SET007 Axioms: SET007+854.ax
%------------------------------------------------------------------------------
% File : SET007+854 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : On the Isomorphism of Fundamental 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 : topalg_3 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 45 ( 0 unt; 0 def)
% Number of atoms : 543 ( 46 equ)
% Maximal formula atoms : 39 ( 12 avg)
% Number of connectives : 558 ( 60 ~; 2 |; 291 &)
% ( 3 <=>; 202 =>; 0 <=; 0 <~>)
% Maximal formula depth : 32 ( 12 avg)
% Maximal term depth : 6 ( 1 avg)
% Number of predicates : 42 ( 41 usr; 0 prp; 1-6 aty)
% Number of functors : 48 ( 48 usr; 6 con; 0-6 aty)
% Number of variables : 207 ( 203 !; 4 ?)
% 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(cc1_topalg_3,axiom,
! [A] :
( l1_pre_topc(A)
=> ( v3_struct_0(A)
=> v1_connsp_1(A) ) ) ).
fof(fc1_topalg_3,axiom,
! [A] :
( ( v2_pre_topc(A)
& v1_connsp_1(A)
& l1_pre_topc(A) )
=> ( v1_pre_topc(g1_pre_topc(u1_struct_0(A),u1_pre_topc(A)))
& v2_pre_topc(g1_pre_topc(u1_struct_0(A),u1_pre_topc(A)))
& v1_connsp_1(g1_pre_topc(u1_struct_0(A),u1_pre_topc(A))) ) ) ).
fof(cc2_topalg_3,axiom,
! [A] :
( l1_pre_topc(A)
=> ( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& v3_realset2(A) )
=> ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& v1_borsuk_2(A)
& v1_connsp_1(A) ) ) ) ).
fof(rc1_topalg_3,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> ? [B] :
( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
& ~ v1_xboole_0(B)
& v1_tops_2(B,A)
& v2_tops_2(B,A)
& v4_taxonom2(B) ) ) ).
fof(fc2_topalg_3,axiom,
! [A,B] :
( ( v2_pre_topc(A)
& v1_connsp_1(A)
& l1_pre_topc(A)
& v2_pre_topc(B)
& v1_connsp_1(B)
& l1_pre_topc(B) )
=> ( v1_pre_topc(k6_borsuk_1(A,B))
& v2_pre_topc(k6_borsuk_1(A,B))
& v1_connsp_1(k6_borsuk_1(A,B)) ) ) ).
fof(cc3_topalg_3,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_borsuk_2(C,A,B,B)
=> ( v1_relat_1(C)
& v5_pre_topc(C,k5_topmetr,A) ) ) ) ).
fof(t1_topalg_3,axiom,
! [A,B,C,D,E] :
( ( v1_funct_1(E)
& v1_funct_2(E,A,B)
& m2_relset_1(E,A,B) )
=> ( ( r2_hidden(C,A)
& r2_hidden(D,B) )
=> ( v1_funct_1(k1_funct_4(E,k3_cqc_lang(C,D)))
& v1_funct_2(k1_funct_4(E,k3_cqc_lang(C,D)),A,B)
& m2_relset_1(k1_funct_4(E,k3_cqc_lang(C,D)),A,B) ) ) ) ).
fof(t2_topalg_3,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ! [B,C] :
( ( v2_funct_1(k7_relat_1(A,B))
& r2_hidden(C,k2_relat_1(k7_relat_1(A,B))) )
=> k1_funct_1(k5_relat_1(k2_funct_1(k7_relat_1(A,B)),A),C) = C ) ) ).
fof(t3_topalg_3,axiom,
! [A,B,C,D,E,F] :
( ( v1_funct_1(F)
& v1_funct_2(F,k2_zfmisc_1(C,D),E)
& m2_relset_1(F,k2_zfmisc_1(C,D),E) )
=> ! [G] :
( ( v1_relat_1(G)
& v1_funct_1(G) )
=> ( ( r2_hidden(A,C)
& r2_hidden(B,D) )
=> ( E = k1_xboole_0
| k1_binop_1(k5_relat_1(F,G),A,B) = k1_funct_1(G,k1_binop_1(F,A,B)) ) ) ) ) ).
