SET007 Axioms: SET007+309.ax
%------------------------------------------------------------------------------
% File : SET007+309 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : A Borsuk Theorem on Homotopy Types
% 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 : borsuk_1 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 159 ( 19 unt; 0 def)
% Number of atoms : 1200 ( 101 equ)
% Maximal formula atoms : 21 ( 7 avg)
% Number of connectives : 1271 ( 230 ~; 4 |; 613 &)
% ( 20 <=>; 404 =>; 0 <=; 0 <~>)
% Maximal formula depth : 23 ( 9 avg)
% Maximal term depth : 7 ( 1 avg)
% Number of predicates : 39 ( 37 usr; 1 prp; 0-3 aty)
% Number of functors : 77 ( 77 usr; 7 con; 0-6 aty)
% Number of variables : 520 ( 489 !; 31 ?)
% 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_borsuk_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ? [B] :
( m1_eqrel_1(B,A)
& ~ v1_xboole_0(B)
& v1_setfam_1(B) ) ) ).
fof(fc1_borsuk_1,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A)
& ~ v3_struct_0(B)
& v2_pre_topc(B)
& l1_pre_topc(B)
& m1_subset_1(C,u1_struct_0(B)) )
=> ( ~ v1_xboole_0(k3_borsuk_1(A,B,C))
& v1_relat_1(k3_borsuk_1(A,B,C))
& v1_funct_1(k3_borsuk_1(A,B,C))
& v1_funct_2(k3_borsuk_1(A,B,C),u1_struct_0(A),u1_struct_0(B))
& v5_pre_topc(k3_borsuk_1(A,B,C),A,B)
& v1_partfun1(k3_borsuk_1(A,B,C),u1_struct_0(A),u1_struct_0(B)) ) ) ).
fof(rc2_borsuk_1,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A)
& ~ v3_struct_0(B)
& v2_pre_topc(B)
& l1_pre_topc(B) )
=> ? [C] :
( m1_relset_1(C,u1_struct_0(A),u1_struct_0(B))
& ~ v1_xboole_0(C)
& v1_relat_1(C)
& v1_funct_1(C)
& v1_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)) ) ) ).
fof(fc2_borsuk_1,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A)
& ~ v3_struct_0(B)
& v2_pre_topc(B)
& l1_pre_topc(B) )
=> ( ~ v3_struct_0(k6_borsuk_1(A,B))
& v1_pre_topc(k6_borsuk_1(A,B))
& v2_pre_topc(k6_borsuk_1(A,B)) ) ) ).
fof(fc3_borsuk_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_struct_0(A) )
=> ( ~ v1_xboole_0(k15_borsuk_1(A))
& v1_setfam_1(k15_borsuk_1(A)) ) ) ).
fof(fc4_borsuk_1,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A)
& ~ v1_xboole_0(B)
& m1_eqrel_1(B,u1_struct_0(A)) )
=> ( ~ v3_struct_0(k16_borsuk_1(A,B))
& v1_pre_topc(k16_borsuk_1(A,B))
& v2_pre_topc(k16_borsuk_1(A,B)) ) ) ).
fof(rc3_borsuk_1,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> ? [B] :
( m1_pre_topc(B,A)
& v1_pre_topc(B)
& v2_pre_topc(B)
& v1_borsuk_1(B,A) ) ) ).
fof(rc4_borsuk_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ? [B] :
( m1_pre_topc(B,A)
& ~ v3_struct_0(B)
& v1_pre_topc(B)
& v2_pre_topc(B)
& v1_borsuk_1(B,A) ) ) ).
fof(rc5_borsuk_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ? [B] :
( m1_borsuk_1(B,A)
& ~ v1_xboole_0(B)
& v1_setfam_1(B)
& v2_borsuk_1(B,A) ) ) ).
fof(fc5_borsuk_1,axiom,
( ~ v3_struct_0(k22_borsuk_1)
& v1_pre_topc(k22_borsuk_1)
& v2_pre_topc(k22_borsuk_1) ) ).
fof(t1_borsuk_1,axiom,
$true ).
fof(t2_borsuk_1,axiom,
$true ).
fof(t3_borsuk_1,axiom,
$true ).
fof(t4_borsuk_1,axiom,
! [A,B] :
( m1_subset_1(B,k1_zfmisc_1(A))
=> k3_funct_2(A,A,k6_partfun1(A),B) = B ) ).
fof(t5_borsuk_1,axiom,
! [A,B,C] :
( ( v1_relat_1(C)
& v1_funct_1(C) )
=> ( r1_tarski(A,k10_relat_1(C,B))
=> r1_tarski(k9_relat_1(C,A),B) ) ) ).
fof(t6_borsuk_1,axiom,
! [A,B,C] : r1_tarski(k9_relat_1(k2_funcop_1(A,B),C),k1_tarski(B)) ).
fof(t7_borsuk_1,axiom,
$true ).
fof(t8_borsuk_1,axiom,
$true ).
fof(t9_borsuk_1,axiom,
! [A,B,C] :
( r1_tarski(A,k2_zfmisc_1(B,C))
=> k1_funct_1(k1_funct_3(k9_funct_3(B,C)),A) = k2_funct_2(k2_zfmisc_1(B,C),B,k9_funct_3(B,C),A) ) ).
fof(t10_borsuk_1,axiom,
! [A,B,C] :
( r1_tarski(A,k2_zfmisc_1(B,C))
=> k1_funct_1(k1_funct_3(k10_funct_3(B,C)),A) = k2_funct_2(k2_zfmisc_1(B,C),C,k10_funct_3(B,C),A) ) ).
fof(t11_borsuk_1,axiom,
$true ).
fof(t12_borsuk_1,axiom,
! [A,B,C] :
( m1_subset_1(C,k1_zfmisc_1(A))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(B))
=> ( k12_mcart_1(A,B,C,D) != k1_xboole_0
=> ( k2_funct_2(k2_zfmisc_1(A,B),A,k9_funct_3(A,B),k12_mcart_1(A,B,C,D)) = C
& k2_funct_2(k2_zfmisc_1(A,B),B,k10_funct_3(A,B),k12_mcart_1(A,B,C,D)) = D ) ) ) ) ).
fof(t13_borsuk_1,axiom,
! [A,B,C] :
( m1_subset_1(C,k1_zfmisc_1(A))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(B))
=> ( k12_mcart_1(A,B,C,D) != k1_xboole_0
=> ( k1_funct_1(k1_funct_3(k9_funct_3(A,B)),k12_mcart_1(A,B,C,D)) = C
& k1_funct_1(k1_funct_3(k10_funct_3(A,B)),k12_mcart_1(A,B,C,D)) = D ) ) ) ) ).
fof(t14_borsuk_1,axiom,
! [A,B,C] :
( m1_subset_1(C,k1_zfmisc_1(k2_zfmisc_1(A,B)))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(k1_zfmisc_1(k2_zfmisc_1(A,B))))
=> ( ! [E] :
( r2_hidden(E,D)
=> ( r1_tarski(E,C)
& ? [F] :
( m1_subset_1(F,k1_zfmisc_1(A))
& ? [G] :
( m1_subset_1(G,k1_zfmisc_1(B))
& E = k12_mcart_1(A,B,F,G) ) ) ) )
=> r1_tarski(k2_zfmisc_1(k3_tarski(k9_relat_1(k1_funct_3(k9_funct_3(A,B)),D)),k1_setfam_1(k9_relat_1(k1_funct_3(k10_funct_3(A,B)),D))),C) ) ) ) ).
