SET007 Axioms: SET007+925.ax
%------------------------------------------------------------------------------
% File : SET007+925 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Niemytzki Plane - an Example of Tychonoff Space Which Is Not T_4
% 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 : topgen_5 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 118 ( 18 unt; 0 def)
% Number of atoms : 838 ( 91 equ)
% Maximal formula atoms : 24 ( 7 avg)
% Number of connectives : 813 ( 93 ~; 12 |; 365 &)
% ( 24 <=>; 319 =>; 0 <=; 0 <~>)
% Maximal formula depth : 22 ( 8 avg)
% Maximal term depth : 7 ( 1 avg)
% Number of predicates : 51 ( 50 usr; 0 prp; 1-3 aty)
% Number of functors : 91 ( 91 usr; 26 con; 0-4 aty)
% Number of variables : 315 ( 288 !; 27 ?)
% 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_topgen_5,axiom,
! [A,B] :
( v2_membered(B)
=> ! [C] :
( m1_relset_1(C,A,B)
=> v1_seq_1(C) ) ) ).
fof(cc2_topgen_5,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(k1_numbers))
=> ( v1_finset_1(A)
=> ( v1_finset_1(A)
& v1_membered(A)
& v2_membered(A)
& v1_seq_4(A)
& v2_seq_4(A)
& v3_seq_4(A) ) ) ) ).
fof(fc1_topgen_5,axiom,
~ v1_xboole_0(k1_topgen_5) ).
fof(fc2_topgen_5,axiom,
~ v1_xboole_0(k2_topgen_5) ).
fof(cc3_topgen_5,axiom,
! [A] :
( l1_pre_topc(A)
=> ( ( v2_pre_topc(A)
& v1_topgen_5(A) )
=> ( v2_pre_topc(A)
& v1_urysohn1(A)
& v4_compts_1(A) ) ) ) ).
fof(cc4_topgen_5,axiom,
! [A] :
( l1_pre_topc(A)
=> ( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& v1_urysohn1(A)
& v5_compts_1(A) )
=> ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& v1_topgen_5(A) ) ) ) ).
fof(cc5_topgen_5,axiom,
! [A,B] :
( ( ~ v1_xboole_0(B)
& v2_membered(B) )
=> ! [C] :
( m1_relset_1(C,A,B)
=> v1_seq_1(C) ) ) ).
fof(fc3_topgen_5,axiom,
! [A,B] :
( ( v1_xreal_0(A)
& v1_xreal_0(B)
& v2_xreal_0(B) )
=> ( ~ v1_xboole_0(k4_topgen_5(A,B))
& v1_relat_1(k4_topgen_5(A,B))
& v1_funct_1(k4_topgen_5(A,B))
& v1_funct_2(k4_topgen_5(A,B),u1_struct_0(k3_topgen_5),u1_struct_0(k5_topmetr))
& v5_pre_topc(k4_topgen_5(A,B),k3_topgen_5,k5_topmetr)
& v1_partfun1(k4_topgen_5(A,B),u1_struct_0(k3_topgen_5),u1_struct_0(k5_topmetr))
& v1_seq_1(k4_topgen_5(A,B)) ) ) ).
fof(fc4_topgen_5,axiom,
! [A,B,C] :
( ( v1_xreal_0(A)
& v1_xreal_0(B)
& v2_xreal_0(B)
& v1_xreal_0(C)
& v2_xreal_0(C) )
=> ( ~ v1_xboole_0(k5_topgen_5(A,B,C))
& v1_relat_1(k5_topgen_5(A,B,C))
& v1_funct_1(k5_topgen_5(A,B,C))
& v1_funct_2(k5_topgen_5(A,B,C),u1_struct_0(k3_topgen_5),u1_struct_0(k5_topmetr))
& v5_pre_topc(k5_topgen_5(A,B,C),k3_topgen_5,k5_topmetr)
& v1_partfun1(k5_topgen_5(A,B,C),u1_struct_0(k3_topgen_5),u1_struct_0(k5_topmetr))
& v1_seq_1(k5_topgen_5(A,B,C)) ) ) ).
fof(t1_topgen_5,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ( r1_partfun1(A,B)
=> ! [C] : k10_relat_1(k1_funct_4(A,B),C) = k2_xboole_0(k10_relat_1(A,C),k10_relat_1(B,C)) ) ) ) ).
fof(t2_topgen_5,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ( r1_xboole_0(k1_relat_1(A),k1_relat_1(B))
=> ! [C] : k10_relat_1(k1_funct_4(A,B),C) = k2_xboole_0(k10_relat_1(A,C),k10_relat_1(B,C)) ) ) ) ).
fof(t3_topgen_5,axiom,
! [A,B,C] :
( ( v1_relat_1(C)
& v1_funct_1(C) )
=> ( r2_hidden(B,k1_relat_1(C))
=> k1_funct_1(k10_funct_6(k3_cqc_lang(A,C)),B) = k3_cqc_lang(A,k1_funct_1(C,B)) ) ) ).
fof(t4_topgen_5,axiom,
! [A,B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ( r2_hidden(A,k1_relat_1(k10_funct_6(B)))
<=> ? [C,D] :
( v1_relat_1(D)
& v1_funct_1(D)
& r2_hidden(C,k1_relat_1(B))
& D = k1_funct_1(B,C)
& r2_hidden(A,k1_relat_1(D)) ) ) ) ).
fof(t5_topgen_5,axiom,
! [A,B,C] :
( ( v1_relat_1(C)
& v1_funct_1(C) )
=> ( r2_hidden(A,k1_relat_1(k1_funct_1(k10_funct_6(C),B)))
<=> ? [D] :
( v1_relat_1(D)
& v1_funct_1(D)
& r2_hidden(A,k1_relat_1(C))
& D = k1_funct_1(C,A)
& r2_hidden(B,k1_relat_1(D)) ) ) ) ).
fof(t6_topgen_5,axiom,
! [A,B,C] :
( ( v1_relat_1(C)
& v1_funct_1(C) )
=> ! [D] :
( ( v1_relat_1(D)
& v1_funct_1(D) )
=> ( ( r2_hidden(A,k1_relat_1(C))
& D = k1_funct_1(C,A)
& r2_hidden(B,k1_relat_1(D)) )
=> k1_funct_1(k1_funct_1(k10_funct_6(C),B),A) = k1_funct_1(D,B) ) ) ) ).
fof(t7_topgen_5,axiom,
! [A,B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C) )
=> ! [D] :
( ( v1_relat_1(D)
& v1_funct_1(D) )
=> ( D = k2_xboole_0(B,C)
=> k1_funct_1(k10_funct_6(D),A) = k2_xboole_0(k1_funct_1(k10_funct_6(B),A),k1_funct_1(k10_funct_6(C),A)) ) ) ) ) ).
fof(t8_topgen_5,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v1_xreal_0(C)
=> ! [D] :
( v1_xreal_0(D)
=> ( r1_xreal_0(C,D)
=> ( r1_xreal_0(B,A)
| k5_subset_1(k1_numbers,k2_rcomp_1(A,C),k1_rcomp_1(B,D)) = k1_rcomp_2(B,C) ) ) ) ) ) ) ).
fof(t9_topgen_5,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v1_xreal_0(C)
=> ! [D] :
( v1_xreal_0(D)
=> ( r1_xreal_0(B,A)
=> ( r1_xreal_0(C,D)
| k5_subset_1(k1_numbers,k2_rcomp_1(A,C),k1_rcomp_1(B,D)) = k2_rcomp_2(A,D) ) ) ) ) ) ) ).
