SET007 Axioms: SET007+896.ax
%------------------------------------------------------------------------------
% File : SET007+896 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : The Fundamental Group of the Circle
% 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_5 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 55 ( 2 unt; 0 def)
% Number of atoms : 486 ( 53 equ)
% Maximal formula atoms : 26 ( 8 avg)
% Number of connectives : 498 ( 67 ~; 1 |; 256 &)
% ( 7 <=>; 167 =>; 0 <=; 0 <~>)
% Maximal formula depth : 31 ( 10 avg)
% Maximal term depth : 6 ( 1 avg)
% Number of predicates : 60 ( 59 usr; 0 prp; 1-6 aty)
% Number of functors : 68 ( 68 usr; 15 con; 0-5 aty)
% Number of variables : 170 ( 165 !; 5 ?)
% 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_5,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k3_gr_cy_1))
=> ( v1_xreal_0(A)
& v1_xcmplx_0(A)
& v1_int_1(A) ) ) ).
fof(fc1_topalg_5,axiom,
( ~ v3_struct_0(k3_gr_cy_1)
& v1_group_1(k3_gr_cy_1)
& v3_group_1(k3_gr_cy_1)
& v4_group_1(k3_gr_cy_1)
& ~ v6_group_1(k3_gr_cy_1) ) ).
fof(fc2_topalg_5,axiom,
! [A] :
( ( ~ v6_group_1(A)
& l1_struct_0(A) )
=> ~ v1_finset_1(u1_struct_0(A)) ) ).
fof(fc3_topalg_5,axiom,
! [A,B] :
( ( v2_pre_topc(A)
& l1_pre_topc(A)
& v3_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_tsep_1(k3_pre_topc(A,B),A) ) ) ).
fof(fc4_topalg_5,axiom,
( ~ v3_struct_0(k22_borsuk_1)
& v1_pre_topc(k22_borsuk_1)
& v2_pre_topc(k22_borsuk_1)
& v2_t_0topsp(k22_borsuk_1)
& v2_compts_1(k22_borsuk_1)
& v3_compts_1(k22_borsuk_1)
& v1_connsp_2(k22_borsuk_1)
& v3_yellow_8(k22_borsuk_1) ) ).
fof(fc5_topalg_5,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v3_pre_topc(A,k5_topmetr)
& m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k5_topmetr))) )
=> ( ~ v3_struct_0(k3_pre_topc(k5_topmetr,A))
& v1_pre_topc(k3_pre_topc(k5_topmetr,A))
& v2_pre_topc(k3_pre_topc(k5_topmetr,A))
& v1_connsp_2(k3_pre_topc(k5_topmetr,A))
& v1_tsep_1(k3_pre_topc(k5_topmetr,A),k5_topmetr) ) ) ).
fof(fc6_topalg_5,axiom,
! [A] :
( v1_xreal_0(A)
=> ( v1_relat_1(k1_topalg_5(A))
& v1_funct_1(k1_topalg_5(A))
& v1_funct_2(k1_topalg_5(A),u1_struct_0(k5_topmetr),u1_struct_0(k3_topalg_2))
& v5_pre_topc(k1_topalg_5(A),k5_topmetr,k3_topalg_2) ) ) ).
fof(fc7_topalg_5,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(k6_borsuk_1(A,B)),u1_struct_0(C))
& v5_pre_topc(D,k6_borsuk_1(A,B),C)
& m1_relset_1(D,u1_struct_0(k6_borsuk_1(A,B)),u1_struct_0(C))
& m1_subset_1(E,u1_struct_0(B)) )
=> ( v1_relat_1(k3_topalg_5(A,B,C,D,E))
& v1_funct_1(k3_topalg_5(A,B,C,D,E))
& v1_funct_2(k3_topalg_5(A,B,C,D,E),u1_struct_0(A),u1_struct_0(C))
& v5_pre_topc(k3_topalg_5(A,B,C,D,E),A,C) ) ) ).