fof(t15_borsuk_1,axiom,
! [A,B,C] :
( m1_subset_1(C,k1_zfmisc_1(k2_zfmisc_1(A,B)))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(k1_zfmisc_1(k2_zfmisc_1(A,B))))
=> ( ! [E] :
( r2_hidden(E,D)
=> ( r1_tarski(E,C)
& ? [F] :
( m1_subset_1(F,k1_zfmisc_1(A))
& ? [G] :
( m1_subset_1(G,k1_zfmisc_1(B))
& E = k12_mcart_1(A,B,F,G) ) ) ) )
=> r1_tarski(k2_zfmisc_1(k1_setfam_1(k9_relat_1(k1_funct_3(k9_funct_3(A,B)),D)),k3_tarski(k9_relat_1(k1_funct_3(k10_funct_3(A,B)),D))),C) ) ) ) ).
fof(t16_borsuk_1,axiom,
! [A,B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,A,B)
& m2_relset_1(C,A,B) )
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(k1_zfmisc_1(A)))
=> k5_setfam_1(B,k2_funct_2(k1_zfmisc_1(A),k1_zfmisc_1(B),k2_funct_3(A,B,C),D)) = k2_funct_2(A,B,C,k5_setfam_1(A,D)) ) ) ) ).
fof(t18_borsuk_1,axiom,
! [A,B] :
( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
=> ( k5_setfam_1(A,B) = A
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(B))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(A))
=> ( D = k3_tarski(C)
=> r1_tarski(k3_subset_1(A,D),k3_tarski(k3_subset_1(B,C))) ) ) ) ) ) ).
fof(t19_borsuk_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( ~ v1_xboole_0(C)
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,A,B)
& m2_relset_1(D,A,B) )
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,A,C)
& m2_relset_1(E,A,C) )
=> ~ ( ! [F] :
( m1_subset_1(F,A)
=> ! [G] :
( m1_subset_1(G,A)
=> ( k8_funct_2(A,B,D,F) = k8_funct_2(A,B,D,G)
=> k8_funct_2(A,C,E,F) = k8_funct_2(A,C,E,G) ) ) )
& ! [F] :
( ( v1_funct_1(F)
& v1_funct_2(F,B,C)
& m2_relset_1(F,B,C) )
=> k7_funct_2(A,B,C,D,F) != E ) ) ) ) ) ) ) ).
fof(t20_borsuk_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( ~ v1_xboole_0(C)
=> ! [D] :
( m1_subset_1(D,B)
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,A,B)
& m2_relset_1(E,A,B) )
=> ! [F] :
( ( v1_funct_1(F)
& v1_funct_2(F,B,C)
& m2_relset_1(F,B,C) )
=> r1_tarski(k3_funct_2(A,B,E,k6_domain_1(B,D)),k3_funct_2(A,C,k7_funct_2(A,B,C,E,F),k6_domain_1(C,k8_funct_2(B,C,F,D)))) ) ) ) ) ) ) ).
fof(t21_borsuk_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( ~ v1_xboole_0(C)
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,A,B)
& m2_relset_1(D,A,B) )
=> ! [E] :
( m1_subset_1(E,A)
=> ! [F] :
( m1_subset_1(F,C)
=> k8_funct_2(k2_zfmisc_1(A,C),k2_zfmisc_1(B,C),k16_funct_3(A,C,B,C,D,k6_partfun1(C)),k1_domain_1(A,C,E,F)) = k1_domain_1(B,C,k8_funct_2(A,B,D,E),F) ) ) ) ) ) ) ).
fof(t22_borsuk_1,axiom,
$true ).
fof(t23_borsuk_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( ~ v1_xboole_0(C)
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,A,B)
& m2_relset_1(D,A,B) )
=> ! [E] :
( m1_subset_1(E,k1_zfmisc_1(A))
=> ! [F] :
( m1_subset_1(F,k1_zfmisc_1(C))
=> k2_funct_2(k2_zfmisc_1(A,C),k2_zfmisc_1(B,C),k16_funct_3(A,C,B,C,D,k6_partfun1(C)),k12_mcart_1(A,C,E,F)) = k12_mcart_1(B,C,k2_funct_2(A,B,D,E),F) ) ) ) ) ) ) ).
fof(t24_borsuk_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( ~ v1_xboole_0(C)
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,A,B)
& m2_relset_1(D,A,B) )
=> ! [E] :
( m1_subset_1(E,B)
=> ! [F] :
( m1_subset_1(F,C)
=> k3_funct_2(k2_zfmisc_1(A,C),k2_zfmisc_1(B,C),k16_funct_3(A,C,B,C,D,k6_partfun1(C)),k6_domain_1(k2_zfmisc_1(B,C),k1_domain_1(B,C,E,F))) = k12_mcart_1(A,C,k3_funct_2(A,B,D,k6_domain_1(B,E)),k6_domain_1(C,F)) ) ) ) ) ) ) ).
fof(t25_borsuk_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(B))
=> m1_subset_1(k3_tarski(C),k1_zfmisc_1(A)) ) ) ) ).
fof(t26_borsuk_1,axiom,
! [A,B] :
( m1_eqrel_1(B,A)
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(B))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(B))
=> k3_tarski(k5_subset_1(B,C,D)) = k3_xboole_0(k3_tarski(C),k3_tarski(D)) ) ) ) ).
fof(t27_borsuk_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_eqrel_1(B,A)
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(B))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(A))
=> ( D = k3_tarski(C)
=> k3_subset_1(A,D) = k3_tarski(k3_subset_1(B,C)) ) ) ) ) ) ).
fof(t28_borsuk_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v3_relat_2(B)
& v8_relat_2(B)
& v1_partfun1(B,A,A)
& m2_relset_1(B,A,A) )
=> ~ v1_xboole_0(k8_eqrel_1(A,B)) ) ) ).
fof(d1_borsuk_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m1_eqrel_1(B,A) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,A,B)
& m2_relset_1(C,A,B) )
=> ( C = k2_borsuk_1(A,B)
<=> ! [D] :
( m1_subset_1(D,A)
=> r2_hidden(D,k8_funct_2(A,B,C,D)) ) ) ) ) ) ).
fof(t29_borsuk_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m1_eqrel_1(B,A) )
=> ! [C] :
( m1_subset_1(C,A)
=> ! [D] :
( m2_subset_1(D,k1_zfmisc_1(A),B)
=> ( r2_hidden(C,D)
=> D = k8_funct_2(A,B,k2_borsuk_1(A,B),C) ) ) ) ) ) ).
fof(t30_borsuk_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m1_eqrel_1(B,A) )
=> ! [C] :
( m2_subset_1(C,k1_zfmisc_1(A),B)
=> C = k3_funct_2(A,B,k2_borsuk_1(A,B),k6_domain_1(B,C)) ) ) ) ).
fof(t31_borsuk_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m1_eqrel_1(B,A) )
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(B))
=> k3_funct_2(A,B,k2_borsuk_1(A,B),C) = k3_tarski(C) ) ) ) ).
fof(t32_borsuk_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m1_eqrel_1(B,A) )
=> ! [C] :
( m2_subset_1(C,k1_zfmisc_1(A),B)
=> ? [D] :
( m1_subset_1(D,A)
& k8_funct_2(A,B,k2_borsuk_1(A,B),D) = C ) ) ) ) ).
fof(t33_borsuk_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m1_eqrel_1(B,A) )
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(A))
=> ( ! [D] :
( m1_subset_1(D,k1_zfmisc_1(A))
=> ( r2_hidden(D,B)
=> ( r1_xboole_0(D,C)
| r1_tarski(D,C) ) ) )
=> C = k3_funct_2(A,B,k2_borsuk_1(A,B),k2_funct_2(A,B,k2_borsuk_1(A,B),C)) ) ) ) ) ).
fof(t34_borsuk_1,axiom,
$true ).
fof(t35_borsuk_1,axiom,
! [A] :
( l1_pre_topc(A)
=> ! [B] :
( m1_pre_topc(B,A)
=> r1_tarski(u1_struct_0(B),u1_struct_0(A)) ) ) ).