fof(t10_topgen_5,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v1_xreal_0(C)
=> ! [D] :
( v1_xreal_0(D)
=> ( ( r1_xreal_0(A,B)
& r1_xreal_0(C,D) )
=> ( r1_xreal_0(C,B)
| k4_subset_1(k1_numbers,k1_rcomp_2(A,C),k2_rcomp_2(B,D)) = k1_rcomp_1(A,D) ) ) ) ) ) ) ).
fof(t11_topgen_5,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v1_xreal_0(C)
=> ! [D] :
( v1_xreal_0(D)
=> ( ( r1_xreal_0(A,B)
& r1_xreal_0(C,D) )
=> ( r1_xreal_0(C,B)
| k5_subset_1(k1_numbers,k1_rcomp_2(A,C),k2_rcomp_2(B,D)) = k2_rcomp_1(B,C) ) ) ) ) ) ) ).
fof(t12_topgen_5,axiom,
! [A,B] :
( r2_tarski(k4_card_3(k10_finseq_1(A,B)),k2_zfmisc_1(A,B))
& k1_card_1(k4_card_3(k10_finseq_1(A,B))) = k2_card_2(k1_card_1(A),k1_card_1(B)) ) ).
fof(t13_topgen_5,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> k7_square_1(k5_toprns_1(np__2,k23_euclid(A,B))) = k2_xcmplx_0(k5_square_1(A),k5_square_1(B)) ) ) ).
fof(t14_topgen_5,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( ~ v3_struct_0(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,A)
& m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A))) )
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,u1_struct_0(k3_pre_topc(A,C)),u1_struct_0(B))
& v5_pre_topc(E,k3_pre_topc(A,C),B)
& m2_relset_1(E,u1_struct_0(k3_pre_topc(A,C)),u1_struct_0(B)) )
=> ! [F] :
( ( v1_funct_1(F)
& v1_funct_2(F,u1_struct_0(k3_pre_topc(A,D)),u1_struct_0(B))
& v5_pre_topc(F,k3_pre_topc(A,D),B)
& m2_relset_1(F,u1_struct_0(k3_pre_topc(A,D)),u1_struct_0(B)) )
=> ( r1_partfun1(E,F)
=> ( v1_funct_1(k1_funct_4(E,F))
& v1_funct_2(k1_funct_4(E,F),u1_struct_0(k3_pre_topc(A,k4_subset_1(u1_struct_0(A),C,D))),u1_struct_0(B))
& v5_pre_topc(k1_funct_4(E,F),k3_pre_topc(A,k4_subset_1(u1_struct_0(A),C,D)),B)
& m2_relset_1(k1_funct_4(E,F),u1_struct_0(k3_pre_topc(A,k4_subset_1(u1_struct_0(A),C,D))),u1_struct_0(B)) ) ) ) ) ) ) ) ) ).
fof(t15_topgen_5,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( ~ v3_struct_0(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,A)
& m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A))) )
=> ( r1_xboole_0(C,D)
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,u1_struct_0(k3_pre_topc(A,C)),u1_struct_0(B))
& v5_pre_topc(E,k3_pre_topc(A,C),B)
& m2_relset_1(E,u1_struct_0(k3_pre_topc(A,C)),u1_struct_0(B)) )
=> ! [F] :
( ( v1_funct_1(F)
& v1_funct_2(F,u1_struct_0(k3_pre_topc(A,D)),u1_struct_0(B))
& v5_pre_topc(F,k3_pre_topc(A,D),B)
& m2_relset_1(F,u1_struct_0(k3_pre_topc(A,D)),u1_struct_0(B)) )
=> ( v1_funct_1(k1_funct_4(E,F))
& v1_funct_2(k1_funct_4(E,F),u1_struct_0(k3_pre_topc(A,k4_subset_1(u1_struct_0(A),C,D))),u1_struct_0(B))
& v5_pre_topc(k1_funct_4(E,F),k3_pre_topc(A,k4_subset_1(u1_struct_0(A),C,D)),B)
& m2_relset_1(k1_funct_4(E,F),u1_struct_0(k3_pre_topc(A,k4_subset_1(u1_struct_0(A),C,D))),u1_struct_0(B)) ) ) ) ) ) ) ) ) ).
fof(t16_topgen_5,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v2_pre_topc(B)
& l1_pre_topc(B) )
=> ! [C] :
( ( v3_pre_topc(C,A)
& v4_pre_topc(C,A)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,u1_struct_0(k3_pre_topc(A,C)),u1_struct_0(B))
& v5_pre_topc(D,k3_pre_topc(A,C),B)
& m2_relset_1(D,u1_struct_0(k3_pre_topc(A,C)),u1_struct_0(B)) )
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,u1_struct_0(k3_pre_topc(A,k3_subset_1(u1_struct_0(A),C))),u1_struct_0(B))
& v5_pre_topc(E,k3_pre_topc(A,k3_subset_1(u1_struct_0(A),C)),B)
& m2_relset_1(E,u1_struct_0(k3_pre_topc(A,k3_subset_1(u1_struct_0(A),C))),u1_struct_0(B)) )
=> ( v1_funct_1(k1_funct_4(D,E))
& v1_funct_2(k1_funct_4(D,E),u1_struct_0(A),u1_struct_0(B))
& v5_pre_topc(k1_funct_4(D,E),A,B)
& m2_relset_1(k1_funct_4(D,E),u1_struct_0(A),u1_struct_0(B)) ) ) ) ) ) ) ).
fof(t17_topgen_5,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k15_euclid(A)))
=> ! [C] :
( ( v1_xreal_0(C)
& v2_xreal_0(C) )
=> r2_hidden(B,k1_topreal9(A,B,C)) ) ) ) ).
fof(t18_topgen_5,axiom,
! [A,B] :
( r2_hidden(k10_finseq_1(A,B),k1_topgen_5)
<=> ( r2_hidden(A,k1_numbers)
& B = np__0 ) ) ).
fof(t19_topgen_5,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ( r2_hidden(k23_euclid(A,B),k1_topgen_5)
<=> B = np__0 ) ) ) ).
fof(t20_topgen_5,axiom,
k1_card_1(k1_topgen_5) = k3_topgen_3 ).
fof(t21_topgen_5,axiom,
! [A,B] :
( r2_hidden(k10_finseq_1(A,B),k2_topgen_5)
<=> ( r2_hidden(A,k1_numbers)
& ? [C] :
( m1_subset_1(C,k1_numbers)
& B = C
& r1_xreal_0(np__0,C) ) ) ) ).
fof(t22_topgen_5,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ( r2_hidden(k23_euclid(A,B),k2_topgen_5)
<=> r1_xreal_0(np__0,B) ) ) ) ).
fof(t23_topgen_5,axiom,
r1_tarski(k1_topgen_5,k2_topgen_5) ).
fof(t24_topgen_5,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v1_xreal_0(C)
=> ( ~ r1_xreal_0(C,np__0)
=> ( r1_tarski(k1_topreal9(np__2,k23_euclid(A,B),C),k2_topgen_5)
<=> r1_xreal_0(C,B) ) ) ) ) ) ).
fof(t25_topgen_5,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v1_xreal_0(C)
=> ( r1_xreal_0(np__0,B)
=> ( r1_xreal_0(C,np__0)
| ( r1_xboole_0(k1_topreal9(np__2,k23_euclid(A,B),C),k1_topgen_5)
<=> r1_xreal_0(C,B) ) ) ) ) ) ) ).
fof(t26_topgen_5,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k15_euclid(A)))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k15_euclid(A)))
=> ! [D] :
( ( v1_xreal_0(D)
& v2_xreal_0(D) )
=> ! [E] :
( ( v1_xreal_0(E)
& v2_xreal_0(E) )
=> ( r1_xreal_0(k5_toprns_1(A,k20_euclid(A,B,C)),k6_xcmplx_0(D,E))
=> r1_tarski(k1_topreal9(A,C,E),k1_topreal9(A,B,D)) ) ) ) ) ) ) ).
