrp_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_topgrp_1(A) )
=> ( v1_funct_1(k5_topgrp_1(A))
& v1_funct_2(k5_topgrp_1(A),u1_struct_0(A),u1_struct_0(A))
& m2_relset_1(k5_topgrp_1(A),u1_struct_0(A),u1_struct_0(A)) ) ) ).
fof(redefinition_k5_topgrp_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_topgrp_1(A) )
=> k5_topgrp_1(A) = k4_group_1(A) ) ).
fof(dt_k6_topgrp_1,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& v2_pre_topc(A)
& v4_topgrp_1(A)
& l1_topgrp_1(A)
& m1_subset_1(B,u1_struct_0(A)) )
=> m1_topgrp_1(k6_topgrp_1(A,B),A) ) ).
fof(redefinition_k6_topgrp_1,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& v2_pre_topc(A)
& v4_topgrp_1(A)
& l1_topgrp_1(A)
& m1_subset_1(B,u1_struct_0(A)) )
=> k6_topgrp_1(A,B) = k1_topgrp_1(A,B) ) ).
fof(dt_k7_topgrp_1,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& v2_pre_topc(A)
& v4_topgrp_1(A)
& l1_topgrp_1(A)
& m1_subset_1(B,u1_struct_0(A)) )
=> m1_topgrp_1(k7_topgrp_1(A,B),A) ) ).
fof(redefinition_k7_topgrp_1,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& v2_pre_topc(A)
& v4_topgrp_1(A)
& l1_topgrp_1(A)
& m1_subset_1(B,u1_struct_0(A)) )
=> k7_topgrp_1(A,B) = k2_topgrp_1(A,B) ) ).
fof(dt_k8_topgrp_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& v2_pre_topc(A)
& v3_topgrp_1(A)
& l1_topgrp_1(A) )
=> m1_topgrp_1(k8_topgrp_1(A),A) ) ).
fof(redefinition_k8_topgrp_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& v2_pre_topc(A)
& v3_topgrp_1(A)
& l1_topgrp_1(A) )
=> k8_topgrp_1(A) = k4_group_1(A) ) ).
fof(dt_g1_topgrp_1,axiom,
! [A,B,C] :
( ( v1_funct_1(B)
& v1_funct_2(B,k2_zfmisc_1(A,A),A)
& m1_relset_1(B,k2_zfmisc_1(A,A),A)
& m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(A))) )
=> ( v2_topgrp_1(g1_topgrp_1(A,B,C))
& l1_topgrp_1(g1_topgrp_1(A,B,C)) ) ) ).
fof(free_g1_topgrp_1,axiom,
! [A,B,C] :
( ( v1_funct_1(B)
& v1_funct_2(B,k2_zfmisc_1(A,A),A)
& m1_relset_1(B,k2_zfmisc_1(A,A),A)
& m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(A))) )
=> ! [D,E,F] :
( g1_topgrp_1(A,B,C) = g1_topgrp_1(D,E,F)
=> ( A = D
& B = E
& C = F ) ) ) ).
fof(t59_topgrp_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& v2_pre_topc(A)
& v3_topgrp_1(A)
& v4_topgrp_1(A)
& l1_topgrp_1(A) )
=> ! [B] :
( m1_yellow_8(B,A,k2_group_1(A))
=> ! [C] :
( ( v1_tops_1(C,A)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
=> m1_cantor_1(a_3_0_topgrp_1(A,B,C),A) ) ) ) ).
fof(fraenkel_a_3_0_topgrp_1,axiom,
! [A,B,C,D] :
( ( ~ v3_struct_0(B)
& v3_group_1(B)
& v4_group_1(B)
& v2_pre_topc(B)
& v3_topgrp_1(B)
& v4_topgrp_1(B)
& l1_topgrp_1(B)
& m1_yellow_8(C,B,k2_group_1(B))
& v1_tops_1(D,B)
& m1_subset_1(D,k1_zfmisc_1(u1_struct_0(B))) )
=> ( r2_hidden(A,a_3_0_topgrp_1(B,C,D))
<=> ? [E,F] :
( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(B)))
& m1_subset_1(F,u1_struct_0(B))
& A = k4_group_2(B,F,E)
& r2_hidden(E,C)
& r2_hidden(F,D) ) ) ) ).
%------------------------------------------------------------------------------