fof(d2_borsuk_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)) )
=> ( v5_pre_topc(C,A,B)
<=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ! [E] :
( m1_connsp_2(E,B,k8_funct_2(u1_struct_0(A),u1_struct_0(B),C,D))
=> ? [F] :
( m1_connsp_2(F,A,D)
& r1_tarski(k4_pre_topc(A,B,C,F),E) ) ) ) ) ) ) ) ).
fof(d3_borsuk_1,axiom,
! [A] :
( l1_struct_0(A)
=> ! [B] :
( ( ~ v3_struct_0(B)
& l1_struct_0(B) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(B))
=> k3_borsuk_1(A,B,C) = k1_borsuk_1(u1_struct_0(B),u1_struct_0(A),C) ) ) ) ).
fof(t36_borsuk_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] :
( m1_subset_1(C,u1_struct_0(A))
=> v5_pre_topc(k3_borsuk_1(B,A,C),B,A) ) ) ) ).
fof(t37_borsuk_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))
& v5_pre_topc(C,A,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)))
=> 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(t38_borsuk_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] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,u1_struct_0(B),u1_struct_0(A))
& v5_pre_topc(D,B,A)
& m2_relset_1(D,u1_struct_0(B),u1_struct_0(A)) )
=> ! [E] :
( m1_connsp_2(E,A,C)
=> m2_connsp_2(k5_pre_topc(B,A,D,E),B,k5_pre_topc(B,A,D,k1_struct_0(A,C))) ) ) ) ) ) ).
fof(t39_borsuk_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)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ! [D] :
( m2_connsp_2(D,A,C)
=> ( r1_tarski(B,C)
=> m2_connsp_2(D,A,B) ) ) ) ) ) ).
fof(t40_borsuk_1,axiom,
$true ).
fof(t41_borsuk_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> v6_compts_1(k1_struct_0(A,B),A) ) ) ).
fof(t42_borsuk_1,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
=> ( v6_compts_1(C,A)
<=> v6_compts_1(D,B) ) ) ) ) ) ) ).
fof(d4_borsuk_1,axiom,
$true ).
fof(t43_borsuk_1,axiom,
$true ).
fof(t44_borsuk_1,axiom,
$true ).
fof(t45_borsuk_1,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( v2_pre_topc(B)
& l1_pre_topc(B) )
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k6_borsuk_1(A,B))))
=> ( v3_pre_topc(C,k6_borsuk_1(A,B))
<=> ? [D] :
( m1_subset_1(D,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(k6_borsuk_1(A,B)))))
& C = k5_setfam_1(u1_struct_0(k6_borsuk_1(A,B)),D)
& ! [E] :
~ ( r2_hidden(E,D)
& ! [F] :
( m1_subset_1(F,k1_zfmisc_1(u1_struct_0(A)))
=> ! [G] :
( m1_subset_1(G,k1_zfmisc_1(u1_struct_0(B)))
=> ~ ( E = k12_mcart_1(u1_struct_0(A),u1_struct_0(B),F,G)
& v3_pre_topc(F,A)
& v3_pre_topc(G,B) ) ) ) ) ) ) ) ) ) ).
fof(t46_borsuk_1,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( v2_pre_topc(B)
& l1_pre_topc(B) )
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(B)))
=> ( ( v3_pre_topc(C,A)
& v3_pre_topc(D,B) )
=> v3_pre_topc(k7_borsuk_1(A,B,C,D),k6_borsuk_1(A,B)) ) ) ) ) ) ).
fof(t47_borsuk_1,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( v2_pre_topc(B)
& l1_pre_topc(B) )
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(B)))
=> k1_tops_1(k6_borsuk_1(A,B),k7_borsuk_1(A,B,C,D)) = k7_borsuk_1(A,B,k1_tops_1(A,C),k1_tops_1(B,D)) ) ) ) ) ).
fof(t48_borsuk_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] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(B))
=> ! [E] :
( m1_connsp_2(E,A,C)
=> ! [F] :
( m1_connsp_2(F,B,D)
=> m1_connsp_2(k7_borsuk_1(A,B,E,F),k6_borsuk_1(A,B),k8_borsuk_1(A,B,C,D)) ) ) ) ) ) ) ).
fof(t49_borsuk_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] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(B)))
=> ! [E] :
( m2_connsp_2(E,A,C)
=> ! [F] :
( m2_connsp_2(F,B,D)
=> m2_connsp_2(k7_borsuk_1(A,B,E,F),k6_borsuk_1(A,B),k7_borsuk_1(A,B,C,D)) ) ) ) ) ) ) ).
fof(t50_borsuk_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] :
( m1_subset_1(C,u1_struct_0(k6_borsuk_1(A,B)))
=> ? [D] :
( m1_subset_1(D,u1_struct_0(A))
& ? [E] :
( m1_subset_1(E,u1_struct_0(B))
& C = k8_borsuk_1(A,B,D,E) ) ) ) ) ) ).
fof(t51_borsuk_1,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( v2_pre_topc(B)
& l1_pre_topc(B) )
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k6_borsuk_1(A,B))))
=> v1_tops_2(k11_borsuk_1(A,B,C),k6_borsuk_1(A,B)) ) ) ) ).
fof(t52_borsuk_1,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( v2_pre_topc(B)
& l1_pre_topc(B) )
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k6_borsuk_1(A,B))))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k6_borsuk_1(A,B))))
=> ( r1_tarski(C,D)
=> r1_tarski(k11_borsuk_1(A,B,C),k11_borsuk_1(A,B,D)) ) ) ) ) ) ).
fof(t53_borsuk_1,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( v2_pre_topc(B)
& l1_pre_topc(B) )
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k6_borsuk_1(A,B))))
=> r1_tarski(k5_setfam_1(u1_struct_0(k6_borsuk_1(A,B)),k11_borsuk_1(A,B,C)),C) ) ) ) ).
fof(t54_borsuk_1,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( v2_pre_topc(B)
& l1_pre_topc(B) )
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k6_borsuk_1(A,B))))
=> ( v3_pre_topc(C,k6_borsuk_1(A,B))
=> C = k5_setfam_1(u1_struct_0(k6_borsuk_1(A,B)),k11_borsuk_1(A,B,C)) ) ) ) ) ).
fof(t55_borsuk_1,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( v2_pre_topc(B)
& l1_pre_topc(B) )
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k6_borsuk_1(A,B))))
=> k1_tops_1(k6_borsuk_1(A,B),C) = k5_setfam_1(u1_struct_0(k6_borsuk_1(A,B)),k11_borsuk_1(A,B,C)) ) ) ) ).
fof(d7_borsuk_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) )
=> k12_borsuk_1(A,B) = k2_funct_3(k2_zfmisc_1(u1_struct_0(A),u1_struct_0(B)),u1_struct_0(A),k9_funct_3(u1_struct_0(A),u1_struct_0(B))) ) ) ).
fof(d8_borsuk_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) )
=> k13_borsuk_1(A,B) = k2_funct_3(k2_zfmisc_1(u1_struct_0(A),u1_struct_0(B)),u1_struct_0(B),k10_funct_3(u1_struct_0(A),u1_struct_0(B))) ) ) ).
fof(t56_borsuk_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] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k6_borsuk_1(A,B))))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(k6_borsuk_1(A,B)))))
=> ( ! [E] :
( r2_hidden(E,D)
=> ( r1_tarski(E,C)
& ? [F] :
( m1_subset_1(F,k1_zfmisc_1(u1_struct_0(A)))
& ? [G] :
( m1_subset_1(G,k1_zfmisc_1(u1_struct_0(B)))
& E = k7_borsuk_1(A,B,F,G) ) ) ) )
=> r1_tarski(k7_borsuk_1(A,B,k5_setfam_1(u1_struct_0(A),k2_funct_2(k1_pcomps_1(u1_struct_0(k6_borsuk_1(A,B))),k1_pcomps_1(u1_struct_0(A)),k12_borsuk_1(A,B),D)),k6_setfam_1(u1_struct_0(B),k2_funct_2(k1_pcomps_1(u1_struct_0(k6_borsuk_1(A,B))),k1_pcomps_1(u1_struct_0(B)),k13_borsuk_1(A,B),D))),C) ) ) ) ) ) ).