fof(t27_topgen_5,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( ( v1_xreal_0(B)
& v2_xreal_0(B) )
=> ! [C] :
( ( v1_xreal_0(C)
& v2_xreal_0(C) )
=> ( r1_xreal_0(B,C)
=> r1_tarski(k1_topreal9(np__2,k23_euclid(A,B),B),k1_topreal9(np__2,k23_euclid(A,C),C)) ) ) ) ) ).
fof(t28_topgen_5,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_topgen_2(C,A)
=> ! [D] :
( m1_topgen_2(D,B)
=> ( C = D
=> g1_pre_topc(u1_struct_0(A),u1_pre_topc(A)) = g1_pre_topc(u1_struct_0(B),u1_pre_topc(B)) ) ) ) ) ) ).
fof(t29_topgen_5,axiom,
( v3_pre_topc(k1_topgen_1(u1_struct_0(k15_euclid(np__2)),k2_topgen_5,k1_topgen_5),k3_topgen_5)
& m1_subset_1(k1_topgen_1(u1_struct_0(k15_euclid(np__2)),k2_topgen_5,k1_topgen_5),k1_zfmisc_1(u1_struct_0(k3_topgen_5))) ) ).
fof(t30_topgen_5,axiom,
( v4_pre_topc(k1_topgen_5,k3_topgen_5)
& m1_subset_1(k1_topgen_5,k1_zfmisc_1(u1_struct_0(k3_topgen_5))) ) ).
fof(t31_topgen_5,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( ( v1_xreal_0(B)
& v2_xreal_0(B) )
=> ( v3_pre_topc(k4_subset_1(u1_struct_0(k15_euclid(np__2)),k1_topreal9(np__2,k23_euclid(A,B),B),k1_struct_0(k15_euclid(np__2),k23_euclid(A,np__0))),k3_topgen_5)
& m1_subset_1(k4_subset_1(u1_struct_0(k15_euclid(np__2)),k1_topreal9(np__2,k23_euclid(A,B),B),k1_struct_0(k15_euclid(np__2),k23_euclid(A,np__0))),k1_zfmisc_1(u1_struct_0(k3_topgen_5))) ) ) ) ).
fof(t32_topgen_5,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( ( v1_xreal_0(B)
& v2_xreal_0(B) )
=> ! [C] :
( ( v1_xreal_0(C)
& v2_xreal_0(C) )
=> ( v3_pre_topc(k5_subset_1(u1_struct_0(k15_euclid(np__2)),k1_topreal9(np__2,k23_euclid(A,B),C),k2_topgen_5),k3_topgen_5)
& m1_subset_1(k5_subset_1(u1_struct_0(k15_euclid(np__2)),k1_topreal9(np__2,k23_euclid(A,B),C),k2_topgen_5),k1_zfmisc_1(u1_struct_0(k3_topgen_5))) ) ) ) ) ).
fof(t33_topgen_5,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( ( v1_xreal_0(C)
& v2_xreal_0(C) )
=> ( r1_xreal_0(C,B)
=> ( v3_pre_topc(k1_topreal9(np__2,k23_euclid(A,B),C),k3_topgen_5)
& m1_subset_1(k1_topreal9(np__2,k23_euclid(A,B),C),k1_zfmisc_1(u1_struct_0(k3_topgen_5))) ) ) ) ) ) ).
fof(t34_topgen_5,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k3_topgen_5))
=> ! [B] :
( ( v1_xreal_0(B)
& v2_xreal_0(B) )
=> ? [C] :
( m1_subset_1(C,u1_struct_0(k15_euclid(np__2)))
& ? [D] :
( v3_pre_topc(D,k3_topgen_5)
& m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k3_topgen_5)))
& r2_hidden(A,D)
& r2_hidden(C,D)
& ! [E] :
( m1_subset_1(E,u1_struct_0(k15_euclid(np__2)))
=> ~ ( r2_hidden(E,D)
& r1_xreal_0(B,k5_toprns_1(np__2,k20_euclid(np__2,E,C))) ) ) ) ) ) ) ).
fof(t35_topgen_5,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( ( v1_xreal_0(C)
& v2_xreal_0(C) )
=> ? [D] :
( v1_rat_1(D)
& ? [E] :
( v1_rat_1(E)
& r2_hidden(k23_euclid(D,E),k1_topreal9(np__2,k23_euclid(A,B),C))
& k23_euclid(D,E) != k23_euclid(A,B) ) ) ) ) ) ).
fof(t36_topgen_5,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k3_topgen_5)))
=> ( A = k3_xboole_0(k1_topgen_1(u1_struct_0(k15_euclid(np__2)),k2_topgen_5,k1_topgen_5),k4_card_3(k10_finseq_1(k3_numbers,k3_numbers)))
=> ! [B] : k6_pre_topc(k3_topgen_5,k1_topgen_1(u1_struct_0(k3_topgen_5),A,k1_tarski(B))) = k2_pre_topc(k3_topgen_5) ) ) ).
fof(t37_topgen_5,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k3_topgen_5)))
=> ( A = k1_topgen_1(u1_struct_0(k15_euclid(np__2)),k2_topgen_5,k1_topgen_5)
=> ! [B] : k6_pre_topc(k3_topgen_5,k1_topgen_1(u1_struct_0(k3_topgen_5),A,k1_tarski(B))) = k2_pre_topc(k3_topgen_5) ) ) ).
fof(t38_topgen_5,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k3_topgen_5)))
=> ( A = k1_topgen_1(u1_struct_0(k15_euclid(np__2)),k2_topgen_5,k1_topgen_5)
=> k6_pre_topc(k3_topgen_5,A) = k2_pre_topc(k3_topgen_5) ) ) ).
fof(t39_topgen_5,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k3_topgen_5)))
=> ( A = k1_topgen_5
=> ( k6_pre_topc(k3_topgen_5,A) = A
& k1_tops_1(k3_topgen_5,A) = k1_xboole_0 ) ) ) ).
fof(t40_topgen_5,axiom,
( v1_tops_1(k3_xboole_0(k1_topgen_1(u1_struct_0(k15_euclid(np__2)),k2_topgen_5,k1_topgen_5),k4_card_3(k10_finseq_1(k3_numbers,k3_numbers))),k3_topgen_5)
& m1_subset_1(k3_xboole_0(k1_topgen_1(u1_struct_0(k15_euclid(np__2)),k2_topgen_5,k1_topgen_5),k4_card_3(k10_finseq_1(k3_numbers,k3_numbers))),k1_zfmisc_1(u1_struct_0(k3_topgen_5))) ) ).
fof(t41_topgen_5,axiom,
( v2_topgen_1(k3_xboole_0(k1_topgen_1(u1_struct_0(k15_euclid(np__2)),k2_topgen_5,k1_topgen_5),k4_card_3(k10_finseq_1(k3_numbers,k3_numbers))),k3_topgen_5)
& m1_subset_1(k3_xboole_0(k1_topgen_1(u1_struct_0(k15_euclid(np__2)),k2_topgen_5,k1_topgen_5),k4_card_3(k10_finseq_1(k3_numbers,k3_numbers))),k1_zfmisc_1(u1_struct_0(k3_topgen_5))) ) ).
fof(t42_topgen_5,axiom,
( v1_tops_1(k1_topgen_1(u1_struct_0(k15_euclid(np__2)),k2_topgen_5,k1_topgen_5),k3_topgen_5)
& m1_subset_1(k1_topgen_1(u1_struct_0(k15_euclid(np__2)),k2_topgen_5,k1_topgen_5),k1_zfmisc_1(u1_struct_0(k3_topgen_5))) ) ).
