SET007 Axioms: SET007+918.ax
%------------------------------------------------------------------------------
% File : SET007+918 : TPTP v9.0.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Tietze Extension Theorem
% 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 : tietze [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 31 ( 0 unt; 0 def)
% Number of atoms : 385 ( 13 equ)
% Maximal formula atoms : 23 ( 12 avg)
% Number of connectives : 397 ( 43 ~; 10 |; 199 &)
% ( 3 <=>; 142 =>; 0 <=; 0 <~>)
% Maximal formula depth : 21 ( 12 avg)
% Maximal term depth : 5 ( 1 avg)
% Number of predicates : 46 ( 45 usr; 0 prp; 1-4 aty)
% Number of functors : 48 ( 48 usr; 7 con; 0-6 aty)
% Number of variables : 126 ( 123 !; 3 ?)
% SPC :
% Comments : The individual reference can be found in [Mat90] by looking for
% the name provided by [Urb08].
% : Translated by MPTP from the Mizar Mathematical Library 4.48.930.
% : These set theory axioms are used in encodings of problems in
% various domains, including ALG, CAT, GRP, LAT, SET, and TOP.
%------------------------------------------------------------------------------
fof(rc1_tietze,axiom,
? [A] :
( m1_relset_1(A,k5_numbers,k1_numbers)
& ~ v1_xboole_0(A)
& v1_relat_1(A)
& v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& v1_series_1(A)
& v5_seqm_3(A)
& v1_partfun1(A,k5_numbers,k1_numbers)
& v1_seq_1(A)
& v4_seq_2(A) ) ).
fof(rc2_tietze,axiom,
! [A,B] :
( ( 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_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(fc1_tietze,axiom,
! [A,B,C,D] :
( ( l1_pre_topc(A)
& l1_pre_topc(B)
& v1_xboole_0(C)
& m1_subset_1(C,k1_zfmisc_1(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_xboole_0(k7_relat_1(D,C))
& v1_relat_1(k7_relat_1(D,C))
& v1_funct_1(k7_relat_1(D,C))
& v2_funct_1(k7_relat_1(D,C))
& v1_membered(k7_relat_1(D,C))
& v2_membered(k7_relat_1(D,C))
& v3_membered(k7_relat_1(D,C))
& v4_membered(k7_relat_1(D,C))
& v5_membered(k7_relat_1(D,C)) ) ) ).
fof(fc2_tietze,axiom,
! [A,B] :
( ( v2_pre_topc(A)
& l1_pre_topc(A)
& v4_pre_topc(B,A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ( v1_pre_topc(k3_pre_topc(A,B))
& v2_pre_topc(k3_pre_topc(A,B))
& v1_borsuk_1(k3_pre_topc(A,B),A) ) ) ).
fof(t1_tietze,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v1_xreal_0(C)
=> ( r1_xreal_0(k17_complex1(k6_xcmplx_0(A,B)),C)
=> ( r1_xreal_0(k6_xcmplx_0(B,C),A)
& r1_xreal_0(A,k2_xcmplx_0(B,C)) ) ) ) ) ) ).
fof(t2_tietze,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ( ~ r1_xreal_0(B,A)
=> r1_xboole_0(k2_limfunc1(A),k3_limfunc1(B)) ) ) ) ).
fof(t3_tietze,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ( r1_xreal_0(A,B)
=> r1_xboole_0(k12_prob_1(A),k4_limfunc1(B)) ) ) ) ).
fof(t4_tietze,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_seq_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v1_seq_1(B) )
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C)
& v1_seq_1(C) )
=> ( r1_tarski(A,B)
=> r1_tarski(k4_seq_1(C,A),k4_seq_1(C,B)) ) ) ) ) ).
fof(t5_tietze,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_seq_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v1_seq_1(B) )
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C)
& v1_seq_1(C) )
=> ( r1_tarski(A,B)
=> r1_tarski(k4_seq_1(A,C),k4_seq_1(B,C)) ) ) ) ) ).