fof(fc8_topalg_5,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(k6_borsuk_1(A,B)),u1_struct_0(C))
& v5_pre_topc(D,k6_borsuk_1(A,B),C)
& m1_relset_1(D,u1_struct_0(k6_borsuk_1(A,B)),u1_struct_0(C))
& m1_subset_1(E,u1_struct_0(A)) )
=> ( v1_relat_1(k4_topalg_5(A,B,C,D,E))
& v1_funct_1(k4_topalg_5(A,B,C,D,E))
& v1_funct_2(k4_topalg_5(A,B,C,D,E),u1_struct_0(B),u1_struct_0(C))
& v5_pre_topc(k4_topalg_5(A,B,C,D,E),B,C) ) ) ).
fof(fc9_topalg_5,axiom,
( v1_relat_1(k7_topalg_5)
& v1_funct_1(k7_topalg_5)
& v2_funct_1(k7_topalg_5)
& v1_funct_2(k7_topalg_5,u1_struct_0(k3_gr_cy_1),u1_struct_0(k3_topalg_1(k8_toprealb(np__2),k9_toprealb)))
& v2_funct_2(k7_topalg_5,u1_struct_0(k3_gr_cy_1),u1_struct_0(k3_topalg_1(k8_toprealb(np__2),k9_toprealb))) ) ).
fof(fc10_topalg_5,axiom,
! [A,B] :
( ( v1_toprealb(A)
& m1_pre_topc(A,k15_euclid(np__2))
& m1_subset_1(B,u1_struct_0(A)) )
=> ( ~ v3_struct_0(k3_topalg_1(A,B))
& v1_group_1(k3_topalg_1(A,B))
& v2_group_1(k3_topalg_1(A,B))
& v3_group_1(k3_topalg_1(A,B))
& v4_group_1(k3_topalg_1(A,B))
& ~ v6_group_1(k3_topalg_1(A,B)) ) ) ).
fof(t1_topalg_5,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v1_xreal_0(C)
=> ( r1_xreal_0(A,B)
=> ( r1_xreal_0(C,np__0)
| ! [D] :
( m1_subset_1(D,u1_struct_0(k2_topmetr(A,B)))
=> ~ ( k9_metric_1(k2_topmetr(A,B),D,C) != k1_rcomp_1(A,B)
& k9_metric_1(k2_topmetr(A,B),D,C) != k1_rcomp_2(A,k2_xcmplx_0(D,C))
& k9_metric_1(k2_topmetr(A,B),D,C) != k2_rcomp_2(k6_xcmplx_0(D,C),B)
& k9_metric_1(k2_topmetr(A,B),D,C) != k2_rcomp_1(k6_xcmplx_0(D,C),k2_xcmplx_0(D,C)) ) ) ) ) ) ) ) ).
fof(t3_topalg_5,axiom,
! [A] :
( l1_pre_topc(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( r2_hidden(C,B)
=> r1_tarski(k6_connsp_3(A,B,C),B) ) ) ) ) ).
fof(t4_topalg_5,axiom,
! [A] :
( ( v2_pre_topc(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
=> k3_pre_topc(A,C) = k3_pre_topc(B,D) ) ) ) ) ) ).
fof(t5_topalg_5,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(B)))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(B)))
=> ! [E] :
( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(A)))
=> ! [F] :
( m1_subset_1(F,k1_zfmisc_1(u1_struct_0(A)))
=> ( ( g1_pre_topc(u1_struct_0(A),u1_pre_topc(A)) = g1_pre_topc(u1_struct_0(B),u1_pre_topc(B))
& C = E
& D = F
& r1_connsp_1(B,C,D) )
=> r1_connsp_1(A,E,F) ) ) ) ) ) ) ) ).
fof(t6_topalg_5,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(A),u1_pre_topc(A)) = g1_pre_topc(u1_struct_0(B),u1_pre_topc(B))
& v1_connsp_1(A) )
=> v1_connsp_1(B) ) ) ) ).
fof(t7_topalg_5,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)))
=> ( ( g1_pre_topc(u1_struct_0(A),u1_pre_topc(A)) = g1_pre_topc(u1_struct_0(B),u1_pre_topc(B))
& C = D
& v2_connsp_1(C,A) )
=> v2_connsp_1(D,B) ) ) ) ) ) ).