fof(t43_topgen_5,axiom,
( v2_topgen_1(k1_topgen_1(u1_struct_0(k15_euclid(np__2)),k2_topgen_5,k1_topgen_5),k3_topgen_5)
& m1_subset_1(k1_topgen_1(u1_struct_0(k15_euclid(np__2)),k2_topgen_5,k1_topgen_5),k1_zfmisc_1(u1_struct_0(k3_topgen_5))) ) ).
fof(t44_topgen_5,axiom,
( v3_tops_1(k1_topgen_5,k3_topgen_5)
& m1_subset_1(k1_topgen_5,k1_zfmisc_1(u1_struct_0(k3_topgen_5))) ) ).
fof(t45_topgen_5,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k3_topgen_5)))
=> ( A = k1_topgen_5
=> v1_xboole_0(k3_topgen_1(k3_topgen_5,A)) ) ) ).
fof(t46_topgen_5,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(k1_topgen_5))
=> ( v4_pre_topc(A,k3_topgen_5)
& m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k3_topgen_5))) ) ) ).
fof(t47_topgen_5,axiom,
( v1_tops_1(k3_numbers,k2_topgen_3)
& m1_subset_1(k3_numbers,k1_zfmisc_1(u1_struct_0(k2_topgen_3))) ) ).
fof(t48_topgen_5,axiom,
v7_topgen_1(k2_topgen_3) ).
fof(t49_topgen_5,axiom,
v7_topgen_1(k3_topgen_5) ).
fof(t50_topgen_5,axiom,
v1_urysohn1(k3_topgen_5) ).
fof(t51_topgen_5,axiom,
~ v5_compts_1(k3_topgen_5) ).
fof(d4_topgen_5,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> ( v1_topgen_5(A)
<=> ( v1_urysohn1(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))
=> ~ ( r2_hidden(C,k3_subset_1(u1_struct_0(A),B))
& ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,u1_struct_0(A),u1_struct_0(k5_topmetr))
& v5_pre_topc(D,A,k5_topmetr)
& m2_relset_1(D,u1_struct_0(A),u1_struct_0(k5_topmetr)) )
=> ~ ( k2_seq_1(u1_struct_0(A),u1_struct_0(k5_topmetr),D,C) = np__0
& r1_tarski(k4_pre_topc(A,k5_topmetr,D,B),k1_seq_4(np__1)) ) ) ) ) ) ) ) ) ).
fof(t52_topgen_5,axiom,
! [A] :
( ( v2_pre_topc(A)
& v1_urysohn1(A)
& l1_pre_topc(A) )
=> ( v1_topgen_5(A)
=> ! [B] :
( m2_cantor_1(B,A)
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
=> ~ ( r2_hidden(C,D)
& r2_hidden(D,B)
& ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,u1_struct_0(A),u1_struct_0(k5_topmetr))
& v5_pre_topc(E,A,k5_topmetr)
& m2_relset_1(E,u1_struct_0(A),u1_struct_0(k5_topmetr)) )
=> ~ ( k2_seq_1(u1_struct_0(A),u1_struct_0(k5_topmetr),E,C) = np__0
& r1_tarski(k4_pre_topc(A,k5_topmetr,E,k3_subset_1(u1_struct_0(A),D)),k1_seq_4(np__1)) ) ) ) ) ) ) ) ) ).
fof(t53_topgen_5,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& m1_pre_topc(B,k3_topmetr) )
=> ! [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] :
( ( 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,k1_zfmisc_1(u1_struct_0(A)))
=> ( ! [F] :
( m1_subset_1(F,u1_struct_0(A))
=> ( r2_hidden(F,E)
<=> r1_xreal_0(k2_seq_1(u1_struct_0(A),u1_struct_0(B),C,F),k2_seq_1(u1_struct_0(A),u1_struct_0(B),D,F)) ) )
=> v4_pre_topc(E,A) ) ) ) ) ) ) ).
fof(t54_topgen_5,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& m1_pre_topc(B,k3_topmetr) )
=> ! [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] :
( ( 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] :
( 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_subset_1(F,u1_struct_0(A))
=> k2_seq_1(u1_struct_0(A),u1_struct_0(B),E,F) = k4_square_1(k2_seq_1(u1_struct_0(A),u1_struct_0(B),C,F),k2_seq_1(u1_struct_0(A),u1_struct_0(B),D,F)) ) ) ) ) ) ) ).
fof(t55_topgen_5,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& m1_pre_topc(B,k3_topmetr) )
=> ! [C] :
( ( ~ v1_xboole_0(C)
& v1_finset_1(C) )
=> ! [D] :
( ( v1_funcop_1(D)
& m1_pboole(D,C) )
=> ~ ( ! [E] :
( r2_hidden(E,C)
=> ( v1_funct_1(k1_funct_1(D,E))
& v1_funct_2(k1_funct_1(D,E),u1_struct_0(A),u1_struct_0(B))
& v5_pre_topc(k1_funct_1(D,E),A,B)
& m2_relset_1(k1_funct_1(D,E),u1_struct_0(A),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_subset_1(F,u1_struct_0(A))
& ? [G] :
( ~ v1_xboole_0(G)
& v1_finset_1(G)
& m1_subset_1(G,k1_zfmisc_1(k1_numbers))
& G = k2_relat_1(k1_funct_1(k10_funct_6(D),F))
& k2_seq_1(u1_struct_0(A),u1_struct_0(B),E,F) != k2_seq_4(G) ) ) ) ) ) ) ) ) ).
fof(t56_topgen_5,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& v1_urysohn1(A)
& l1_pre_topc(A) )
=> ! [B] :
( m2_cantor_1(B,A)
=> ( ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
=> ~ ( r2_hidden(C,D)
& r2_hidden(D,B)
& ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,u1_struct_0(A),u1_struct_0(k5_topmetr))
& v5_pre_topc(E,A,k5_topmetr)
& m2_relset_1(E,u1_struct_0(A),u1_struct_0(k5_topmetr)) )
=> ~ ( k2_seq_1(u1_struct_0(A),u1_struct_0(k5_topmetr),E,C) = np__0
& r1_tarski(k4_pre_topc(A,k5_topmetr,E,k3_subset_1(u1_struct_0(A),D)),k1_seq_4(np__1)) ) ) ) ) )
=> v1_topgen_5(A) ) ) ) ).
fof(t57_topgen_5,axiom,
v1_urysohn1(k2_topgen_3) ).
fof(t58_topgen_5,axiom,
! [A] :
( v1_xreal_0(A)
=> ( v4_pre_topc(k12_prob_1(A),k2_topgen_3)
& m1_subset_1(k12_prob_1(A),k1_zfmisc_1(u1_struct_0(k2_topgen_3))) ) ) ).
fof(t59_topgen_5,axiom,
! [A] :
( v1_xreal_0(A)
=> ( v4_pre_topc(k2_limfunc1(A),k2_topgen_3)
& m1_subset_1(k2_limfunc1(A),k1_zfmisc_1(u1_struct_0(k2_topgen_3))) ) ) ).
fof(t60_topgen_5,axiom,
! [A] :
( v1_xreal_0(A)
=> ( v4_pre_topc(k3_limfunc1(A),k2_topgen_3)
& m1_subset_1(k3_limfunc1(A),k1_zfmisc_1(u1_struct_0(k2_topgen_3))) ) ) ).
fof(t61_topgen_5,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ( v4_pre_topc(k1_rcomp_2(A,B),k2_topgen_3)
& m1_subset_1(k1_rcomp_2(A,B),k1_zfmisc_1(u1_struct_0(k2_topgen_3))) ) ) ) ).