fof(d1_tietze,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_seq_1(A) )
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( r1_tietze(A,B,C)
<=> ! [D] :
( r2_hidden(D,k3_xboole_0(C,k1_relat_1(A)))
=> r1_xreal_0(k18_complex1(k1_seq_1(A,D)),B) ) ) ) ) ).
fof(t6_tietze,axiom,
! [A] :
( ( v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( v2_pre_topc(B)
& l1_pre_topc(B) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
& m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
=> v5_pre_topc(C,A,B) ) ) ) ).
fof(t7_tietze,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& v1_series_1(A)
& m2_relset_1(A,k5_numbers,k1_numbers) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_numbers)
& v1_series_1(B)
& m2_relset_1(B,k5_numbers,k1_numbers) )
=> ( ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> r1_xreal_0(k2_seq_1(k5_numbers,k1_numbers,A,C),k2_seq_1(k5_numbers,k1_numbers,B,C)) )
=> r1_xreal_0(k2_series_1(A),k2_series_1(B)) ) ) ) ).
fof(t8_tietze,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& m2_relset_1(A,k5_numbers,k1_numbers) )
=> ( v2_series_1(A)
=> r1_xreal_0(k18_complex1(k2_series_1(A)),k2_series_1(k7_partfun3(k5_numbers,A))) ) ) ).
fof(t9_tietze,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& m2_relset_1(A,k5_numbers,k1_numbers) )
=> ! [B] :
( ( v1_xreal_0(B)
& v2_xreal_0(B) )
=> ! [C] :
( ( v1_xreal_0(C)
& v2_xreal_0(C) )
=> ( ! [D] :
( v4_ordinal2(D)
=> r1_xreal_0(k17_complex1(k5_real_1(k2_seq_1(k5_numbers,k1_numbers,A,D),k2_seq_1(k5_numbers,k1_numbers,A,k2_xcmplx_0(D,np__1)))),k3_xcmplx_0(B,k3_power(C,D))) )
=> ( r1_xreal_0(np__1,C)
| ( v4_seq_2(A)
& ! [D] :
( v4_ordinal2(D)
=> r1_xreal_0(k17_complex1(k5_real_1(k2_seq_2(A),k2_seq_1(k5_numbers,k1_numbers,A,D))),k7_xcmplx_0(k3_xcmplx_0(B,k3_power(C,D)),k6_xcmplx_0(np__1,C))) ) ) ) ) ) ) ) ).
fof(t10_tietze,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& m2_relset_1(A,k5_numbers,k1_numbers) )
=> ! [B] :
( ( v1_xreal_0(B)
& v2_xreal_0(B) )
=> ! [C] :
( ( v1_xreal_0(C)
& v2_xreal_0(C) )
=> ( ! [D] :
( v4_ordinal2(D)
=> r1_xreal_0(k17_complex1(k5_real_1(k2_seq_1(k5_numbers,k1_numbers,A,D),k2_seq_1(k5_numbers,k1_numbers,A,k2_xcmplx_0(D,np__1)))),k3_xcmplx_0(B,k3_power(C,D))) )
=> ( r1_xreal_0(np__1,C)
| ( r1_xreal_0(k6_xcmplx_0(k2_seq_1(k5_numbers,k1_numbers,A,np__0),k7_xcmplx_0(B,k6_xcmplx_0(np__1,C))),k2_seq_2(A))
& r1_xreal_0(k2_seq_2(A),k2_xcmplx_0(k2_seq_1(k5_numbers,k1_numbers,A,np__0),k7_xcmplx_0(B,k6_xcmplx_0(np__1,C)))) ) ) ) ) ) ) ).