fof(t57_borsuk_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] :
( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(k6_borsuk_1(A,B)))))
=> ! [D] :
~ ( r2_hidden(D,k2_funct_2(k1_pcomps_1(u1_struct_0(k6_borsuk_1(A,B))),k1_pcomps_1(u1_struct_0(A)),k12_borsuk_1(A,B),C))
& ! [E] :
( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(k6_borsuk_1(A,B))))
=> ~ ( r2_hidden(E,C)
& D = k2_funct_2(k2_zfmisc_1(u1_struct_0(A),u1_struct_0(B)),u1_struct_0(A),k9_funct_3(u1_struct_0(A),u1_struct_0(B)),E) ) ) ) ) ) ) ).
fof(t58_borsuk_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] :
( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(k6_borsuk_1(A,B)))))
=> ! [D] :
~ ( r2_hidden(D,k2_funct_2(k1_pcomps_1(u1_struct_0(k6_borsuk_1(A,B))),k1_pcomps_1(u1_struct_0(B)),k13_borsuk_1(A,B),C))
& ! [E] :
( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(k6_borsuk_1(A,B))))
=> ~ ( r2_hidden(E,C)
& D = k2_funct_2(k2_zfmisc_1(u1_struct_0(A),u1_struct_0(B)),u1_struct_0(B),k10_funct_3(u1_struct_0(A),u1_struct_0(B)),E) ) ) ) ) ) ) ).
fof(t59_borsuk_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] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k6_borsuk_1(A,B))))
=> ( v3_pre_topc(C,k6_borsuk_1(A,B))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
=> ! [E] :
( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(B)))
=> ( ( D = k2_funct_2(k2_zfmisc_1(u1_struct_0(A),u1_struct_0(B)),u1_struct_0(A),k9_funct_3(u1_struct_0(A),u1_struct_0(B)),C)
=> v3_pre_topc(D,A) )
& ( E = k2_funct_2(k2_zfmisc_1(u1_struct_0(A),u1_struct_0(B)),u1_struct_0(B),k10_funct_3(u1_struct_0(A),u1_struct_0(B)),C)
=> v3_pre_topc(E,B) ) ) ) ) ) ) ) ) ).
fof(t60_borsuk_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] :
( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(k6_borsuk_1(A,B)))))
=> ( v1_tops_2(C,k6_borsuk_1(A,B))
=> ( v1_tops_2(k14_borsuk_1(u1_struct_0(k6_borsuk_1(A,B)),u1_struct_0(A),k12_borsuk_1(A,B),C),A)
& v1_tops_2(k14_borsuk_1(u1_struct_0(k6_borsuk_1(A,B)),u1_struct_0(B),k13_borsuk_1(A,B),C),B) ) ) ) ) ) ).
fof(t61_borsuk_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] :
( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(k6_borsuk_1(A,B)))))
=> ( ( k14_borsuk_1(u1_struct_0(k6_borsuk_1(A,B)),u1_struct_0(A),k12_borsuk_1(A,B),C) = k1_xboole_0
| k14_borsuk_1(u1_struct_0(k6_borsuk_1(A,B)),u1_struct_0(B),k13_borsuk_1(A,B),C) = k1_xboole_0 )
=> C = k1_xboole_0 ) ) ) ) ).
fof(t62_borsuk_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] :
( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(k6_borsuk_1(A,B)))))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
=> ! [E] :
( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(B)))
=> ( r1_compts_1(k6_borsuk_1(A,B),C,k7_borsuk_1(A,B,D,E))
=> ( ( E != k1_xboole_0
=> r1_compts_1(A,k14_borsuk_1(u1_struct_0(k6_borsuk_1(A,B)),u1_struct_0(A),k12_borsuk_1(A,B),C),D) )
& ( D != k1_xboole_0
=> r1_compts_1(B,k14_borsuk_1(u1_struct_0(k6_borsuk_1(A,B)),u1_struct_0(B),k13_borsuk_1(A,B),C),E) ) ) ) ) ) ) ) ) ).
fof(t63_borsuk_1,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( v2_pre_topc(B)
& l1_pre_topc(B) )
=> ! [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)))
=> ~ ( r1_compts_1(A,C,D)
& ! [E] :
( m1_subset_1(E,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
=> ~ ( r1_tarski(E,C)
& r1_compts_1(A,E,D)
& ! [F] :
~ ( r2_hidden(F,E)
& r1_xboole_0(F,D) ) ) ) ) ) ) ) ) ).
fof(t64_borsuk_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] :
( 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(k6_borsuk_1(A,B)))))
=> ~ ( v1_finset_1(C)
& r1_tarski(C,k14_borsuk_1(u1_struct_0(k6_borsuk_1(A,B)),u1_struct_0(A),k12_borsuk_1(A,B),D))
& ! [E] :
( m1_subset_1(E,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(k6_borsuk_1(A,B)))))
=> ~ ( r1_tarski(E,D)
& v1_finset_1(E)
& C = k14_borsuk_1(u1_struct_0(k6_borsuk_1(A,B)),u1_struct_0(A),k12_borsuk_1(A,B),E) ) ) ) ) ) ) ) ).
fof(t65_borsuk_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] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(B)))
=> ( k7_borsuk_1(A,B,C,D) != k1_xboole_0
=> ( k8_funct_2(k1_zfmisc_1(u1_struct_0(k6_borsuk_1(A,B))),k1_pcomps_1(u1_struct_0(A)),k12_borsuk_1(A,B),k7_borsuk_1(A,B,C,D)) = C
& k8_funct_2(k1_zfmisc_1(u1_struct_0(k6_borsuk_1(A,B))),k1_pcomps_1(u1_struct_0(B)),k13_borsuk_1(A,B),k7_borsuk_1(A,B,C,D)) = D ) ) ) ) ) ) ).
fof(t66_borsuk_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) )
=> ( k1_funct_1(k12_borsuk_1(A,B),k1_xboole_0) = k1_xboole_0
& k1_funct_1(k13_borsuk_1(A,B),k1_xboole_0) = k1_xboole_0 ) ) ) ).
fof(t67_borsuk_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] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(B)))
=> ( v6_compts_1(D,B)
=> ! [E] :
( m2_connsp_2(E,k6_borsuk_1(B,A),k7_borsuk_1(B,A,D,k1_struct_0(A,C)))
=> ? [F] :
( m2_connsp_2(F,B,D)
& ? [G] :
( m1_connsp_2(G,A,C)
& r1_tarski(k10_borsuk_1(B,A,D,C,F,G),E) ) ) ) ) ) ) ) ) ).
fof(d9_borsuk_1,axiom,
! [A] :
( l1_struct_0(A)
=> k15_borsuk_1(A) = k8_eqrel_1(u1_struct_0(A),k6_partfun1(u1_struct_0(A))) ) ).
fof(t68_borsuk_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)))
=> ~ ( r2_hidden(B,k15_borsuk_1(A))
& ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> B != k1_struct_0(A,C) ) ) ) ) ).
fof(t69_borsuk_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m1_eqrel_1(B,u1_struct_0(A)) )
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(B))
=> ( r2_hidden(k3_tarski(C),u1_pre_topc(A))
<=> r2_hidden(C,u1_pre_topc(k16_borsuk_1(A,B))) ) ) ) ) ).
fof(d11_borsuk_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m1_eqrel_1(B,u1_struct_0(A)) )
=> k17_borsuk_1(A,B) = k2_borsuk_1(u1_struct_0(A),B) ) ) ).
fof(t70_borsuk_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m1_eqrel_1(B,u1_struct_0(A)) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> r2_hidden(C,k8_funct_2(u1_struct_0(A),u1_struct_0(k16_borsuk_1(A,B)),k17_borsuk_1(A,B),C)) ) ) ) ).