fof(t4_topalg_3,axiom,
! [A,B,C,D] :
( ( v1_funct_1(D)
& v1_funct_2(D,A,k2_tarski(B,C))
& m2_relset_1(D,A,k2_tarski(B,C)) )
=> A = k4_subset_1(A,k3_funct_2(A,k2_tarski(B,C),D,k1_tarski(B)),k3_funct_2(A,k2_tarski(B,C),D,k1_tarski(C))) ) ).
fof(t5_topalg_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_struct_0(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& l1_struct_0(B) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(B))
=> k8_funct_2(u1_struct_0(A),u1_struct_0(B),k3_borsuk_1(A,B,D),C) = D ) ) ) ) ).
fof(t6_topalg_3,axiom,
! [A] :
( ( ~ v3_struct_0(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)))
=> ( C = k1_struct_0(A,B)
=> k1_tex_2(A,B) = k3_pre_topc(A,C) ) ) ) ) ).
fof(t7_topalg_3,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)))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(g1_pre_topc(u1_struct_0(A),u1_pre_topc(A)))))
=> ! [E] :
( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(g1_pre_topc(u1_struct_0(A),u1_pre_topc(A)))))
=> ( ( B = D
& C = E )
=> ( r1_connsp_1(A,B,C)
<=> r1_connsp_1(g1_pre_topc(u1_struct_0(A),u1_pre_topc(A)),D,E) ) ) ) ) ) ) ) ).
fof(t8_topalg_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ( v1_connsp_1(A)
<=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,u1_struct_0(A),u1_struct_0(k2_pcomps_1(k2_tarski(np__0,np__1))))
& m2_relset_1(B,u1_struct_0(A),u1_struct_0(k2_pcomps_1(k2_tarski(np__0,np__1)))) )
=> ~ ( v5_pre_topc(B,A,k2_pcomps_1(k2_tarski(np__0,np__1)))
& v2_funct_2(B,u1_struct_0(A),u1_struct_0(k2_pcomps_1(k2_tarski(np__0,np__1)))) ) ) ) ) ).
fof(t9_topalg_3,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> ( v1_connsp_1(g1_pre_topc(u1_struct_0(A),u1_pre_topc(A)))
=> v1_connsp_1(A) ) ) ).
fof(t10_topalg_3,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) )
=> ( ( r1_borsuk_3(A,B)
& v1_borsuk_2(A) )
=> v1_borsuk_2(B) ) ) ) ).
fof(t11_topalg_3,axiom,
! [A] :
( m1_pre_topc(A,k15_euclid(np__2))
=> ( ( v1_topreal2(u1_struct_0(A))
& m1_subset_1(u1_struct_0(A),k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
=> v1_borsuk_2(A) ) ) ).
fof(t12_topalg_3,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))))
& B = k1_tarski(u1_struct_0(A))
& r1_pre_topc(A,B)
& v1_tops_2(B,A) ) ) ).
fof(t13_topalg_3,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( v1_tops_2(B,A)
& v4_taxonom2(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)))
=> ! [D] :
( ( v2_connsp_1(C,A)
& r2_hidden(D,B)
& r1_compts_1(A,B,C) )
=> ( r1_xboole_0(D,C)
| r1_tarski(C,D) ) ) ) ) ) ).
fof(t14_topalg_3,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( v2_pre_topc(B)
& l1_pre_topc(B) )
=> g1_pre_topc(u1_struct_0(k6_borsuk_1(A,B)),u1_pre_topc(k6_borsuk_1(A,B))) = k6_borsuk_1(g1_pre_topc(u1_struct_0(A),u1_pre_topc(A)),g1_pre_topc(u1_struct_0(B),u1_pre_topc(B))) ) ) ).