fof(t11_tietze,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( m1_seqfunc(C,A,k1_numbers)
=> ( r1_seqfunc(A,k1_numbers,C,B)
=> ! [D] :
( ( v1_xreal_0(D)
& v2_xreal_0(D) )
=> ! [E] :
( ( v1_xreal_0(E)
& v2_xreal_0(E) )
=> ( ! [F] :
( v4_ordinal2(F)
=> r1_tietze(k7_seq_1(A,k1_numbers,k1_seqfunc(A,k1_numbers,C,F),k1_seqfunc(A,k1_numbers,C,k2_xcmplx_0(F,np__1))),k3_xcmplx_0(D,k3_power(E,F)),B) )
=> ( r1_xreal_0(np__1,E)
| ( r3_seqfunc(A,C,B)
& ! [F] :
( v4_ordinal2(F)
=> r1_tietze(k7_seq_1(A,k1_numbers,k11_seqfunc(A,C,B),k1_seqfunc(A,k1_numbers,C,F)),k7_xcmplx_0(k3_xcmplx_0(D,k3_power(E,F)),k6_xcmplx_0(np__1,E)),B) ) ) ) ) ) ) ) ) ) ) ).
fof(t12_tietze,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( m1_seqfunc(C,A,k1_numbers)
=> ( r1_seqfunc(A,k1_numbers,C,B)
=> ! [D] :
( ( v1_xreal_0(D)
& v2_xreal_0(D) )
=> ! [E] :
( ( v1_xreal_0(E)
& v2_xreal_0(E) )
=> ( ! [F] :
( v4_ordinal2(F)
=> r1_tietze(k7_seq_1(A,k1_numbers,k1_seqfunc(A,k1_numbers,C,F),k1_seqfunc(A,k1_numbers,C,k2_xcmplx_0(F,np__1))),k3_xcmplx_0(D,k3_power(E,F)),B) )
=> ( r1_xreal_0(np__1,E)
| ! [F] :
( m1_subset_1(F,B)
=> ( r1_xreal_0(k6_xcmplx_0(k2_seq_1(A,k1_numbers,k1_seqfunc(A,k1_numbers,C,np__0),F),k7_xcmplx_0(D,k6_xcmplx_0(np__1,E))),k2_seq_1(A,k1_numbers,k11_seqfunc(A,C,B),F))
& r1_xreal_0(k2_seq_1(A,k1_numbers,k11_seqfunc(A,C,B),F),k2_xcmplx_0(k2_seq_1(A,k1_numbers,k1_seqfunc(A,k1_numbers,C,np__0),F),k7_xcmplx_0(D,k6_xcmplx_0(np__1,E)))) ) ) ) ) ) ) ) ) ) ) ).
fof(t13_tietze,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( m1_seqfunc(C,A,k1_numbers)
=> ( r1_seqfunc(A,k1_numbers,C,B)
=> ! [D] :
( ( v1_xreal_0(D)
& v2_xreal_0(D) )
=> ! [E] :
( ( v1_xreal_0(E)
& v2_xreal_0(E) )
=> ! [F] :
( ( v1_funct_1(F)
& v1_funct_2(F,B,k1_numbers)
& m2_relset_1(F,B,k1_numbers) )
=> ( ! [G] :
( v4_ordinal2(G)
=> r1_tietze(k4_seq_1(k1_seqfunc(A,k1_numbers,C,G),F),k3_xcmplx_0(D,k3_power(E,G)),B) )
=> ( r1_xreal_0(np__1,E)
| ( r2_seqfunc(A,C,B)
& k11_seqfunc(A,C,B) = F ) ) ) ) ) ) ) ) ) ) ).
fof(t14_tietze,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)
& m1_pre_topc(C,A) )
=> ! [D] :
( ( ~ v3_struct_0(D)
& m1_pre_topc(D,A) )
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,u1_struct_0(C),u1_struct_0(B))
& m2_relset_1(E,u1_struct_0(C),u1_struct_0(B)) )
=> ! [F] :
( ( v1_funct_1(F)
& v1_funct_2(F,u1_struct_0(D),u1_struct_0(B))
& m2_relset_1(F,u1_struct_0(D),u1_struct_0(B)) )
=> ( ( r1_tsep_1(A,C,D)
| k3_tmap_1(A,B,C,k2_tsep_1(A,C,D),E) = k3_tmap_1(A,B,D,k2_tsep_1(A,C,D),F) )
=> ! [G] :
( m1_subset_1(G,u1_struct_0(A))
=> ( ( r2_hidden(G,u1_struct_0(C))
=> k1_funct_1(k10_tmap_1(A,B,C,D,E,F),G) = k1_funct_1(E,G) )
& ( r2_hidden(G,u1_struct_0(D))
=> k1_funct_1(k10_tmap_1(A,B,C,D,E,F),G) = k1_funct_1(F,G) ) ) ) ) ) ) ) ) ) ) ).