fof(t62_topgen_5,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_rat_1(B)
=> ~ ( ~ r1_xreal_0(B,A)
& ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(k2_topgen_3),u1_struct_0(k5_topmetr))
& v5_pre_topc(C,k2_topgen_3,k5_topmetr)
& m2_relset_1(C,u1_struct_0(k2_topgen_3),u1_struct_0(k5_topmetr)) )
=> ~ ! [D] :
( m1_subset_1(D,u1_struct_0(k2_topgen_3))
=> ( ( r2_hidden(D,k1_rcomp_2(A,B))
=> k2_seq_1(u1_struct_0(k2_topgen_3),u1_struct_0(k5_topmetr),C,D) = np__0 )
& ( ~ r2_hidden(D,k1_rcomp_2(A,B))
=> k2_seq_1(u1_struct_0(k2_topgen_3),u1_struct_0(k5_topmetr),C,D) = np__1 ) ) ) ) ) ) ) ).
fof(t63_topgen_5,axiom,
v1_topgen_5(k2_topgen_3) ).
fof(d5_topgen_5,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( ( v1_xreal_0(B)
& v2_xreal_0(B) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(k3_topgen_5),u1_struct_0(k5_topmetr))
& m2_relset_1(C,u1_struct_0(k3_topgen_5),u1_struct_0(k5_topmetr)) )
=> ( C = k4_topgen_5(A,B)
<=> ( k2_seq_1(u1_struct_0(k3_topgen_5),u1_struct_0(k5_topmetr),C,k23_euclid(A,np__0)) = np__0
& ! [D] :
( v1_xreal_0(D)
=> ! [E] :
( ( v1_xreal_0(E)
& ~ v3_xreal_0(E) )
=> ( ~ ( ~ ( D = A
& E = np__0 )
& ~ r2_hidden(k23_euclid(D,E),k1_topreal9(np__2,k23_euclid(A,B),B))
& k2_seq_1(u1_struct_0(k3_topgen_5),u1_struct_0(k5_topmetr),C,k23_euclid(D,E)) != np__1 )
& ( r2_hidden(k23_euclid(D,E),k1_topreal9(np__2,k23_euclid(A,B),B))
=> k2_seq_1(u1_struct_0(k3_topgen_5),u1_struct_0(k5_topmetr),C,k23_euclid(D,E)) = k7_xcmplx_0(k7_square_1(k5_toprns_1(np__2,k20_euclid(np__2,k23_euclid(A,np__0),k23_euclid(D,E)))),k3_xcmplx_0(k3_xcmplx_0(np__2,B),E)) ) ) ) ) ) ) ) ) ) ).
fof(t64_topgen_5,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k15_euclid(np__2)))
=> ( r1_xreal_0(np__0,k22_euclid(A))
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( ( v1_xreal_0(C)
& v2_xreal_0(C) )
=> ( k2_seq_1(u1_struct_0(k3_topgen_5),u1_struct_0(k5_topmetr),k4_topgen_5(B,C),A) = np__0
=> A = k23_euclid(B,np__0) ) ) ) ) ) ).
fof(t65_topgen_5,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( ( v1_xreal_0(C)
& v2_xreal_0(C) )
=> ( A != B
=> k2_seq_1(u1_struct_0(k3_topgen_5),u1_struct_0(k5_topmetr),k4_topgen_5(A,C),k23_euclid(B,np__0)) = np__1 ) ) ) ) ).
fof(t66_topgen_5,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k15_euclid(np__2)))
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( ( v1_xreal_0(C)
& v2_xreal_0(C) )
=> ! [D] :
( ( v1_xreal_0(D)
& v2_xreal_0(D) )
=> ( ( r1_xreal_0(C,np__1)
& k5_toprns_1(np__2,k20_euclid(np__2,A,k23_euclid(B,k3_xcmplx_0(D,C)))) = k3_xcmplx_0(D,C) )
=> ( k22_euclid(A) = np__0
| k2_seq_1(u1_struct_0(k3_topgen_5),u1_struct_0(k5_topmetr),k4_topgen_5(B,D),A) = C ) ) ) ) ) ) ).
fof(t67_topgen_5,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k15_euclid(np__2)))
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v1_xreal_0(C)
=> ! [D] :
( ( v1_xreal_0(D)
& v2_xreal_0(D) )
=> ~ ( r1_xreal_0(np__0,C)
& r1_xreal_0(C,np__1)
& ~ r1_xreal_0(k3_xcmplx_0(D,C),k5_toprns_1(np__2,k20_euclid(np__2,A,k23_euclid(B,k3_xcmplx_0(D,C)))))
& r1_xreal_0(C,k2_seq_1(u1_struct_0(k3_topgen_5),u1_struct_0(k5_topmetr),k4_topgen_5(B,D),A)) ) ) ) ) ) ).
fof(t68_topgen_5,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k15_euclid(np__2)))
=> ( r1_xreal_0(np__0,k22_euclid(A))
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v1_xreal_0(C)
=> ! [D] :
( ( v1_xreal_0(D)
& v2_xreal_0(D) )
=> ~ ( r1_xreal_0(np__0,C)
& ~ r1_xreal_0(np__1,C)
& ~ r1_xreal_0(k5_toprns_1(np__2,k20_euclid(np__2,A,k23_euclid(B,k3_xcmplx_0(D,C)))),k3_xcmplx_0(D,C))
& r1_xreal_0(k2_seq_1(u1_struct_0(k3_topgen_5),u1_struct_0(k5_topmetr),k4_topgen_5(B,D),A),C) ) ) ) ) ) ) ).
fof(t69_topgen_5,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k15_euclid(np__2)))
=> ( r1_xreal_0(np__0,k22_euclid(A))
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v1_xreal_0(C)
=> ! [D] :
( v1_xreal_0(D)
=> ! [E] :
( ( v1_xreal_0(E)
& v2_xreal_0(E) )
=> ~ ( r1_xreal_0(np__0,C)
& r1_xreal_0(D,np__1)
& r2_hidden(k2_seq_1(u1_struct_0(k3_topgen_5),u1_struct_0(k5_topmetr),k4_topgen_5(B,E),A),k2_rcomp_1(C,D))
& ! [F] :
( ( v1_xreal_0(F)
& v2_xreal_0(F) )
=> ~ ( r1_xreal_0(F,k22_euclid(A))
& r1_tarski(k1_topreal9(np__2,A,F),k5_pre_topc(k3_topgen_5,k5_topmetr,k4_topgen_5(B,E),k2_rcomp_1(C,D))) ) ) ) ) ) ) ) ) ) ).
fof(t70_topgen_5,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( ( v1_xreal_0(B)
& v2_xreal_0(B) )
=> ! [C] :
( ( v1_xreal_0(C)
& v2_xreal_0(C) )
=> r1_tarski(k1_topreal9(np__2,k23_euclid(A,k3_xcmplx_0(C,B)),k3_xcmplx_0(C,B)),k5_pre_topc(k3_topgen_5,k5_topmetr,k4_topgen_5(A,C),k2_rcomp_1(np__0,B))) ) ) ) ).
fof(t71_topgen_5,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( ( v1_xreal_0(B)
& v2_xreal_0(B) )
=> ! [C] :
( ( v1_xreal_0(C)
& v2_xreal_0(C) )
=> r1_tarski(k4_subset_1(u1_struct_0(k15_euclid(np__2)),k1_topreal9(np__2,k23_euclid(A,k3_xcmplx_0(C,B)),k3_xcmplx_0(C,B)),k1_struct_0(k15_euclid(np__2),k23_euclid(A,np__0))),k5_pre_topc(k3_topgen_5,k5_topmetr,k4_topgen_5(A,C),k1_rcomp_2(np__0,B))) ) ) ) ).