fof(t15_topalg_3,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)))
=> k6_pre_topc(k6_borsuk_1(A,B),k7_borsuk_1(A,B,C,D)) = k7_borsuk_1(A,B,k6_pre_topc(A,C),k6_pre_topc(B,D)) ) ) ) ) ).
fof(t16_topalg_3,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( v2_pre_topc(B)
& l1_pre_topc(B) )
=> ! [C] :
( ( v4_pre_topc(C,A)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
=> ! [D] :
( ( v4_pre_topc(D,B)
& m1_subset_1(D,k1_zfmisc_1(u1_struct_0(B))) )
=> v4_pre_topc(k7_borsuk_1(A,B,C,D),k6_borsuk_1(A,B)) ) ) ) ) ).
fof(t17_topalg_3,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)))
=> ( ( v2_connsp_1(C,A)
& v2_connsp_1(D,B) )
=> v2_connsp_1(k7_borsuk_1(A,B,C,D),k6_borsuk_1(A,B)) ) ) ) ) ) ).
fof(t18_topalg_3,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( v2_pre_topc(B)
& l1_pre_topc(B) )
=> ! [C] :
( ( ~ v3_struct_0(C)
& v2_pre_topc(C)
& l1_pre_topc(C) )
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,u1_struct_0(k6_borsuk_1(A,B)),u1_struct_0(C))
& m2_relset_1(E,u1_struct_0(k6_borsuk_1(A,B)),u1_struct_0(C)) )
=> ! [F] :
( ( v1_funct_1(F)
& v1_funct_2(F,u1_struct_0(k6_borsuk_1(k3_pre_topc(A,D),B)),u1_struct_0(C))
& m2_relset_1(F,u1_struct_0(k6_borsuk_1(k3_pre_topc(A,D),B)),u1_struct_0(C)) )
=> ( ( F = k8_relset_1(u1_struct_0(k6_borsuk_1(A,B)),u1_struct_0(C),E,k2_zfmisc_1(D,u1_struct_0(B)))
& v5_pre_topc(E,k6_borsuk_1(A,B),C) )
=> v5_pre_topc(F,k6_borsuk_1(k3_pre_topc(A,D),B),C) ) ) ) ) ) ) ) ).
fof(t19_topalg_3,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( v2_pre_topc(B)
& l1_pre_topc(B) )
=> ! [C] :
( ( ~ v3_struct_0(C)
& v2_pre_topc(C)
& l1_pre_topc(C) )
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(B)))
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,u1_struct_0(k6_borsuk_1(A,B)),u1_struct_0(C))
& m2_relset_1(E,u1_struct_0(k6_borsuk_1(A,B)),u1_struct_0(C)) )
=> ! [F] :
( ( v1_funct_1(F)
& v1_funct_2(F,u1_struct_0(k6_borsuk_1(A,k3_pre_topc(B,D))),u1_struct_0(C))
& m2_relset_1(F,u1_struct_0(k6_borsuk_1(A,k3_pre_topc(B,D))),u1_struct_0(C)) )
=> ( ( F = k8_relset_1(u1_struct_0(k6_borsuk_1(A,B)),u1_struct_0(C),E,k2_zfmisc_1(u1_struct_0(A),D))
& v5_pre_topc(E,k6_borsuk_1(A,B),C) )
=> v5_pre_topc(F,k6_borsuk_1(A,k3_pre_topc(B,D)),C) ) ) ) ) ) ) ) ).