fof(t8_topalg_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_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(B))
=> ! [E] :
( m1_connsp_2(E,A,C)
=> ( ( g1_pre_topc(u1_struct_0(A),u1_pre_topc(A)) = g1_pre_topc(u1_struct_0(B),u1_pre_topc(B))
& C = D )
=> m1_connsp_2(E,B,D) ) ) ) ) ) ) ).
fof(t9_topalg_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_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)
=> ( ( g1_pre_topc(u1_struct_0(A),u1_pre_topc(A)) = g1_pre_topc(u1_struct_0(B),u1_pre_topc(B))
& C = D )
=> m2_connsp_2(E,B,D) ) ) ) ) ) ) ).
fof(t10_topalg_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_subset_1(C,k1_zfmisc_1(u1_struct_0(B)))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(B)))
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,u1_struct_0(A),u1_struct_0(B))
& m2_relset_1(E,u1_struct_0(A),u1_struct_0(B)) )
=> ( ( v3_tops_2(E,A,B)
& r4_connsp_1(B,D,C) )
=> r4_connsp_1(A,k5_pre_topc(A,B,E,D),k5_pre_topc(A,B,E,C)) ) ) ) ) ) ) ).
fof(t11_topalg_5,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_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
=> ! [D] :
( ( ~ v1_xboole_0(D)
& m1_subset_1(D,k1_zfmisc_1(u1_struct_0(B))) )
=> ( ( C = D
& v2_connsp_2(C,A) )
=> v2_connsp_2(D,B) ) ) ) ) ) ).
fof(t12_topalg_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) )
=> ( ( g1_pre_topc(u1_struct_0(A),u1_pre_topc(A)) = g1_pre_topc(u1_struct_0(B),u1_pre_topc(B))
& v1_connsp_2(A) )
=> v1_connsp_2(B) ) ) ) ).
fof(t13_topalg_5,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ( v1_connsp_2(A)
<=> v2_connsp_2(k2_pre_topc(A),A) ) ) ).
fof(t14_topalg_5,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v1_tsep_1(B,A)
& m1_pre_topc(B,A) )
=> ( v1_connsp_2(A)
=> v1_connsp_2(B) ) ) ) ).
fof(t15_topalg_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) )
=> ( ( r1_borsuk_3(A,B)
& v1_connsp_2(A) )
=> v1_connsp_2(B) ) ) ) ).
fof(t16_topalg_5,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ( ? [B] :
( m1_cantor_1(B,A)
& ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ( r2_hidden(C,B)
=> v2_connsp_1(C,A) ) ) )
=> v1_connsp_2(A) ) ) ).
fof(t17_topalg_5,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ( r1_xreal_0(A,B)
=> v1_connsp_2(k4_topmetr(A,B)) ) ) ) ).
fof(d1_topalg_5,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,u1_struct_0(k5_topmetr),u1_struct_0(k3_topalg_2))
& m2_relset_1(B,u1_struct_0(k5_topmetr),u1_struct_0(k3_topalg_2)) )
=> ( B = k1_topalg_5(A)
<=> ! [C] :
( m1_subset_1(C,u1_struct_0(k5_topmetr))
=> k8_funct_2(u1_struct_0(k5_topmetr),u1_struct_0(k3_topalg_2),B,C) = k3_xcmplx_0(A,C) ) ) ) ) ).
fof(d2_topalg_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] :
( ( ~ v3_struct_0(C)
& v2_pre_topc(C)
& l1_pre_topc(C) )
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,u1_struct_0(k6_borsuk_1(A,B)),u1_struct_0(C))
& m2_relset_1(D,u1_struct_0(k6_borsuk_1(A,B)),u1_struct_0(C)) )
=> ! [E] :
( m1_subset_1(E,u1_struct_0(B))
=> ! [F] :
( ( v1_funct_1(F)
& v1_funct_2(F,u1_struct_0(A),u1_struct_0(C))
& m2_relset_1(F,u1_struct_0(A),u1_struct_0(C)) )
=> ( F = k3_topalg_5(A,B,C,D,E)
<=> ! [G] :
( m1_subset_1(G,u1_struct_0(A))
=> k8_funct_2(u1_struct_0(A),u1_struct_0(C),F,G) = k1_binop_1(D,G,E) ) ) ) ) ) ) ) ) ).