fof(t72_topgen_5,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k15_euclid(np__2)))
=> ( r1_xreal_0(np__0,k22_euclid(A))
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v1_xreal_0(C)
=> ! [D] :
( ( v1_xreal_0(D)
& v2_xreal_0(D) )
=> ( r1_xreal_0(C,np__1)
=> ( r1_xreal_0(k2_seq_1(u1_struct_0(k3_topgen_5),u1_struct_0(k5_topmetr),k4_topgen_5(B,D),A),np__0)
| r1_xreal_0(C,k2_seq_1(u1_struct_0(k3_topgen_5),u1_struct_0(k5_topmetr),k4_topgen_5(B,D),A))
| r2_hidden(A,k1_topreal9(np__2,k23_euclid(B,k3_xcmplx_0(D,C)),k3_xcmplx_0(D,C))) ) ) ) ) ) ) ) ).
fof(t73_topgen_5,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k15_euclid(np__2)))
=> ( ~ r1_xreal_0(k22_euclid(A),np__0)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v1_xreal_0(C)
=> ! [D] :
( ( v1_xreal_0(D)
& v2_xreal_0(D) )
=> ~ ( r1_xreal_0(np__0,C)
& ~ r1_xreal_0(k2_seq_1(u1_struct_0(k3_topgen_5),u1_struct_0(k5_topmetr),k4_topgen_5(B,D),A),C)
& ! [E] :
( ( v1_xreal_0(E)
& v2_xreal_0(E) )
=> ~ ( r1_xreal_0(E,k22_euclid(A))
& r1_tarski(k1_topreal9(np__2,A,E),k5_pre_topc(k3_topgen_5,k5_topmetr,k4_topgen_5(B,D),k2_rcomp_2(C,np__1))) ) ) ) ) ) ) ) ) ).
fof(t74_topgen_5,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k15_euclid(np__2)))
=> ( k22_euclid(A) = np__0
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( ( v1_xreal_0(C)
& v2_xreal_0(C) )
=> ~ ( k2_seq_1(u1_struct_0(k3_topgen_5),u1_struct_0(k5_topmetr),k4_topgen_5(B,C),A) = np__1
& ! [D] :
( ( v1_xreal_0(D)
& v2_xreal_0(D) )
=> ~ r1_tarski(k4_subset_1(u1_struct_0(k15_euclid(np__2)),k1_topreal9(np__2,k23_euclid(k21_euclid(A),D),D),k1_struct_0(k15_euclid(np__2),A)),k5_pre_topc(k3_topgen_5,k5_topmetr,k4_topgen_5(B,C),k1_seq_4(np__1))) ) ) ) ) ) ) ).
fof(t77_topgen_5,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] :
( ( r2_hidden(B,D)
& r2_hidden(C,D)
& m1_cantor_1(k2_xboole_0(D,k1_tarski(k4_subset_1(u1_struct_0(A),B,C))),A) )
=> m1_cantor_1(D,A) ) ) ) ) ).
fof(t79_topgen_5,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(k5_topmetr))
& m2_relset_1(B,u1_struct_0(A),u1_struct_0(k5_topmetr)) )
=> ( v5_pre_topc(B,A,k5_topmetr)
<=> ! [C] :
( v1_xreal_0(C)
=> ! [D] :
( v1_xreal_0(D)
=> ( ( r1_xreal_0(np__0,C)
& r1_xreal_0(D,np__1) )
=> ( r1_xreal_0(np__1,C)
| r1_xreal_0(D,np__0)
| ( v3_pre_topc(k5_pre_topc(A,k5_topmetr,B,k1_rcomp_2(np__0,D)),A)
& v3_pre_topc(k5_pre_topc(A,k5_topmetr,B,k2_rcomp_2(C,np__1)),A) ) ) ) ) ) ) ) ) ).
fof(t80_topgen_5,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k3_topgen_5)))
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ! [C] :
( ( v1_xreal_0(C)
& v2_xreal_0(C) )
=> ~ ( A = k4_subset_1(u1_struct_0(k15_euclid(np__2)),k1_topreal9(np__2,k23_euclid(B,C),C),k1_struct_0(k15_euclid(np__2),k23_euclid(B,np__0)))
& ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,u1_struct_0(k3_topgen_5),u1_struct_0(k5_topmetr))
& v5_pre_topc(D,k3_topgen_5,k5_topmetr)
& m2_relset_1(D,u1_struct_0(k3_topgen_5),u1_struct_0(k5_topmetr)) )
=> ~ ( k2_seq_1(u1_struct_0(k3_topgen_5),u1_struct_0(k5_topmetr),D,k23_euclid(B,np__0)) = np__0
& ! [E] :
( v1_xreal_0(E)
=> ! [F] :
( v1_xreal_0(F)
=> ( ( r2_hidden(k23_euclid(E,F),k3_subset_1(u1_struct_0(k3_topgen_5),A))
=> k2_seq_1(u1_struct_0(k3_topgen_5),u1_struct_0(k5_topmetr),D,k23_euclid(E,F)) = np__1 )
& ( r2_hidden(k23_euclid(E,F),k1_topgen_1(u1_struct_0(k3_topgen_5),A,k1_struct_0(k15_euclid(np__2),k23_euclid(B,np__0))))
=> k2_seq_1(u1_struct_0(k3_topgen_5),u1_struct_0(k5_topmetr),D,k23_euclid(E,F)) = k7_xcmplx_0(k7_square_1(k5_toprns_1(np__2,k20_euclid(np__2,k23_euclid(B,np__0),k23_euclid(E,F)))),k3_xcmplx_0(k3_xcmplx_0(np__2,C),F)) ) ) ) ) ) ) ) ) ) ) ).
fof(d6_topgen_5,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( ( v1_xreal_0(C)
& v2_xreal_0(C) )
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,u1_struct_0(k3_topgen_5),u1_struct_0(k5_topmetr))
& m2_relset_1(D,u1_struct_0(k3_topgen_5),u1_struct_0(k5_topmetr)) )
=> ( D = k5_topgen_5(A,B,C)
<=> ! [E] :
( v1_xreal_0(E)
=> ! [F] :
( ( v1_xreal_0(F)
& ~ v3_xreal_0(F) )
=> ( ( ~ r2_hidden(k23_euclid(E,F),k1_topreal9(np__2,k23_euclid(A,B),C))
=> k2_seq_1(u1_struct_0(k3_topgen_5),u1_struct_0(k5_topmetr),D,k23_euclid(E,F)) = np__1 )
& ( r2_hidden(k23_euclid(E,F),k1_topreal9(np__2,k23_euclid(A,B),C))
=> k2_seq_1(u1_struct_0(k3_topgen_5),u1_struct_0(k5_topmetr),D,k23_euclid(E,F)) = k7_xcmplx_0(k5_toprns_1(np__2,k20_euclid(np__2,k23_euclid(A,B),k23_euclid(E,F))),C) ) ) ) ) ) ) ) ) ) ).
fof(t81_topgen_5,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k15_euclid(np__2)))
=> ( r1_xreal_0(np__0,k22_euclid(A))
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( ( v1_xreal_0(C)
& ~ v3_xreal_0(C) )
=> ! [D] :
( ( v1_xreal_0(D)
& v2_xreal_0(D) )
=> ( k2_seq_1(u1_struct_0(k3_topgen_5),u1_struct_0(k5_topmetr),k5_topgen_5(B,C,D),A) = np__0
<=> A = k23_euclid(B,C) ) ) ) ) ) ) ).