fof(t71_borsuk_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m1_eqrel_1(B,u1_struct_0(A)) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k16_borsuk_1(A,B)))
=> ? [D] :
( m1_subset_1(D,u1_struct_0(A))
& k8_funct_2(u1_struct_0(A),u1_struct_0(k16_borsuk_1(A,B)),k17_borsuk_1(A,B),D) = C ) ) ) ) ).
fof(t72_borsuk_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m1_eqrel_1(B,u1_struct_0(A)) )
=> k2_relat_1(k17_borsuk_1(A,B)) = u1_struct_0(k16_borsuk_1(A,B)) ) ) ).
fof(t73_borsuk_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& m1_pre_topc(B,A) )
=> ! [C] :
( ( ~ v1_xboole_0(C)
& m1_eqrel_1(C,u1_struct_0(B)) )
=> r1_tarski(C,k18_borsuk_1(A,B,C)) ) ) ) ).
fof(t74_borsuk_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& m1_pre_topc(B,A) )
=> ! [C] :
( ( ~ v1_xboole_0(C)
& m1_eqrel_1(C,u1_struct_0(B)) )
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
=> ~ ( r2_hidden(D,k18_borsuk_1(A,B,C))
& ~ r2_hidden(D,C)
& ! [E] :
( m1_subset_1(E,u1_struct_0(A))
=> ~ ( ~ r2_hidden(E,k2_pre_topc(B))
& D = k1_struct_0(A,E) ) ) ) ) ) ) ) ).
fof(t75_borsuk_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& m1_pre_topc(B,A) )
=> ! [C] :
( ( ~ v1_xboole_0(C)
& m1_eqrel_1(C,u1_struct_0(B)) )
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ( ~ r2_hidden(D,u1_struct_0(B))
=> r2_hidden(k1_struct_0(A,D),k18_borsuk_1(A,B,C)) ) ) ) ) ) ).
fof(t76_borsuk_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& m1_pre_topc(B,A) )
=> ! [C] :
( ( ~ v1_xboole_0(C)
& m1_eqrel_1(C,u1_struct_0(B)) )
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ( r2_hidden(D,u1_struct_0(B))
=> k8_funct_2(u1_struct_0(A),u1_struct_0(k16_borsuk_1(A,k18_borsuk_1(A,B,C))),k17_borsuk_1(A,k18_borsuk_1(A,B,C)),D) = k1_funct_1(k17_borsuk_1(B,C),D) ) ) ) ) ) ).
fof(t77_borsuk_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& m1_pre_topc(B,A) )
=> ! [C] :
( ( ~ v1_xboole_0(C)
& m1_eqrel_1(C,u1_struct_0(B)) )
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ( ~ r2_hidden(D,u1_struct_0(B))
=> k8_funct_2(u1_struct_0(A),u1_struct_0(k16_borsuk_1(A,k18_borsuk_1(A,B,C))),k17_borsuk_1(A,k18_borsuk_1(A,B,C)),D) = k1_struct_0(A,D) ) ) ) ) ) ).
fof(t78_borsuk_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& m1_pre_topc(B,A) )
=> ! [C] :
( ( ~ v1_xboole_0(C)
& m1_eqrel_1(C,u1_struct_0(B)) )
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(A))
=> ( k8_funct_2(u1_struct_0(A),u1_struct_0(k16_borsuk_1(A,k18_borsuk_1(A,B,C))),k17_borsuk_1(A,k18_borsuk_1(A,B,C)),D) = k8_funct_2(u1_struct_0(A),u1_struct_0(k16_borsuk_1(A,k18_borsuk_1(A,B,C))),k17_borsuk_1(A,k18_borsuk_1(A,B,C)),E)
=> ( r2_hidden(D,u1_struct_0(B))
| D = E ) ) ) ) ) ) ) ).
fof(t79_borsuk_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& m1_pre_topc(B,A) )
=> ! [C] :
( ( ~ v1_xboole_0(C)
& m1_eqrel_1(C,u1_struct_0(B)) )
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ( r2_hidden(k8_funct_2(u1_struct_0(A),u1_struct_0(k16_borsuk_1(A,k18_borsuk_1(A,B,C))),k17_borsuk_1(A,k18_borsuk_1(A,B,C)),D),u1_struct_0(k16_borsuk_1(B,C)))
=> r2_hidden(D,u1_struct_0(B)) ) ) ) ) ) ).
fof(t80_borsuk_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& m1_pre_topc(B,A) )
=> ! [C] :
( ( ~ v1_xboole_0(C)
& m1_eqrel_1(C,u1_struct_0(B)) )
=> ! [D] :
( r2_hidden(D,u1_struct_0(B))
=> r2_hidden(k1_funct_1(k17_borsuk_1(A,k18_borsuk_1(A,B,C)),D),u1_struct_0(k16_borsuk_1(B,C))) ) ) ) ) ).
fof(d13_borsuk_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m1_eqrel_1(B,u1_struct_0(A)) )
=> ( m1_borsuk_1(B,A)
<=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ( r2_hidden(C,B)
=> ! [D] :
( m2_connsp_2(D,A,C)
=> ? [E] :
( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(A)))
& v3_pre_topc(E,A)
& r1_tarski(C,E)
& r1_tarski(E,D)
& ! [F] :
( m1_subset_1(F,k1_zfmisc_1(u1_struct_0(A)))
=> ( r2_hidden(F,B)
=> ( r1_xboole_0(F,E)
| r1_tarski(F,E) ) ) ) ) ) ) ) ) ) ) ).
fof(t81_borsuk_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_borsuk_1(B,A)
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k16_borsuk_1(A,B)))
=> ! [D] :
( m2_connsp_2(D,A,k5_pre_topc(A,k16_borsuk_1(A,B),k17_borsuk_1(A,B),k1_struct_0(k16_borsuk_1(A,B),C)))
=> m1_connsp_2(k4_pre_topc(A,k16_borsuk_1(A,B),k17_borsuk_1(A,B),D),k16_borsuk_1(A,B),C) ) ) ) ) ).
fof(t82_borsuk_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> m1_borsuk_1(k15_borsuk_1(A),A) ) ).
fof(d14_borsuk_1,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_pre_topc(B,A)
=> ( v1_borsuk_1(B,A)
<=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ( C = u1_struct_0(B)
=> v4_pre_topc(C,A) ) ) ) ) ) ).
fof(d15_borsuk_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_borsuk_1(B,A)
=> ( v2_borsuk_1(B,A)
<=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ( r2_hidden(C,B)
=> v6_compts_1(C,A) ) ) ) ) ) ).
fof(d16_borsuk_1,axiom,
! [A] :
( l1_pre_topc(A)
=> ( A = k22_borsuk_1
<=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k5_pcomps_1(k8_metric_1))))
=> ( B = k1_rcomp_1(np__0,np__1)
=> A = k3_pre_topc(k5_pcomps_1(k8_metric_1),B) ) ) ) ) ).
fof(t83_borsuk_1,axiom,
u1_struct_0(k22_borsuk_1) = k1_rcomp_1(np__0,np__1) ).
fof(d17_borsuk_1,axiom,
k23_borsuk_1 = np__0 ).
fof(d18_borsuk_1,axiom,
k24_borsuk_1 = np__1 ).
fof(d19_borsuk_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& m1_pre_topc(B,A) )
=> ! [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_borsuk_1(C,A,B)
<=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ( r2_hidden(D,u1_struct_0(B))
=> k8_funct_2(u1_struct_0(A),u1_struct_0(B),C,D) = D ) ) ) ) ) ) ).
fof(d20_borsuk_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& m1_pre_topc(B,A) )
=> ( r1_borsuk_1(A,B)
<=> ? [C] :
( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
& v5_pre_topc(C,A,B)
& m2_relset_1(C,u1_struct_0(A),u1_struct_0(B))
& v3_borsuk_1(C,A,B) ) ) ) ) ).