fof(d3_topalg_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] :
( ( ~ v3_struct_0(C)
& v2_pre_topc(C)
& l1_pre_topc(C) )
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,u1_struct_0(k6_borsuk_1(A,B)),u1_struct_0(C))
& m2_relset_1(D,u1_struct_0(k6_borsuk_1(A,B)),u1_struct_0(C)) )
=> ! [E] :
( m1_subset_1(E,u1_struct_0(A))
=> ! [F] :
( ( v1_funct_1(F)
& v1_funct_2(F,u1_struct_0(B),u1_struct_0(C))
& m2_relset_1(F,u1_struct_0(B),u1_struct_0(C)) )
=> ( F = k4_topalg_5(A,B,C,D,E)
<=> ! [G] :
( m1_subset_1(G,u1_struct_0(B))
=> k8_funct_2(u1_struct_0(B),u1_struct_0(C),F,G) = k1_binop_1(D,E,G) ) ) ) ) ) ) ) ) ).
fof(t18_topalg_5,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(A))
=> ! [D] :
( m1_borsuk_2(D,A,B,C)
=> ! [E] :
( m1_borsuk_2(E,A,B,C)
=> ! [F] :
( m1_borsuk_6(F,A,B,C,D,E)
=> ! [G] :
( m1_subset_1(G,u1_struct_0(k5_topmetr))
=> ( v5_pre_topc(F,k6_borsuk_1(k5_topmetr,k5_topmetr),A)
=> v5_pre_topc(k3_topalg_5(k5_topmetr,k5_topmetr,A,F,G),k5_topmetr,A) ) ) ) ) ) ) ) ) ).
fof(t19_topalg_5,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(A))
=> ! [D] :
( m1_borsuk_2(D,A,B,C)
=> ! [E] :
( m1_borsuk_2(E,A,B,C)
=> ! [F] :
( m1_borsuk_6(F,A,B,C,D,E)
=> ! [G] :
( m1_subset_1(G,u1_struct_0(k5_topmetr))
=> ( v5_pre_topc(F,k6_borsuk_1(k5_topmetr,k5_topmetr),A)
=> v5_pre_topc(k4_topalg_5(k5_topmetr,k5_topmetr,A,F,G),k5_topmetr,A) ) ) ) ) ) ) ) ) ).
fof(d4_topalg_5,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,u1_struct_0(k5_topmetr),u1_struct_0(k8_toprealb(np__2)))
& m2_relset_1(B,u1_struct_0(k5_topmetr),u1_struct_0(k8_toprealb(np__2))) )
=> ( B = k5_topalg_5(A)
<=> ! [C] :
( m1_subset_1(C,u1_struct_0(k5_topmetr))
=> k8_funct_2(u1_struct_0(k5_topmetr),u1_struct_0(k8_toprealb(np__2)),B,C) = k23_euclid(k22_sin_cos(k3_xcmplx_0(k3_xcmplx_0(k4_real_1(np__2,k32_sin_cos),A),C)),k19_sin_cos(k3_xcmplx_0(k3_xcmplx_0(k4_real_1(np__2,k32_sin_cos),A),C))) ) ) ) ) ).
fof(t20_topalg_5,axiom,
! [A] :
( v1_xreal_0(A)
=> k5_topalg_5(A) = k4_borsuk_1(k5_topmetr,k3_topalg_2,k8_toprealb(np__2),k2_topalg_5(A),k12_toprealb) ) ).