fof(t82_topgen_5,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( ( v1_xreal_0(B)
& ~ v3_xreal_0(B) )
=> ! [C] :
( ( v1_xreal_0(C)
& v2_xreal_0(C) )
=> ! [D] :
( ( v1_xreal_0(D)
& v2_xreal_0(D) )
=> ( r1_xreal_0(D,np__1)
=> k5_pre_topc(k3_topgen_5,k5_topmetr,k5_topgen_5(A,B,C),k1_rcomp_2(np__0,D)) = k5_subset_1(u1_struct_0(k15_euclid(np__2)),k1_topreal9(np__2,k23_euclid(A,B),k3_xcmplx_0(C,D)),k2_topgen_5) ) ) ) ) ) ).
fof(t83_topgen_5,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k15_euclid(np__2)))
=> ( ~ r1_xreal_0(k22_euclid(A),np__0)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( ( v1_xreal_0(C)
& ~ v3_xreal_0(C) )
=> ! [D] :
( ( v1_xreal_0(D)
& v2_xreal_0(D) )
=> ! [E] :
( ( v1_xreal_0(E)
& v2_xreal_0(E) )
=> ( ~ r1_xreal_0(k2_seq_1(u1_struct_0(k3_topgen_5),u1_struct_0(k5_topmetr),k5_topgen_5(B,D,E),A),C)
=> ( ~ r1_xreal_0(k5_toprns_1(np__2,k20_euclid(np__2,k23_euclid(B,D),A)),k3_xcmplx_0(E,C))
& r1_tarski(k5_subset_1(u1_struct_0(k15_euclid(np__2)),k1_topreal9(np__2,A,k6_xcmplx_0(k5_toprns_1(np__2,k20_euclid(np__2,k23_euclid(B,D),A)),k3_xcmplx_0(E,C))),k2_topgen_5),k5_pre_topc(k3_topgen_5,k5_topmetr,k5_topgen_5(B,D,E),k2_rcomp_2(C,np__1))) ) ) ) ) ) ) ) ) ).
fof(t84_topgen_5,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k15_euclid(np__2)))
=> ( k22_euclid(A) = np__0
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( ( v1_xreal_0(C)
& ~ v3_xreal_0(C) )
=> ! [D] :
( ( v1_xreal_0(D)
& v2_xreal_0(D) )
=> ! [E] :
( ( v1_xreal_0(E)
& v2_xreal_0(E) )
=> ( ~ r1_xreal_0(k2_seq_1(u1_struct_0(k3_topgen_5),u1_struct_0(k5_topmetr),k5_topgen_5(B,D,E),A),C)
=> ( ~ r1_xreal_0(k5_toprns_1(np__2,k20_euclid(np__2,k23_euclid(B,D),A)),k3_xcmplx_0(E,C))
& ? [F] :
( v1_xreal_0(F)
& v2_xreal_0(F)
& F = k7_xcmplx_0(k6_xcmplx_0(k5_toprns_1(np__2,k20_euclid(np__2,k23_euclid(B,D),A)),k3_xcmplx_0(E,C)),np__2)
& r1_tarski(k4_subset_1(u1_struct_0(k15_euclid(np__2)),k1_topreal9(np__2,k23_euclid(k21_euclid(A),F),F),k1_struct_0(k15_euclid(np__2),A)),k5_pre_topc(k3_topgen_5,k5_topmetr,k5_topgen_5(B,D,E),k2_rcomp_2(C,np__1))) ) ) ) ) ) ) ) ) ) ).
fof(t85_topgen_5,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k3_topgen_5)))
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ! [C] :
( m1_subset_1(C,k1_numbers)
=> ! [D] :
( ( v1_xreal_0(D)
& v2_xreal_0(D) )
=> ~ ( ~ r1_xreal_0(C,np__0)
& A = k5_subset_1(u1_struct_0(k15_euclid(np__2)),k1_topreal9(np__2,k23_euclid(B,C),D),k2_topgen_5)
& ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,u1_struct_0(k3_topgen_5),u1_struct_0(k5_topmetr))
& v5_pre_topc(E,k3_topgen_5,k5_topmetr)
& m2_relset_1(E,u1_struct_0(k3_topgen_5),u1_struct_0(k5_topmetr)) )
=> ~ ( k2_seq_1(u1_struct_0(k3_topgen_5),u1_struct_0(k5_topmetr),E,k23_euclid(B,C)) = np__0
& ! [F] :
( v1_xreal_0(F)
=> ! [G] :
( v1_xreal_0(G)
=> ( ( r2_hidden(k23_euclid(F,G),k3_subset_1(u1_struct_0(k3_topgen_5),A))
=> k2_seq_1(u1_struct_0(k3_topgen_5),u1_struct_0(k5_topmetr),E,k23_euclid(F,G)) = np__1 )
& ( r2_hidden(k23_euclid(F,G),A)
=> k2_seq_1(u1_struct_0(k3_topgen_5),u1_struct_0(k5_topmetr),E,k23_euclid(F,G)) = k7_xcmplx_0(k5_toprns_1(np__2,k20_euclid(np__2,k23_euclid(B,C),k23_euclid(F,G))),D) ) ) ) ) ) ) ) ) ) ) ) ).
fof(t86_topgen_5,axiom,
v1_urysohn1(k3_topgen_5) ).
fof(t87_topgen_5,axiom,
v1_topgen_5(k3_topgen_5) ).
fof(s1_topgen_5,axiom,
( ( r1_tarski(f3_s1_topgen_5,f1_s1_topgen_5)
& ! [A] :
( m1_subset_1(A,f1_s1_topgen_5)
=> ( r2_hidden(A,f3_s1_topgen_5)
=> ( ( p1_s1_topgen_5(A)
=> r2_hidden(f4_s1_topgen_5(A),f2_s1_topgen_5) )
& ( ~ p1_s1_topgen_5(A)
=> r2_hidden(f5_s1_topgen_5(A),f2_s1_topgen_5) ) ) ) ) )
=> ? [A] :
( v1_funct_1(A)
& v1_funct_2(A,f3_s1_topgen_5,f2_s1_topgen_5)
& m2_relset_1(A,f3_s1_topgen_5,f2_s1_topgen_5)
& ! [B] :
( m1_subset_1(B,f1_s1_topgen_5)
=> ( r2_hidden(B,f3_s1_topgen_5)
=> ( ( p1_s1_topgen_5(B)
=> k1_funct_1(A,B) = f4_s1_topgen_5(B) )
& ( ~ p1_s1_topgen_5(B)
=> k1_funct_1(A,B) = f5_s1_topgen_5(B) ) ) ) ) ) ) ).
fof(s2_topgen_5,axiom,
( ( r1_tarski(f3_s2_topgen_5,f1_s2_topgen_5)
& ! [A] :
( m1_subset_1(A,f1_s2_topgen_5)
=> ( r2_hidden(A,f3_s2_topgen_5)
=> ( ( p1_s2_topgen_5(A)
=> r2_hidden(f4_s2_topgen_5(A),f2_s2_topgen_5) )
& ( p2_s2_topgen_5(A)
=> ( p1_s2_topgen_5(A)
| r2_hidden(f5_s2_topgen_5(A),f2_s2_topgen_5) ) )
& ~ ( ~ p1_s2_topgen_5(A)
& ~ p2_s2_topgen_5(A)
& ~ r2_hidden(f6_s2_topgen_5(A),f2_s2_topgen_5) ) ) ) ) )
=> ? [A] :
( v1_funct_1(A)
& v1_funct_2(A,f3_s2_topgen_5,f2_s2_topgen_5)
& m2_relset_1(A,f3_s2_topgen_5,f2_s2_topgen_5)
& ! [B] :
( m1_subset_1(B,f1_s2_topgen_5)
=> ( r2_hidden(B,f3_s2_topgen_5)
=> ( ( p1_s2_topgen_5(B)
=> k1_funct_1(A,B) = f4_s2_topgen_5(B) )
& ( p2_s2_topgen_5(B)
=> ( p1_s2_topgen_5(B)
| k1_funct_1(A,B) = f5_s2_topgen_5(B) ) )
& ~ ( ~ p1_s2_topgen_5(B)
& ~ p2_s2_topgen_5(B)
& k1_funct_1(A,B) != f6_s2_topgen_5(B) ) ) ) ) ) ) ).