fof(t20_topalg_3,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] :
( ( ~ v3_struct_0(C)
& v2_pre_topc(C)
& l1_pre_topc(C) )
=> ! [D] :
( ( ~ v3_struct_0(D)
& v2_pre_topc(D)
& l1_pre_topc(D) )
=> ! [E] :
( ( ~ v3_struct_0(E)
& v2_pre_topc(E)
& l1_pre_topc(E) )
=> ! [F] :
( ( v1_funct_1(F)
& v1_funct_2(F,u1_struct_0(k6_borsuk_1(E,C)),u1_struct_0(A))
& m2_relset_1(F,u1_struct_0(k6_borsuk_1(E,C)),u1_struct_0(A)) )
=> ! [G] :
( ( v1_funct_1(G)
& v1_funct_2(G,u1_struct_0(k6_borsuk_1(E,D)),u1_struct_0(A))
& m2_relset_1(G,u1_struct_0(k6_borsuk_1(E,D)),u1_struct_0(A)) )
=> ! [H] :
( ( v4_pre_topc(H,B)
& m1_subset_1(H,k1_zfmisc_1(u1_struct_0(B))) )
=> ! [I] :
( ( v4_pre_topc(I,B)
& m1_subset_1(I,k1_zfmisc_1(u1_struct_0(B))) )
=> ~ ( m1_pre_topc(C,B)
& m1_pre_topc(D,B)
& H = k2_pre_topc(C)
& I = k2_pre_topc(D)
& k2_xboole_0(k2_pre_topc(C),k2_pre_topc(D)) = k2_pre_topc(B)
& v5_pre_topc(F,k6_borsuk_1(E,C),A)
& v5_pre_topc(G,k6_borsuk_1(E,D),A)
& ! [J] :
( r2_hidden(J,k3_xboole_0(k2_pre_topc(k6_borsuk_1(E,C)),k2_pre_topc(k6_borsuk_1(E,D))))
=> k1_funct_1(F,J) = k1_funct_1(G,J) )
& ! [J] :
( ( v1_funct_1(J)
& v1_funct_2(J,u1_struct_0(k6_borsuk_1(E,B)),u1_struct_0(A))
& m2_relset_1(J,u1_struct_0(k6_borsuk_1(E,B)),u1_struct_0(A)) )
=> ~ ( J = k1_funct_4(F,G)
& v5_pre_topc(J,k6_borsuk_1(E,B),A) ) ) ) ) ) ) ) ) ) ) ) ) ).
fof(t21_topalg_3,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] :
( ( ~ v3_struct_0(C)
& v2_pre_topc(C)
& l1_pre_topc(C) )
=> ! [D] :
( ( ~ v3_struct_0(D)
& v2_pre_topc(D)
& l1_pre_topc(D) )
=> ! [E] :
( ( ~ v3_struct_0(E)
& v2_pre_topc(E)
& l1_pre_topc(E) )
=> ! [F] :
( ( v1_funct_1(F)
& v1_funct_2(F,u1_struct_0(k6_borsuk_1(C,E)),u1_struct_0(A))
& m2_relset_1(F,u1_struct_0(k6_borsuk_1(C,E)),u1_struct_0(A)) )
=> ! [G] :
( ( v1_funct_1(G)
& v1_funct_2(G,u1_struct_0(k6_borsuk_1(D,E)),u1_struct_0(A))
& m2_relset_1(G,u1_struct_0(k6_borsuk_1(D,E)),u1_struct_0(A)) )
=> ! [H] :
( ( v4_pre_topc(H,B)
& m1_subset_1(H,k1_zfmisc_1(u1_struct_0(B))) )
=> ! [I] :
( ( v4_pre_topc(I,B)
& m1_subset_1(I,k1_zfmisc_1(u1_struct_0(B))) )
=> ~ ( m1_pre_topc(C,B)
& m1_pre_topc(D,B)
& H = k2_pre_topc(C)
& I = k2_pre_topc(D)
& k2_xboole_0(k2_pre_topc(C),k2_pre_topc(D)) = k2_pre_topc(B)
& v5_pre_topc(F,k6_borsuk_1(C,E),A)
& v5_pre_topc(G,k6_borsuk_1(D,E),A)
& ! [J] :
( r2_hidden(J,k3_xboole_0(k2_pre_topc(k6_borsuk_1(C,E)),k2_pre_topc(k6_borsuk_1(D,E))))
=> k1_funct_1(F,J) = k1_funct_1(G,J) )
& ! [J] :
( ( v1_funct_1(J)
& v1_funct_2(J,u1_struct_0(k6_borsuk_1(B,E)),u1_struct_0(A))
& m2_relset_1(J,u1_struct_0(k6_borsuk_1(B,E)),u1_struct_0(A)) )
=> ~ ( J = k1_funct_4(F,G)
& v5_pre_topc(J,k6_borsuk_1(B,E),A) ) ) ) ) ) ) ) ) ) ) ) ) ).