fof(d21_borsuk_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& m1_pre_topc(B,A) )
=> ( r2_borsuk_1(A,B)
<=> ? [C] :
( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(k6_borsuk_1(A,k22_borsuk_1)),u1_struct_0(A))
& v5_pre_topc(C,k6_borsuk_1(A,k22_borsuk_1),A)
& m2_relset_1(C,u1_struct_0(k6_borsuk_1(A,k22_borsuk_1)),u1_struct_0(A))
& ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ( k8_funct_2(u1_struct_0(k6_borsuk_1(A,k22_borsuk_1)),u1_struct_0(A),C,k8_borsuk_1(A,k22_borsuk_1,D,k23_borsuk_1)) = D
& r2_hidden(k8_funct_2(u1_struct_0(k6_borsuk_1(A,k22_borsuk_1)),u1_struct_0(A),C,k8_borsuk_1(A,k22_borsuk_1,D,k24_borsuk_1)),u1_struct_0(B))
& ( r2_hidden(D,u1_struct_0(B))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(k22_borsuk_1))
=> k8_funct_2(u1_struct_0(k6_borsuk_1(A,k22_borsuk_1)),u1_struct_0(A),C,k8_borsuk_1(A,k22_borsuk_1,D,E)) = D ) ) ) ) ) ) ) ) ).
fof(t84_borsuk_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v1_borsuk_1(B,A)
& m1_pre_topc(B,A) )
=> ! [C] :
( ( v2_borsuk_1(C,B)
& m1_borsuk_1(C,B) )
=> ( r1_borsuk_1(A,B)
=> r1_borsuk_1(k16_borsuk_1(A,k20_borsuk_1(A,B,C)),k21_borsuk_1(A,B,C)) ) ) ) ) ).
fof(t85_borsuk_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v1_borsuk_1(B,A)
& m1_pre_topc(B,A) )
=> ! [C] :
( ( v2_borsuk_1(C,B)
& m1_borsuk_1(C,B) )
=> ( r2_borsuk_1(A,B)
=> r2_borsuk_1(k16_borsuk_1(A,k20_borsuk_1(A,B,C)),k21_borsuk_1(A,B,C)) ) ) ) ) ).
fof(dt_m1_borsuk_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_borsuk_1(B,A)
=> ( ~ v1_xboole_0(B)
& m1_eqrel_1(B,u1_struct_0(A)) ) ) ) ).
fof(existence_m1_borsuk_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ? [B] : m1_borsuk_1(B,A) ) ).
fof(dt_k1_borsuk_1,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(C,A) )
=> ( v1_funct_1(k1_borsuk_1(A,B,C))
& v1_funct_2(k1_borsuk_1(A,B,C),B,A)
& m2_relset_1(k1_borsuk_1(A,B,C),B,A) ) ) ).
fof(redefinition_k1_borsuk_1,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(C,A) )
=> k1_borsuk_1(A,B,C) = k2_funcop_1(B,C) ) ).
fof(dt_k2_borsuk_1,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& ~ v1_xboole_0(B)
& m1_eqrel_1(B,A) )
=> ( v1_funct_1(k2_borsuk_1(A,B))
& v1_funct_2(k2_borsuk_1(A,B),A,B)
& m2_relset_1(k2_borsuk_1(A,B),A,B) ) ) ).
fof(dt_k3_borsuk_1,axiom,
! [A,B,C] :
( ( l1_struct_0(A)
& ~ v3_struct_0(B)
& l1_struct_0(B)
& m1_subset_1(C,u1_struct_0(B)) )
=> ( v1_funct_1(k3_borsuk_1(A,B,C))
& v1_funct_2(k3_borsuk_1(A,B,C),u1_struct_0(A),u1_struct_0(B))
& m2_relset_1(k3_borsuk_1(A,B,C),u1_struct_0(A),u1_struct_0(B)) ) ) ).
fof(dt_k4_borsuk_1,axiom,
! [A,B,C,D,E] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A)
& ~ v3_struct_0(B)
& v2_pre_topc(B)
& l1_pre_topc(B)
& ~ v3_struct_0(C)
& v2_pre_topc(C)
& l1_pre_topc(C)
& v1_funct_1(D)
& v1_funct_2(D,u1_struct_0(A),u1_struct_0(B))
& v5_pre_topc(D,A,B)
& m1_relset_1(D,u1_struct_0(A),u1_struct_0(B))
& v1_funct_1(E)
& v1_funct_2(E,u1_struct_0(B),u1_struct_0(C))
& v5_pre_topc(E,B,C)
& m1_relset_1(E,u1_struct_0(B),u1_struct_0(C)) )
=> ( v1_funct_1(k4_borsuk_1(A,B,C,D,E))
& v1_funct_2(k4_borsuk_1(A,B,C,D,E),u1_struct_0(A),u1_struct_0(C))
& v5_pre_topc(k4_borsuk_1(A,B,C,D,E),A,C)
& m2_relset_1(k4_borsuk_1(A,B,C,D,E),u1_struct_0(A),u1_struct_0(C)) ) ) ).
fof(redefinition_k4_borsuk_1,axiom,
! [A,B,C,D,E] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A)
& ~ v3_struct_0(B)
& v2_pre_topc(B)
& l1_pre_topc(B)
& ~ v3_struct_0(C)
& v2_pre_topc(C)
& l1_pre_topc(C)
& v1_funct_1(D)
& v1_funct_2(D,u1_struct_0(A),u1_struct_0(B))
& v5_pre_topc(D,A,B)
& m1_relset_1(D,u1_struct_0(A),u1_struct_0(B))
& v1_funct_1(E)
& v1_funct_2(E,u1_struct_0(B),u1_struct_0(C))
& v5_pre_topc(E,B,C)
& m1_relset_1(E,u1_struct_0(B),u1_struct_0(C)) )
=> k4_borsuk_1(A,B,C,D,E) = k5_relat_1(D,E) ) ).
fof(dt_k5_borsuk_1,axiom,
! [A,B,C,D,E] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A)
& ~ v3_struct_0(B)
& v2_pre_topc(B)
& l1_pre_topc(B)
& m1_subset_1(C,u1_struct_0(B))
& v1_funct_1(D)
& v1_funct_2(D,u1_struct_0(A),u1_struct_0(B))
& v5_pre_topc(D,A,B)
& m1_relset_1(D,u1_struct_0(A),u1_struct_0(B))
& m1_connsp_2(E,B,C) )
=> m2_connsp_2(k5_borsuk_1(A,B,C,D,E),A,k5_pre_topc(A,B,D,k1_struct_0(B,C))) ) ).
fof(redefinition_k5_borsuk_1,axiom,
! [A,B,C,D,E] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A)
& ~ v3_struct_0(B)
& v2_pre_topc(B)
& l1_pre_topc(B)
& m1_subset_1(C,u1_struct_0(B))
& v1_funct_1(D)
& v1_funct_2(D,u1_struct_0(A),u1_struct_0(B))
& v5_pre_topc(D,A,B)
& m1_relset_1(D,u1_struct_0(A),u1_struct_0(B))
& m1_connsp_2(E,B,C) )
=> k5_borsuk_1(A,B,C,D,E) = k10_relat_1(D,E) ) ).
fof(dt_k6_borsuk_1,axiom,
! [A,B] :
( ( v2_pre_topc(A)
& l1_pre_topc(A)
& v2_pre_topc(B)
& l1_pre_topc(B) )
=> ( v1_pre_topc(k6_borsuk_1(A,B))
& v2_pre_topc(k6_borsuk_1(A,B))
& l1_pre_topc(k6_borsuk_1(A,B)) ) ) ).
fof(dt_k7_borsuk_1,axiom,
! [A,B,C,D] :
( ( v2_pre_topc(A)
& l1_pre_topc(A)
& v2_pre_topc(B)
& l1_pre_topc(B)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
& m1_subset_1(D,k1_zfmisc_1(u1_struct_0(B))) )
=> m1_subset_1(k7_borsuk_1(A,B,C,D),k1_zfmisc_1(u1_struct_0(k6_borsuk_1(A,B)))) ) ).