fof(t21_topalg_5,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(k8_toprealb(np__2)))))
=> ( ( r1_pre_topc(k8_toprealb(np__2),A)
& v1_tops_2(A,k8_toprealb(np__2)) )
=> ! [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(k6_borsuk_1(B,k5_topmetr)),u1_struct_0(k8_toprealb(np__2)))
& v5_pre_topc(C,k6_borsuk_1(B,k5_topmetr),k8_toprealb(np__2))
& m2_relset_1(C,u1_struct_0(k6_borsuk_1(B,k5_topmetr)),u1_struct_0(k8_toprealb(np__2))) )
=> ! [D] :
( m1_subset_1(D,u1_struct_0(B))
=> ? [E] :
( ~ v1_xboole_0(E)
& m2_finseq_1(E,k1_numbers)
& k2_seq_1(k5_numbers,k1_numbers,E,np__1) = np__0
& k2_seq_1(k5_numbers,k1_numbers,E,k3_finseq_1(E)) = np__1
& v1_seqm_3(E)
& ? [F] :
( v3_pre_topc(F,B)
& m1_subset_1(F,k1_zfmisc_1(u1_struct_0(B)))
& r2_hidden(D,F)
& ! [G] :
( v4_ordinal2(G)
=> ~ ( r2_hidden(G,k4_finseq_1(E))
& r2_hidden(k2_xcmplx_0(G,np__1),k4_finseq_1(E))
& ! [H] :
( ( ~ v1_xboole_0(H)
& m1_subset_1(H,k1_zfmisc_1(u1_struct_0(k8_toprealb(np__2)))) )
=> ~ ( r2_hidden(H,A)
& r1_tarski(k4_pre_topc(k6_borsuk_1(B,k5_topmetr),k8_toprealb(np__2),C,k2_zfmisc_1(F,k1_rcomp_1(k2_seq_1(k5_numbers,k1_numbers,E,G),k2_seq_1(k5_numbers,k1_numbers,E,k2_xcmplx_0(G,np__1))))),H) ) ) ) ) ) ) ) ) ) ) ) ).
fof(t22_topalg_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(k6_borsuk_1(A,k5_topmetr)),u1_struct_0(k8_toprealb(np__2)))
& m2_relset_1(B,u1_struct_0(k6_borsuk_1(A,k5_topmetr)),u1_struct_0(k8_toprealb(np__2))) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(k6_borsuk_1(A,k1_tex_2(k22_borsuk_1,k23_borsuk_1))),u1_struct_0(k3_topalg_2))
& m2_relset_1(C,u1_struct_0(k6_borsuk_1(A,k1_tex_2(k22_borsuk_1,k23_borsuk_1))),u1_struct_0(k3_topalg_2)) )
=> ~ ( v5_pre_topc(B,k6_borsuk_1(A,k5_topmetr),k8_toprealb(np__2))
& v5_pre_topc(C,k6_borsuk_1(A,k1_tex_2(k22_borsuk_1,k23_borsuk_1)),k3_topalg_2)
& k2_partfun1(u1_struct_0(k6_borsuk_1(A,k5_topmetr)),u1_struct_0(k8_toprealb(np__2)),B,k2_zfmisc_1(u1_struct_0(A),k1_tarski(np__0))) = k7_funct_2(u1_struct_0(k6_borsuk_1(A,k1_tex_2(k22_borsuk_1,k23_borsuk_1))),u1_struct_0(k3_topalg_2),u1_struct_0(k8_toprealb(np__2)),C,k12_toprealb)
& ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,u1_struct_0(k6_borsuk_1(A,k5_topmetr)),u1_struct_0(k3_topalg_2))
& m2_relset_1(D,u1_struct_0(k6_borsuk_1(A,k5_topmetr)),u1_struct_0(k3_topalg_2)) )
=> ~ ( v5_pre_topc(D,k6_borsuk_1(A,k5_topmetr),k3_topalg_2)
& B = k7_funct_2(u1_struct_0(k6_borsuk_1(A,k5_topmetr)),u1_struct_0(k3_topalg_2),u1_struct_0(k8_toprealb(np__2)),D,k12_toprealb)
& k2_partfun1(u1_struct_0(k6_borsuk_1(A,k5_topmetr)),u1_struct_0(k3_topalg_2),D,k2_zfmisc_1(u1_struct_0(A),k1_tarski(np__0))) = C
& ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,u1_struct_0(k6_borsuk_1(A,k5_topmetr)),u1_struct_0(k3_topalg_2))
& m2_relset_1(E,u1_struct_0(k6_borsuk_1(A,k5_topmetr)),u1_struct_0(k3_topalg_2)) )
=> ( ( v5_pre_topc(E,k6_borsuk_1(A,k5_topmetr),k3_topalg_2)
& B = k7_funct_2(u1_struct_0(k6_borsuk_1(A,k5_topmetr)),u1_struct_0(k3_topalg_2),u1_struct_0(k8_toprealb(np__2)),E,k12_toprealb)
& k2_partfun1(u1_struct_0(k6_borsuk_1(A,k5_topmetr)),u1_struct_0(k3_topalg_2),E,k2_zfmisc_1(u1_struct_0(A),k1_tarski(np__0))) = C )
=> D = E ) ) ) ) ) ) ) ) ).