fof(t22_topalg_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k22_borsuk_1))
=> ! [D] :
( ( v5_seqm_3(D)
& m1_borsuk_2(D,A,B,B) )
=> k8_funct_2(u1_struct_0(k22_borsuk_1),u1_struct_0(A),D,C) = B ) ) ) ) ).
fof(t23_topalg_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_borsuk_2(C,A,B,B)
=> ( k1_funct_1(C,np__0) = B
& k1_funct_1(C,np__1) = B ) ) ) ) ).
fof(t24_topalg_3,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,u1_struct_0(A))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(A))
=> ( r1_borsuk_6(A,D,E)
=> r1_borsuk_6(B,k8_funct_2(u1_struct_0(A),u1_struct_0(B),C,D),k8_funct_2(u1_struct_0(A),u1_struct_0(B),C,E)) ) ) ) ) ) ) ).
fof(t25_topalg_3,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,u1_struct_0(A))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(A))
=> ! [F] :
( m1_borsuk_2(F,A,D,E)
=> ( r1_borsuk_6(A,D,E)
=> m1_borsuk_2(k7_funct_2(u1_struct_0(k5_topmetr),u1_struct_0(A),u1_struct_0(B),F,C),B,k8_funct_2(u1_struct_0(A),u1_struct_0(B),C,D),k8_funct_2(u1_struct_0(A),u1_struct_0(B),C,E)) ) ) ) ) ) ) ) ).
fof(t26_topalg_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& v1_borsuk_2(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,u1_struct_0(A))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(A))
=> ! [F] :
( m1_borsuk_2(F,A,D,E)
=> m1_borsuk_2(k4_borsuk_1(k5_topmetr,A,B,F,C),B,k8_funct_2(u1_struct_0(A),u1_struct_0(B),C,D),k8_funct_2(u1_struct_0(A),u1_struct_0(B),C,E)) ) ) ) ) ) ) ).
fof(t27_topalg_3,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,u1_struct_0(A))
=> ! [E] :
( m1_borsuk_2(E,A,D,D)
=> m1_borsuk_2(k4_borsuk_1(k5_topmetr,A,B,E,C),B,k8_funct_2(u1_struct_0(A),u1_struct_0(B),C,D),k8_funct_2(u1_struct_0(A),u1_struct_0(B),C,D)) ) ) ) ) ) ).
fof(t28_topalg_3,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,u1_struct_0(A))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(A))
=> ! [F] :
( m1_borsuk_2(F,A,D,E)
=> ! [G] :
( m1_borsuk_2(G,A,D,E)
=> ! [H] :
( m1_borsuk_2(H,B,k8_funct_2(u1_struct_0(A),u1_struct_0(B),C,D),k8_funct_2(u1_struct_0(A),u1_struct_0(B),C,E))
=> ! [I] :
( m1_borsuk_2(I,B,k8_funct_2(u1_struct_0(A),u1_struct_0(B),C,D),k8_funct_2(u1_struct_0(A),u1_struct_0(B),C,E))
=> ( ( r3_borsuk_2(A,D,E,F,G)
& H = k7_funct_2(u1_struct_0(k5_topmetr),u1_struct_0(A),u1_struct_0(B),F,C)
& I = k7_funct_2(u1_struct_0(k5_topmetr),u1_struct_0(A),u1_struct_0(B),G,C) )
=> r3_borsuk_2(B,k8_funct_2(u1_struct_0(A),u1_struct_0(B),C,D),k8_funct_2(u1_struct_0(A),u1_struct_0(B),C,E),H,I) ) ) ) ) ) ) ) ) ) ) ).