fof(redefinition_k7_borsuk_1,axiom,
! [A,B,C,D] :
( ( v2_pre_topc(A)
& l1_pre_topc(A)
& v2_pre_topc(B)
& l1_pre_topc(B)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
& m1_subset_1(D,k1_zfmisc_1(u1_struct_0(B))) )
=> k7_borsuk_1(A,B,C,D) = k2_zfmisc_1(C,D) ) ).
fof(dt_k8_borsuk_1,axiom,
! [A,B,C,D] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A)
& ~ v3_struct_0(B)
& v2_pre_topc(B)
& l1_pre_topc(B)
& m1_subset_1(C,u1_struct_0(A))
& m1_subset_1(D,u1_struct_0(B)) )
=> m1_subset_1(k8_borsuk_1(A,B,C,D),u1_struct_0(k6_borsuk_1(A,B))) ) ).
fof(redefinition_k8_borsuk_1,axiom,
! [A,B,C,D] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A)
& ~ v3_struct_0(B)
& v2_pre_topc(B)
& l1_pre_topc(B)
& m1_subset_1(C,u1_struct_0(A))
& m1_subset_1(D,u1_struct_0(B)) )
=> k8_borsuk_1(A,B,C,D) = k4_tarski(C,D) ) ).
fof(dt_k9_borsuk_1,axiom,
! [A,B,C,D,E,F] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A)
& ~ v3_struct_0(B)
& v2_pre_topc(B)
& l1_pre_topc(B)
& m1_subset_1(C,u1_struct_0(A))
& m1_subset_1(D,u1_struct_0(B))
& m1_connsp_2(E,A,C)
& m1_connsp_2(F,B,D) )
=> m1_connsp_2(k9_borsuk_1(A,B,C,D,E,F),k6_borsuk_1(A,B),k8_borsuk_1(A,B,C,D)) ) ).
fof(redefinition_k9_borsuk_1,axiom,
! [A,B,C,D,E,F] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A)
& ~ v3_struct_0(B)
& v2_pre_topc(B)
& l1_pre_topc(B)
& m1_subset_1(C,u1_struct_0(A))
& m1_subset_1(D,u1_struct_0(B))
& m1_connsp_2(E,A,C)
& m1_connsp_2(F,B,D) )
=> k9_borsuk_1(A,B,C,D,E,F) = k2_zfmisc_1(E,F) ) ).
fof(dt_k10_borsuk_1,axiom,
! [A,B,C,D,E,F] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A)
& ~ v3_struct_0(B)
& v2_pre_topc(B)
& l1_pre_topc(B)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
& m1_subset_1(D,u1_struct_0(B))
& m2_connsp_2(E,A,C)
& m1_connsp_2(F,B,D) )
=> m2_connsp_2(k10_borsuk_1(A,B,C,D,E,F),k6_borsuk_1(A,B),k7_borsuk_1(A,B,C,k1_struct_0(B,D))) ) ).
fof(redefinition_k10_borsuk_1,axiom,
! [A,B,C,D,E,F] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A)
& ~ v3_struct_0(B)
& v2_pre_topc(B)
& l1_pre_topc(B)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
& m1_subset_1(D,u1_struct_0(B))
& m2_connsp_2(E,A,C)
& m1_connsp_2(F,B,D) )
=> k10_borsuk_1(A,B,C,D,E,F) = k2_zfmisc_1(E,F) ) ).
fof(dt_k11_borsuk_1,axiom,
! [A,B,C] :
( ( v2_pre_topc(A)
& l1_pre_topc(A)
& v2_pre_topc(B)
& l1_pre_topc(B)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k6_borsuk_1(A,B)))) )
=> m1_subset_1(k11_borsuk_1(A,B,C),k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(k6_borsuk_1(A,B))))) ) ).
fof(dt_k12_borsuk_1,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A)
& ~ v3_struct_0(B)
& v2_pre_topc(B)
& l1_pre_topc(B) )
=> ( v1_funct_1(k12_borsuk_1(A,B))
& v1_funct_2(k12_borsuk_1(A,B),k1_pcomps_1(u1_struct_0(k6_borsuk_1(A,B))),k1_pcomps_1(u1_struct_0(A)))
& m2_relset_1(k12_borsuk_1(A,B),k1_pcomps_1(u1_struct_0(k6_borsuk_1(A,B))),k1_pcomps_1(u1_struct_0(A))) ) ) ).
fof(dt_k13_borsuk_1,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A)
& ~ v3_struct_0(B)
& v2_pre_topc(B)
& l1_pre_topc(B) )
=> ( v1_funct_1(k13_borsuk_1(A,B))
& v1_funct_2(k13_borsuk_1(A,B),k1_pcomps_1(u1_struct_0(k6_borsuk_1(A,B))),k1_pcomps_1(u1_struct_0(B)))
& m2_relset_1(k13_borsuk_1(A,B),k1_pcomps_1(u1_struct_0(k6_borsuk_1(A,B))),k1_pcomps_1(u1_struct_0(B))) ) ) ).
fof(dt_k14_borsuk_1,axiom,
! [A,B,C,D] :
( ( v1_funct_1(C)
& v1_funct_2(C,k1_pcomps_1(A),k1_pcomps_1(B))
& m1_relset_1(C,k1_pcomps_1(A),k1_pcomps_1(B))
& m1_subset_1(D,k1_zfmisc_1(k1_zfmisc_1(A))) )
=> m1_subset_1(k14_borsuk_1(A,B,C,D),k1_zfmisc_1(k1_zfmisc_1(B))) ) ).
fof(redefinition_k14_borsuk_1,axiom,
! [A,B,C,D] :
( ( v1_funct_1(C)
& v1_funct_2(C,k1_pcomps_1(A),k1_pcomps_1(B))
& m1_relset_1(C,k1_pcomps_1(A),k1_pcomps_1(B))
& m1_subset_1(D,k1_zfmisc_1(k1_zfmisc_1(A))) )
=> k14_borsuk_1(A,B,C,D) = k9_relat_1(C,D) ) ).
fof(dt_k15_borsuk_1,axiom,
! [A] :
( l1_struct_0(A)
=> m1_eqrel_1(k15_borsuk_1(A),u1_struct_0(A)) ) ).
fof(dt_k16_borsuk_1,axiom,
! [A,B] :
( ( v2_pre_topc(A)
& l1_pre_topc(A)
& m1_eqrel_1(B,u1_struct_0(A)) )
=> ( v1_pre_topc(k16_borsuk_1(A,B))
& v2_pre_topc(k16_borsuk_1(A,B))
& l1_pre_topc(k16_borsuk_1(A,B)) ) ) ).
fof(dt_k17_borsuk_1,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A)
& ~ v1_xboole_0(B)
& m1_eqrel_1(B,u1_struct_0(A)) )
=> ( v1_funct_1(k17_borsuk_1(A,B))
& v1_funct_2(k17_borsuk_1(A,B),u1_struct_0(A),u1_struct_0(k16_borsuk_1(A,B)))
& v5_pre_topc(k17_borsuk_1(A,B),A,k16_borsuk_1(A,B))
& m2_relset_1(k17_borsuk_1(A,B),u1_struct_0(A),u1_struct_0(k16_borsuk_1(A,B))) ) ) ).
fof(dt_k18_borsuk_1,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A)
& ~ v3_struct_0(B)
& m1_pre_topc(B,A)
& ~ v1_xboole_0(C)
& m1_eqrel_1(C,u1_struct_0(B)) )
=> ( ~ v1_xboole_0(k18_borsuk_1(A,B,C))
& m1_eqrel_1(k18_borsuk_1(A,B,C),u1_struct_0(A)) ) ) ).