fof(dt_k1_topgen_5,axiom,
m1_subset_1(k1_topgen_5,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) ).
fof(dt_k2_topgen_5,axiom,
m1_subset_1(k2_topgen_5,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) ).
fof(dt_k3_topgen_5,axiom,
( ~ v3_struct_0(k3_topgen_5)
& v1_pre_topc(k3_topgen_5)
& v2_pre_topc(k3_topgen_5)
& l1_pre_topc(k3_topgen_5) ) ).
fof(dt_k4_topgen_5,axiom,
! [A,B] :
( ( v1_xreal_0(A)
& v1_xreal_0(B)
& v2_xreal_0(B) )
=> ( v1_funct_1(k4_topgen_5(A,B))
& v1_funct_2(k4_topgen_5(A,B),u1_struct_0(k3_topgen_5),u1_struct_0(k5_topmetr))
& m2_relset_1(k4_topgen_5(A,B),u1_struct_0(k3_topgen_5),u1_struct_0(k5_topmetr)) ) ) ).
fof(dt_k5_topgen_5,axiom,
! [A,B,C] :
( ( v1_xreal_0(A)
& v1_xreal_0(B)
& v1_xreal_0(C)
& v2_xreal_0(C) )
=> ( v1_funct_1(k5_topgen_5(A,B,C))
& v1_funct_2(k5_topgen_5(A,B,C),u1_struct_0(k3_topgen_5),u1_struct_0(k5_topmetr))
& m2_relset_1(k5_topgen_5(A,B,C),u1_struct_0(k3_topgen_5),u1_struct_0(k5_topmetr)) ) ) ).
fof(d1_topgen_5,axiom,
k1_topgen_5 = a_0_0_topgen_5 ).
fof(d2_topgen_5,axiom,
k2_topgen_5 = a_0_1_topgen_5 ).
fof(d3_topgen_5,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v1_pre_topc(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ( A = k3_topgen_5
<=> ( u1_struct_0(A) = k2_topgen_5
& ? [B] :
( m1_topgen_2(B,A)
& ! [C] :
( m1_subset_1(C,k1_numbers)
=> k1_funct_1(B,k23_euclid(C,np__0)) = a_1_0_topgen_5(C) )
& ! [C] :
( m1_subset_1(C,k1_numbers)
=> ! [D] :
( m1_subset_1(D,k1_numbers)
=> ( ~ r1_xreal_0(D,np__0)
=> k1_funct_1(B,k23_euclid(C,D)) = a_2_0_topgen_5(C,D) ) ) ) ) ) ) ) ).
fof(t75_topgen_5,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_pre_topc(B,A)
=> ! [C] :
( m1_cantor_1(C,A)
=> m1_cantor_1(a_3_0_topgen_5(A,B,C),B) ) ) ) ).
fof(t76_topgen_5,axiom,
m1_cantor_1(a_0_2_topgen_5,k3_topmetr) ).
fof(t78_topgen_5,axiom,
m1_cantor_1(k2_xboole_0(k2_xboole_0(a_0_3_topgen_5,a_0_4_topgen_5),a_0_5_topgen_5),k5_topmetr) ).
fof(fraenkel_a_0_0_topgen_5,axiom,
! [A] :
( r2_hidden(A,a_0_0_topgen_5)
<=> ? [B] :
( m1_subset_1(B,k1_numbers)
& A = k23_euclid(B,np__0) ) ) ).
fof(fraenkel_a_0_1_topgen_5,axiom,
! [A] :
( r2_hidden(A,a_0_1_topgen_5)
<=> ? [B,C] :
( m1_subset_1(B,k1_numbers)
& m1_subset_1(C,k1_numbers)
& A = k23_euclid(B,C)
& r1_xreal_0(np__0,C) ) ) ).
fof(fraenkel_a_1_0_topgen_5,axiom,
! [A,B] :
( m1_subset_1(B,k1_numbers)
=> ( r2_hidden(A,a_1_0_topgen_5(B))
<=> ? [C] :
( m1_subset_1(C,k1_numbers)
& A = k4_subset_1(u1_struct_0(k15_euclid(np__2)),k1_topreal9(np__2,k23_euclid(B,C),C),k1_struct_0(k15_euclid(np__2),k23_euclid(B,np__0)))
& ~ r1_xreal_0(C,np__0) ) ) ) ).
fof(fraenkel_a_2_0_topgen_5,axiom,
! [A,B,C] :
( ( m1_subset_1(B,k1_numbers)
& m1_subset_1(C,k1_numbers) )
=> ( r2_hidden(A,a_2_0_topgen_5(B,C))
<=> ? [D] :
( m1_subset_1(D,k1_numbers)
& A = k5_subset_1(u1_struct_0(k15_euclid(np__2)),k1_topreal9(np__2,k23_euclid(B,C),D),k2_topgen_5)
& ~ r1_xreal_0(D,np__0) ) ) ) ).
fof(fraenkel_a_3_0_topgen_5,axiom,
! [A,B,C,D] :
( ( ~ v3_struct_0(B)
& v2_pre_topc(B)
& l1_pre_topc(B)
& m1_pre_topc(C,B)
& m1_cantor_1(D,B) )
=> ( r2_hidden(A,a_3_0_topgen_5(B,C,D))
<=> ? [E] :
( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(B)))
& A = k3_xboole_0(E,k2_pre_topc(C))
& r2_hidden(E,D)
& ~ r1_xboole_0(E,k2_pre_topc(C)) ) ) ) ).
fof(fraenkel_a_0_2_topgen_5,axiom,
! [A] :
( r2_hidden(A,a_0_2_topgen_5)
<=> ? [B,C] :
( m1_subset_1(B,k1_numbers)
& m1_subset_1(C,k1_numbers)
& A = k2_rcomp_1(B,C)
& ~ r1_xreal_0(C,B) ) ) ).
fof(fraenkel_a_0_3_topgen_5,axiom,
! [A] :
( r2_hidden(A,a_0_3_topgen_5)
<=> ? [B] :
( m1_subset_1(B,k1_numbers)
& A = k1_rcomp_2(np__0,B)
& ~ r1_xreal_0(B,np__0)
& r1_xreal_0(B,np__1) ) ) ).
fof(fraenkel_a_0_4_topgen_5,axiom,
! [A] :
( r2_hidden(A,a_0_4_topgen_5)
<=> ? [B] :
( m1_subset_1(B,k1_numbers)
& A = k2_rcomp_2(B,np__1)
& r1_xreal_0(np__0,B)
& ~ r1_xreal_0(np__1,B) ) ) ).
fof(fraenkel_a_0_5_topgen_5,axiom,
! [A] :
( r2_hidden(A,a_0_5_topgen_5)
<=> ? [B,C] :
( m1_subset_1(B,k1_numbers)
& m1_subset_1(C,k1_numbers)
& A = k2_rcomp_1(B,C)
& r1_xreal_0(np__0,B)
& ~ r1_xreal_0(C,B)
& r1_xreal_0(C,np__1) ) ) ).
%------------------------------------------------------------------------------