fof(t29_topalg_3,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,u1_struct_0(A))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(A))
=> ! [F] :
( m1_borsuk_2(F,A,D,E)
=> ! [G] :
( m1_borsuk_2(G,A,D,E)
=> ! [H] :
( m1_borsuk_2(H,B,k8_funct_2(u1_struct_0(A),u1_struct_0(B),C,D),k8_funct_2(u1_struct_0(A),u1_struct_0(B),C,E))
=> ! [I] :
( m1_borsuk_2(I,B,k8_funct_2(u1_struct_0(A),u1_struct_0(B),C,D),k8_funct_2(u1_struct_0(A),u1_struct_0(B),C,E))
=> ! [J] :
( m1_borsuk_6(J,A,D,E,F,G)
=> ( ( r3_borsuk_2(A,D,E,F,G)
& H = k7_funct_2(u1_struct_0(k5_topmetr),u1_struct_0(A),u1_struct_0(B),F,C)
& I = k7_funct_2(u1_struct_0(k5_topmetr),u1_struct_0(A),u1_struct_0(B),G,C) )
=> m1_borsuk_6(k7_funct_2(u1_struct_0(k6_borsuk_1(k5_topmetr,k5_topmetr)),u1_struct_0(A),u1_struct_0(B),J,C),B,k8_funct_2(u1_struct_0(A),u1_struct_0(B),C,D),k8_funct_2(u1_struct_0(A),u1_struct_0(B),C,E),H,I) ) ) ) ) ) ) ) ) ) ) ) ).
fof(t30_topalg_3,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,u1_struct_0(A))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(A))
=> ! [F] :
( m1_subset_1(F,u1_struct_0(A))
=> ! [G] :
( m1_borsuk_2(G,A,D,E)
=> ! [H] :
( m1_borsuk_2(H,A,E,F)
=> ! [I] :
( m1_borsuk_2(I,B,k8_funct_2(u1_struct_0(A),u1_struct_0(B),C,D),k8_funct_2(u1_struct_0(A),u1_struct_0(B),C,E))
=> ! [J] :
( m1_borsuk_2(J,B,k8_funct_2(u1_struct_0(A),u1_struct_0(B),C,E),k8_funct_2(u1_struct_0(A),u1_struct_0(B),C,F))
=> ( ( r1_borsuk_6(A,D,E)
& r1_borsuk_6(A,E,F)
& I = k7_funct_2(u1_struct_0(k5_topmetr),u1_struct_0(A),u1_struct_0(B),G,C)
& J = k7_funct_2(u1_struct_0(k5_topmetr),u1_struct_0(A),u1_struct_0(B),H,C) )
=> k1_borsuk_2(B,k8_funct_2(u1_struct_0(A),u1_struct_0(B),C,D),k8_funct_2(u1_struct_0(A),u1_struct_0(B),C,E),k8_funct_2(u1_struct_0(A),u1_struct_0(B),C,F),I,J) = k7_funct_2(u1_struct_0(k5_topmetr),u1_struct_0(A),u1_struct_0(B),k1_borsuk_2(A,D,E,F,G,H),C) ) ) ) ) ) ) ) ) ) ) ) ).
fof(t31_topalg_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k3_topalg_1(A,B)))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(k3_topalg_1(A,B)))
=> ! [E] :
( m1_borsuk_2(E,A,B,B)
=> ! [F] :
( m1_borsuk_2(F,A,B,B)
=> ( ( C = k6_eqrel_1(k1_topalg_1(A,B),k2_topalg_1(A,B),E)
& D = k6_eqrel_1(k1_topalg_1(A,B),k2_topalg_1(A,B),F) )
=> k1_group_1(k3_topalg_1(A,B),C,D) = k6_eqrel_1(k1_topalg_1(A,B),k2_topalg_1(A,B),k1_borsuk_2(A,B,B,B,E,F)) ) ) ) ) ) ) ) ).