fof(t23_topalg_5,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k8_toprealb(np__2)))
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k8_toprealb(np__2)))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k3_topalg_2))
=> ! [D] :
( m1_borsuk_2(D,k8_toprealb(np__2),A,B)
=> ~ ( r2_hidden(C,k5_pre_topc(k3_topalg_2,k8_toprealb(np__2),k12_toprealb,k1_struct_0(k8_toprealb(np__2),A)))
& ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,u1_struct_0(k5_topmetr),u1_struct_0(k3_topalg_2))
& m2_relset_1(E,u1_struct_0(k5_topmetr),u1_struct_0(k3_topalg_2)) )
=> ~ ( k2_seq_1(u1_struct_0(k5_topmetr),u1_struct_0(k3_topalg_2),E,np__0) = C
& D = k7_funct_2(u1_struct_0(k5_topmetr),u1_struct_0(k3_topalg_2),u1_struct_0(k8_toprealb(np__2)),E,k12_toprealb)
& v5_pre_topc(E,k5_topmetr,k3_topalg_2)
& ! [F] :
( ( v1_funct_1(F)
& v1_funct_2(F,u1_struct_0(k5_topmetr),u1_struct_0(k3_topalg_2))
& m2_relset_1(F,u1_struct_0(k5_topmetr),u1_struct_0(k3_topalg_2)) )
=> ( ( v5_pre_topc(F,k5_topmetr,k3_topalg_2)
& D = k7_funct_2(u1_struct_0(k5_topmetr),u1_struct_0(k3_topalg_2),u1_struct_0(k8_toprealb(np__2)),F,k12_toprealb)
& k2_seq_1(u1_struct_0(k5_topmetr),u1_struct_0(k3_topalg_2),F,np__0) = C )
=> E = F ) ) ) ) ) ) ) ) ) ).
fof(t24_topalg_5,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k8_toprealb(np__2)))
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k8_toprealb(np__2)))
=> ! [C] :
( m1_borsuk_2(C,k8_toprealb(np__2),A,B)
=> ! [D] :
( m1_borsuk_2(D,k8_toprealb(np__2),A,B)
=> ! [E] :
( m1_borsuk_6(E,k8_toprealb(np__2),A,B,C,D)
=> ! [F] :
( m1_subset_1(F,u1_struct_0(k3_topalg_2))
=> ~ ( r4_borsuk_2(k8_toprealb(np__2),A,B,C,D)
& r2_hidden(F,k5_pre_topc(k3_topalg_2,k8_toprealb(np__2),k12_toprealb,k1_struct_0(k8_toprealb(np__2),A)))
& ! [G] :
( m1_subset_1(G,u1_struct_0(k3_topalg_2))
=> ! [H] :
( m1_borsuk_2(H,k3_topalg_2,F,G)
=> ! [I] :
( m1_borsuk_2(I,k3_topalg_2,F,G)
=> ! [J] :
( m1_borsuk_6(J,k3_topalg_2,F,G,H,I)
=> ~ ( r4_borsuk_2(k3_topalg_2,F,G,H,I)
& E = k4_borsuk_1(k6_borsuk_1(k5_topmetr,k5_topmetr),k3_topalg_2,k8_toprealb(np__2),J,k12_toprealb)
& r2_hidden(G,k5_pre_topc(k3_topalg_2,k8_toprealb(np__2),k12_toprealb,k1_struct_0(k8_toprealb(np__2),B)))
& ! [K] :
( m1_borsuk_6(K,k3_topalg_2,F,G,H,I)
=> ( E = k4_borsuk_1(k6_borsuk_1(k5_topmetr,k5_topmetr),k3_topalg_2,k8_toprealb(np__2),K,k12_toprealb)
=> J = K ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
fof(d5_topalg_5,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,u1_struct_0(k3_gr_cy_1),u1_struct_0(k3_topalg_1(k8_toprealb(np__2),k9_toprealb)))
& m2_relset_1(A,u1_struct_0(k3_gr_cy_1),u1_struct_0(k3_topalg_1(k8_toprealb(np__2),k9_toprealb))) )
=> ( A = k7_topalg_5
<=> ! [B] :
( v1_int_1(B)
=> k1_funct_1(A,B) = k6_eqrel_1(k1_topalg_1(k8_toprealb(np__2),k9_toprealb),k2_topalg_1(k8_toprealb(np__2),k9_toprealb),k6_topalg_5(B)) ) ) ) ).