fof(dt_k19_borsuk_1,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A)
& ~ v3_struct_0(B)
& v1_borsuk_1(B,A)
& m1_pre_topc(B,A)
& m1_borsuk_1(C,B) )
=> m1_borsuk_1(k19_borsuk_1(A,B,C),A) ) ).
fof(redefinition_k19_borsuk_1,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A)
& ~ v3_struct_0(B)
& v1_borsuk_1(B,A)
& m1_pre_topc(B,A)
& m1_borsuk_1(C,B) )
=> k19_borsuk_1(A,B,C) = k18_borsuk_1(A,B,C) ) ).
fof(dt_k20_borsuk_1,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A)
& ~ v3_struct_0(B)
& v1_borsuk_1(B,A)
& m1_pre_topc(B,A)
& v2_borsuk_1(C,B)
& m1_borsuk_1(C,B) )
=> ( v2_borsuk_1(k20_borsuk_1(A,B,C),A)
& m1_borsuk_1(k20_borsuk_1(A,B,C),A) ) ) ).
fof(redefinition_k20_borsuk_1,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A)
& ~ v3_struct_0(B)
& v1_borsuk_1(B,A)
& m1_pre_topc(B,A)
& v2_borsuk_1(C,B)
& m1_borsuk_1(C,B) )
=> k20_borsuk_1(A,B,C) = k18_borsuk_1(A,B,C) ) ).
fof(dt_k21_borsuk_1,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A)
& ~ v3_struct_0(B)
& v1_borsuk_1(B,A)
& m1_pre_topc(B,A)
& v2_borsuk_1(C,B)
& m1_borsuk_1(C,B) )
=> ( v1_pre_topc(k21_borsuk_1(A,B,C))
& v1_borsuk_1(k21_borsuk_1(A,B,C),k16_borsuk_1(A,k20_borsuk_1(A,B,C)))
& m1_pre_topc(k21_borsuk_1(A,B,C),k16_borsuk_1(A,k20_borsuk_1(A,B,C))) ) ) ).
fof(redefinition_k21_borsuk_1,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A)
& ~ v3_struct_0(B)
& v1_borsuk_1(B,A)
& m1_pre_topc(B,A)
& v2_borsuk_1(C,B)
& m1_borsuk_1(C,B) )
=> k21_borsuk_1(A,B,C) = k16_borsuk_1(B,C) ) ).
fof(dt_k22_borsuk_1,axiom,
l1_pre_topc(k22_borsuk_1) ).
fof(dt_k23_borsuk_1,axiom,
m1_subset_1(k23_borsuk_1,u1_struct_0(k22_borsuk_1)) ).
fof(dt_k24_borsuk_1,axiom,
m1_subset_1(k24_borsuk_1,u1_struct_0(k22_borsuk_1)) ).
fof(t17_borsuk_1,axiom,
! [A,B] :
( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
=> k3_tarski(k5_setfam_1(A,B)) = k3_tarski(a_2_0_borsuk_1(A,B)) ) ).
fof(d5_borsuk_1,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( v2_pre_topc(B)
& l1_pre_topc(B) )
=> ! [C] :
( ( v1_pre_topc(C)
& v2_pre_topc(C)
& l1_pre_topc(C) )
=> ( C = k6_borsuk_1(A,B)
<=> ( u1_struct_0(C) = k2_zfmisc_1(u1_struct_0(A),u1_struct_0(B))
& u1_pre_topc(C) = a_3_0_borsuk_1(A,B,C) ) ) ) ) ) ).
fof(d6_borsuk_1,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( v2_pre_topc(B)
& l1_pre_topc(B) )
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k6_borsuk_1(A,B))))
=> k11_borsuk_1(A,B,C) = a_3_1_borsuk_1(A,B,C) ) ) ) ).
fof(d10_borsuk_1,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_eqrel_1(B,u1_struct_0(A))
=> ! [C] :
( ( v1_pre_topc(C)
& v2_pre_topc(C)
& l1_pre_topc(C) )
=> ( C = k16_borsuk_1(A,B)
<=> ( u1_struct_0(C) = B
& u1_pre_topc(C) = a_2_2_borsuk_1(A,B) ) ) ) ) ) ).
fof(d12_borsuk_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& m1_pre_topc(B,A) )
=> ! [C] :
( ( ~ v1_xboole_0(C)
& m1_eqrel_1(C,u1_struct_0(B)) )
=> k18_borsuk_1(A,B,C) = k2_xboole_0(C,a_2_3_borsuk_1(A,B)) ) ) ) ).
fof(fraenkel_a_2_0_borsuk_1,axiom,
! [A,B,C] :
( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(B)))
=> ( r2_hidden(A,a_2_0_borsuk_1(B,C))
<=> ? [D] :
( m1_subset_1(D,k1_zfmisc_1(B))
& A = k3_tarski(D)
& r2_hidden(D,C) ) ) ) ).
fof(fraenkel_a_3_0_borsuk_1,axiom,
! [A,B,C,D] :
( ( v2_pre_topc(B)
& l1_pre_topc(B)
& v2_pre_topc(C)
& l1_pre_topc(C)
& v1_pre_topc(D)
& v2_pre_topc(D)
& l1_pre_topc(D) )
=> ( r2_hidden(A,a_3_0_borsuk_1(B,C,D))
<=> ? [E] :
( m1_subset_1(E,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(D))))
& A = k5_setfam_1(u1_struct_0(D),E)
& r1_tarski(E,a_2_1_borsuk_1(B,C)) ) ) ) ).
fof(fraenkel_a_2_1_borsuk_1,axiom,
! [A,B,C] :
( ( v2_pre_topc(B)
& l1_pre_topc(B)
& v2_pre_topc(C)
& l1_pre_topc(C) )
=> ( r2_hidden(A,a_2_1_borsuk_1(B,C))
<=> ? [D,E] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(B)))
& m1_subset_1(E,k1_zfmisc_1(u1_struct_0(C)))
& A = k12_mcart_1(u1_struct_0(B),u1_struct_0(C),D,E)
& r2_hidden(D,u1_pre_topc(B))
& r2_hidden(E,u1_pre_topc(C)) ) ) ) ).
fof(fraenkel_a_3_1_borsuk_1,axiom,
! [A,B,C,D] :
( ( v2_pre_topc(B)
& l1_pre_topc(B)
& v2_pre_topc(C)
& l1_pre_topc(C)
& m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k6_borsuk_1(B,C)))) )
=> ( r2_hidden(A,a_3_1_borsuk_1(B,C,D))
<=> ? [E,F] :
( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(B)))
& m1_subset_1(F,k1_zfmisc_1(u1_struct_0(C)))
& A = k7_borsuk_1(B,C,E,F)
& r1_tarski(k7_borsuk_1(B,C,E,F),D)
& v3_pre_topc(E,B)
& v3_pre_topc(F,C) ) ) ) ).
fof(fraenkel_a_2_2_borsuk_1,axiom,
! [A,B,C] :
( ( v2_pre_topc(B)
& l1_pre_topc(B)
& m1_eqrel_1(C,u1_struct_0(B)) )
=> ( r2_hidden(A,a_2_2_borsuk_1(B,C))
<=> ? [D] :
( m1_subset_1(D,k1_zfmisc_1(C))
& A = D
& r2_hidden(k3_tarski(D),u1_pre_topc(B)) ) ) ) ).
fof(fraenkel_a_2_3_borsuk_1,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(B)
& v2_pre_topc(B)
& l1_pre_topc(B)
& ~ v3_struct_0(C)
& m1_pre_topc(C,B) )
=> ( r2_hidden(A,a_2_3_borsuk_1(B,C))
<=> ? [D] :
( m1_subset_1(D,u1_struct_0(B))
& A = k1_struct_0(B,D)
& ~ r2_hidden(D,u1_struct_0(C)) ) ) ) ).
%------------------------------------------------------------------------------