fof(t15_tietze,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)
& m1_pre_topc(C,A) )
=> ! [D] :
( ( ~ v3_struct_0(D)
& m1_pre_topc(D,A) )
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,u1_struct_0(C),u1_struct_0(B))
& m2_relset_1(E,u1_struct_0(C),u1_struct_0(B)) )
=> ! [F] :
( ( v1_funct_1(F)
& v1_funct_2(F,u1_struct_0(D),u1_struct_0(B))
& m2_relset_1(F,u1_struct_0(D),u1_struct_0(B)) )
=> ( ( r1_tsep_1(A,C,D)
| k3_tmap_1(A,B,C,k2_tsep_1(A,C,D),E) = k3_tmap_1(A,B,D,k2_tsep_1(A,C,D),F) )
=> r1_tarski(k1_pscomp_1(u1_struct_0(k1_tsep_1(A,C,D)),u1_struct_0(B),k10_tmap_1(A,B,C,D,E,F)),k4_subset_1(u1_struct_0(B),k1_pscomp_1(u1_struct_0(C),u1_struct_0(B),E),k1_pscomp_1(u1_struct_0(D),u1_struct_0(B),F))) ) ) ) ) ) ) ) ).
fof(t16_tietze,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)
& m1_pre_topc(C,A) )
=> ! [D] :
( ( ~ v3_struct_0(D)
& m1_pre_topc(D,A) )
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,u1_struct_0(C),u1_struct_0(B))
& m2_relset_1(E,u1_struct_0(C),u1_struct_0(B)) )
=> ! [F] :
( ( v1_funct_1(F)
& v1_funct_2(F,u1_struct_0(D),u1_struct_0(B))
& m2_relset_1(F,u1_struct_0(D),u1_struct_0(B)) )
=> ( ( r1_tsep_1(A,C,D)
| k3_tmap_1(A,B,C,k2_tsep_1(A,C,D),E) = k3_tmap_1(A,B,D,k2_tsep_1(A,C,D),F) )
=> ( ! [G] :
( m1_subset_1(G,k1_zfmisc_1(u1_struct_0(C)))
=> k4_pre_topc(k1_tsep_1(A,C,D),B,k10_tmap_1(A,B,C,D,E,F),G) = k4_pre_topc(C,B,E,G) )
& ! [G] :
( m1_subset_1(G,k1_zfmisc_1(u1_struct_0(D)))
=> k4_pre_topc(k1_tsep_1(A,C,D),B,k10_tmap_1(A,B,C,D,E,F),G) = k4_pre_topc(D,B,F,G) ) ) ) ) ) ) ) ) ) ).
fof(t17_tietze,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B,C] :
( ( v1_relat_1(C)
& v1_funct_1(C)
& v1_seq_1(C) )
=> ! [D] :
( ( v1_relat_1(D)
& v1_funct_1(D)
& v1_seq_1(D) )
=> ( ( r1_tarski(C,D)
& r1_tietze(D,A,B) )
=> r1_tietze(C,A,B) ) ) ) ) ).
fof(t18_tietze,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B,C] :
( ( v1_relat_1(C)
& v1_funct_1(C)
& v1_seq_1(C) )
=> ! [D] :
( ( v1_relat_1(D)
& v1_funct_1(D)
& v1_seq_1(D) )
=> ( ( k7_relat_1(C,B) = k7_relat_1(D,B)
& r1_tietze(C,A,B) )
=> ( ( ~ r1_tarski(B,k1_relat_1(C))
& ~ r1_tarski(k1_relat_1(D),k1_relat_1(C)) )
| r1_tietze(D,A,B) ) ) ) ) ) ).