fof(d1_topalg_3,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(A),u1_struct_0(B))
& m2_relset_1(D,u1_struct_0(A),u1_struct_0(B)) )
=> ( v5_pre_topc(D,A,B)
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,u1_struct_0(k3_topalg_1(A,C)),u1_struct_0(k3_topalg_1(B,k8_funct_2(u1_struct_0(A),u1_struct_0(B),D,C))))
& m2_relset_1(E,u1_struct_0(k3_topalg_1(A,C)),u1_struct_0(k3_topalg_1(B,k8_funct_2(u1_struct_0(A),u1_struct_0(B),D,C)))) )
=> ( E = k1_topalg_3(A,B,C,D)
<=> ! [F] :
( m1_subset_1(F,u1_struct_0(k3_topalg_1(A,C)))
=> ? [G] :
( m1_borsuk_2(G,A,C,C)
& ? [H] :
( m1_borsuk_2(H,B,k8_funct_2(u1_struct_0(A),u1_struct_0(B),D,C),k8_funct_2(u1_struct_0(A),u1_struct_0(B),D,C))
& F = k6_eqrel_1(k1_topalg_1(A,C),k2_topalg_1(A,C),G)
& H = k7_funct_2(u1_struct_0(k5_topmetr),u1_struct_0(A),u1_struct_0(B),G,D)
& k8_funct_2(u1_struct_0(k3_topalg_1(A,C)),u1_struct_0(k3_topalg_1(B,k8_funct_2(u1_struct_0(A),u1_struct_0(B),D,C))),E,F) = k6_eqrel_1(k1_topalg_1(B,k8_funct_2(u1_struct_0(A),u1_struct_0(B),D,C)),k2_topalg_1(B,k8_funct_2(u1_struct_0(A),u1_struct_0(B),D,C)),H) ) ) ) ) ) ) ) ) ) ) ).
fof(t32_topalg_3,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(A),u1_struct_0(B))
& v5_pre_topc(D,A,B)
& m2_relset_1(D,u1_struct_0(A),u1_struct_0(B)) )
=> ! [E] :
( m1_subset_1(E,u1_struct_0(k3_topalg_1(A,C)))
=> ! [F] :
( m1_borsuk_2(F,A,C,C)
=> ! [G] :
( m1_borsuk_2(G,B,k8_funct_2(u1_struct_0(A),u1_struct_0(B),D,C),k8_funct_2(u1_struct_0(A),u1_struct_0(B),D,C))
=> ( ( E = k6_eqrel_1(k1_topalg_1(A,C),k2_topalg_1(A,C),F)
& G = k4_borsuk_1(k5_topmetr,A,B,F,D) )
=> k8_funct_2(u1_struct_0(k3_topalg_1(A,C)),u1_struct_0(k3_topalg_1(B,k8_funct_2(u1_struct_0(A),u1_struct_0(B),D,C))),k1_topalg_3(A,B,C,D),E) = k6_eqrel_1(k1_topalg_1(B,k8_funct_2(u1_struct_0(A),u1_struct_0(B),D,C)),k2_topalg_1(B,k8_funct_2(u1_struct_0(A),u1_struct_0(B),D,C)),G) ) ) ) ) ) ) ) ) ).
fof(t33_topalg_3,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(A),u1_struct_0(B))
& v5_pre_topc(D,A,B)
& m2_relset_1(D,u1_struct_0(A),u1_struct_0(B)) )
=> ( v3_tops_2(D,A,B)
=> v4_group_6(k2_topalg_3(A,B,C,D),k3_topalg_1(A,C),k3_topalg_1(B,k8_funct_2(u1_struct_0(A),u1_struct_0(B),D,C))) ) ) ) ) ) ).