fof(t19_tietze,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( ( ~ v3_struct_0(B)
& v2_pre_topc(B)
& l1_pre_topc(B) )
=> ! [C] :
( ( v4_pre_topc(C,B)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(B))) )
=> ( v5_compts_1(B)
=> ( r1_xreal_0(A,np__0)
| ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,u1_struct_0(k3_pre_topc(B,C)),u1_struct_0(k3_topmetr))
& v5_pre_topc(D,k3_pre_topc(B,C),k3_topmetr)
& m2_relset_1(D,u1_struct_0(k3_pre_topc(B,C)),u1_struct_0(k3_topmetr)) )
=> ~ ( r1_tietze(D,A,C)
& ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,u1_struct_0(B),u1_struct_0(k3_topmetr))
& v5_pre_topc(E,B,k3_topmetr)
& m2_relset_1(E,u1_struct_0(B),u1_struct_0(k3_topmetr)) )
=> ~ ( r1_tietze(E,k7_xcmplx_0(A,np__3),k4_relset_1(u1_struct_0(B),u1_struct_0(k3_topmetr),E))
& r1_tietze(k4_seq_1(D,E),k7_xcmplx_0(k3_xcmplx_0(np__2,A),np__3),C) ) ) ) ) ) ) ) ) ) ).
fof(t20_tietze,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ( ! [B] :
( ( ~ v1_xboole_0(B)
& v4_pre_topc(B,A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ! [C] :
( ( ~ v1_xboole_0(C)
& v4_pre_topc(C,A)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
=> ~ ( r1_subset_1(B,C)
& ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,u1_struct_0(A),u1_struct_0(k3_topmetr))
& v5_pre_topc(D,A,k3_topmetr)
& m2_relset_1(D,u1_struct_0(A),u1_struct_0(k3_topmetr)) )
=> ~ ( k4_pre_topc(A,k3_topmetr,D,B) = k1_tarski(np__0)
& k4_pre_topc(A,k3_topmetr,D,C) = k1_tarski(np__1) ) ) ) ) )
=> v5_compts_1(A) ) ) ).
fof(t21_tietze,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,u1_struct_0(A),u1_struct_0(k3_topmetr))
& m2_relset_1(B,u1_struct_0(A),u1_struct_0(k3_topmetr)) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( r1_tmap_1(A,k3_topmetr,B,C)
<=> ! [D] :
( v1_xreal_0(D)
=> ~ ( ~ r1_xreal_0(D,np__0)
& ! [E] :
( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(A)))
=> ~ ( v3_pre_topc(E,A)
& r2_hidden(C,E)
& ! [F] :
( m1_subset_1(F,u1_struct_0(A))
=> ~ ( r2_hidden(F,E)
& r1_xreal_0(D,k18_complex1(k5_real_1(k2_seq_1(u1_struct_0(A),u1_struct_0(k3_topmetr),B,F),k2_seq_1(u1_struct_0(A),u1_struct_0(k3_topmetr),B,C)))) ) ) ) ) ) ) ) ) ) ) ).
fof(t22_tietze,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_seqfunc(B,u1_struct_0(A),k1_numbers)
=> ( ( r3_seqfunc(u1_struct_0(A),B,u1_struct_0(A))
& ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( v1_funct_1(k1_seqfunc(u1_struct_0(A),k1_numbers,B,C))
& v1_funct_2(k1_seqfunc(u1_struct_0(A),k1_numbers,B,C),u1_struct_0(A),u1_struct_0(k3_topmetr))
& v5_pre_topc(k1_seqfunc(u1_struct_0(A),k1_numbers,B,C),A,k3_topmetr)
& m2_relset_1(k1_seqfunc(u1_struct_0(A),k1_numbers,B,C),u1_struct_0(A),u1_struct_0(k3_topmetr)) ) ) )
=> ( v1_funct_1(k11_seqfunc(u1_struct_0(A),B,u1_struct_0(A)))
& v1_funct_2(k11_seqfunc(u1_struct_0(A),B,u1_struct_0(A)),u1_struct_0(A),u1_struct_0(k3_topmetr))
& v5_pre_topc(k11_seqfunc(u1_struct_0(A),B,u1_struct_0(A)),A,k3_topmetr)
& m2_relset_1(k11_seqfunc(u1_struct_0(A),B,u1_struct_0(A)),u1_struct_0(A),u1_struct_0(k3_topmetr)) ) ) ) ) ).
fof(t23_tietze,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,u1_struct_0(A),u1_struct_0(k3_topmetr))
& m2_relset_1(B,u1_struct_0(A),u1_struct_0(k3_topmetr)) )
=> ! [C] :
( ( v1_xreal_0(C)
& v2_xreal_0(C) )
=> ( r1_tietze(B,C,u1_struct_0(A))
<=> ( v1_funct_1(B)
& v1_funct_2(B,u1_struct_0(A),u1_struct_0(k4_topmetr(k4_xcmplx_0(C),C)))
& m2_relset_1(B,u1_struct_0(A),u1_struct_0(k4_topmetr(k4_xcmplx_0(C),C))) ) ) ) ) ) ).
fof(t24_tietze,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B,C] :
( ( v1_relat_1(C)
& v1_funct_1(C)
& v1_seq_1(C) )
=> ! [D] :
( ( v1_relat_1(D)
& v1_funct_1(D)
& v1_seq_1(D) )
=> ( r1_tietze(k4_seq_1(C,D),A,B)
=> r1_tietze(k4_seq_1(D,C),A,B) ) ) ) ) ).
fof(t25_tietze,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ( v5_compts_1(A)
=> ! [B] :
( ( v4_pre_topc(B,A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(k3_pre_topc(A,B)),u1_struct_0(k4_topmetr(k1_real_1(np__1),np__1)))
& m2_relset_1(C,u1_struct_0(k3_pre_topc(A,B)),u1_struct_0(k4_topmetr(k1_real_1(np__1),np__1))) )
=> ~ ( v5_pre_topc(C,k3_pre_topc(A,B),k4_topmetr(k1_real_1(np__1),np__1))
& ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,u1_struct_0(A),u1_struct_0(k4_topmetr(k1_real_1(np__1),np__1)))
& v5_pre_topc(D,A,k4_topmetr(k1_real_1(np__1),np__1))
& m2_relset_1(D,u1_struct_0(A),u1_struct_0(k4_topmetr(k1_real_1(np__1),np__1))) )
=> k2_partfun1(u1_struct_0(A),u1_struct_0(k4_topmetr(k1_real_1(np__1),np__1)),D,B) != C ) ) ) ) ) ) ).
fof(t26_tietze,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ( ! [B] :
( ( ~ v1_xboole_0(B)
& v4_pre_topc(B,A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(k3_pre_topc(A,B)),u1_struct_0(k4_topmetr(k1_real_1(np__1),np__1)))
& v5_pre_topc(C,k3_pre_topc(A,B),k4_topmetr(k1_real_1(np__1),np__1))
& m2_relset_1(C,u1_struct_0(k3_pre_topc(A,B)),u1_struct_0(k4_topmetr(k1_real_1(np__1),np__1))) )
=> ? [D] :
( v1_funct_1(D)
& v1_funct_2(D,u1_struct_0(A),u1_struct_0(k4_topmetr(k1_real_1(np__1),np__1)))
& v5_pre_topc(D,A,k4_topmetr(k1_real_1(np__1),np__1))
& m2_relset_1(D,u1_struct_0(A),u1_struct_0(k4_topmetr(k1_real_1(np__1),np__1)))
& k2_partfun1(u1_struct_0(A),u1_struct_0(k4_topmetr(k1_real_1(np__1),np__1)),D,B) = C ) ) )
=> v5_compts_1(A) ) ) ).
%------------------------------------------------------------------------------