fof(t34_topalg_3,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] :
( ( v1_funct_1(E)
& v1_funct_2(E,u1_struct_0(A),u1_struct_0(B))
& v5_pre_topc(E,A,B)
& m2_relset_1(E,u1_struct_0(A),u1_struct_0(B)) )
=> ! [F] :
( m1_borsuk_2(F,B,D,k8_funct_2(u1_struct_0(A),u1_struct_0(B),E,C))
=> ! [G] :
( ( v1_funct_1(G)
& v1_funct_2(G,u1_struct_0(k3_topalg_1(A,C)),u1_struct_0(k3_topalg_1(B,D)))
& v1_group_6(G,k3_topalg_1(A,C),k3_topalg_1(B,D))
& m2_relset_1(G,u1_struct_0(k3_topalg_1(A,C)),u1_struct_0(k3_topalg_1(B,D))) )
=> ( ( v3_tops_2(E,A,B)
& r1_borsuk_6(B,k8_funct_2(u1_struct_0(A),u1_struct_0(B),E,C),D)
& G = k7_funct_2(u1_struct_0(k3_topalg_1(A,C)),u1_struct_0(k3_topalg_1(B,k8_funct_2(u1_struct_0(A),u1_struct_0(B),E,C))),u1_struct_0(k3_topalg_1(B,D)),k2_topalg_3(A,B,C,E),k4_topalg_1(B,D,k8_funct_2(u1_struct_0(A),u1_struct_0(B),E,C),F)) )
=> v4_group_6(G,k3_topalg_1(A,C),k3_topalg_1(B,D)) ) ) ) ) ) ) ) ) ).
fof(t35_topalg_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v2_pre_topc(B)
& v1_borsuk_2(B)
& l1_pre_topc(B) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(B))
=> ( r1_borsuk_3(A,B)
=> r2_group_6(k3_topalg_1(A,C),k3_topalg_1(B,D)) ) ) ) ) ) ).
fof(dt_k1_topalg_3,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))
& v1_funct_1(D)
& v1_funct_2(D,u1_struct_0(A),u1_struct_0(B))
& m1_relset_1(D,u1_struct_0(A),u1_struct_0(B)) )
=> ( v1_funct_1(k1_topalg_3(A,B,C,D))
& v1_funct_2(k1_topalg_3(A,B,C,D),u1_struct_0(k3_topalg_1(A,C)),u1_struct_0(k3_topalg_1(B,k8_funct_2(u1_struct_0(A),u1_struct_0(B),D,C))))
& m2_relset_1(k1_topalg_3(A,B,C,D),u1_struct_0(k3_topalg_1(A,C)),u1_struct_0(k3_topalg_1(B,k8_funct_2(u1_struct_0(A),u1_struct_0(B),D,C)))) ) ) ).
fof(dt_k2_topalg_3,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))
& 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(k2_topalg_3(A,B,C,D))
& v1_funct_2(k2_topalg_3(A,B,C,D),u1_struct_0(k3_topalg_1(A,C)),u1_struct_0(k3_topalg_1(B,k8_funct_2(u1_struct_0(A),u1_struct_0(B),D,C))))
& v1_group_6(k2_topalg_3(A,B,C,D),k3_topalg_1(A,C),k3_topalg_1(B,k8_funct_2(u1_struct_0(A),u1_struct_0(B),D,C)))
& m2_relset_1(k2_topalg_3(A,B,C,D),u1_struct_0(k3_topalg_1(A,C)),u1_struct_0(k3_topalg_1(B,k8_funct_2(u1_struct_0(A),u1_struct_0(B),D,C)))) ) ) ).
fof(redefinition_k2_topalg_3,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))
& 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)) )
=> k2_topalg_3(A,B,C,D) = k1_topalg_3(A,B,C,D) ) ).
%------------------------------------------------------------------------------