fof(t25_topalg_5,axiom,
! [A] :
( v1_int_1(A)
=> ! [B] :
( m1_borsuk_2(B,k3_topalg_2,k4_toprealb(np__0),k4_toprealb(A))
=> k1_funct_1(k7_topalg_5,A) = k6_eqrel_1(k1_topalg_1(k8_toprealb(np__2),k9_toprealb),k2_topalg_1(k8_toprealb(np__2),k9_toprealb),k4_borsuk_1(k5_topmetr,k3_topalg_2,k8_toprealb(np__2),B,k12_toprealb)) ) ) ).
fof(t26_topalg_5,axiom,
v4_group_6(k8_topalg_5,k3_gr_cy_1,k3_topalg_1(k8_toprealb(np__2),k9_toprealb)) ).
fof(t27_topalg_5,axiom,
! [A] :
( ( v1_toprealb(A)
& m1_pre_topc(A,k15_euclid(np__2)) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> r2_group_6(k3_gr_cy_1,k3_topalg_1(A,B)) ) ) ).
fof(dt_k1_topalg_5,axiom,
! [A] :
( v1_xreal_0(A)
=> ( v1_funct_1(k1_topalg_5(A))
& v1_funct_2(k1_topalg_5(A),u1_struct_0(k5_topmetr),u1_struct_0(k3_topalg_2))
& m2_relset_1(k1_topalg_5(A),u1_struct_0(k5_topmetr),u1_struct_0(k3_topalg_2)) ) ) ).
fof(dt_k2_topalg_5,axiom,
! [A] :
( v1_xreal_0(A)
=> m1_borsuk_2(k2_topalg_5(A),k3_topalg_2,k4_toprealb(np__0),k4_toprealb(A)) ) ).
fof(redefinition_k2_topalg_5,axiom,
! [A] :
( v1_xreal_0(A)
=> k2_topalg_5(A) = k1_topalg_5(A) ) ).
fof(dt_k3_topalg_5,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(k6_borsuk_1(A,B)),u1_struct_0(C))
& m1_relset_1(D,u1_struct_0(k6_borsuk_1(A,B)),u1_struct_0(C))
& m1_subset_1(E,u1_struct_0(B)) )
=> ( v1_funct_1(k3_topalg_5(A,B,C,D,E))
& v1_funct_2(k3_topalg_5(A,B,C,D,E),u1_struct_0(A),u1_struct_0(C))
& m2_relset_1(k3_topalg_5(A,B,C,D,E),u1_struct_0(A),u1_struct_0(C)) ) ) ).
fof(dt_k4_topalg_5,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(k6_borsuk_1(A,B)),u1_struct_0(C))
& m1_relset_1(D,u1_struct_0(k6_borsuk_1(A,B)),u1_struct_0(C))
& m1_subset_1(E,u1_struct_0(A)) )
=> ( v1_funct_1(k4_topalg_5(A,B,C,D,E))
& v1_funct_2(k4_topalg_5(A,B,C,D,E),u1_struct_0(B),u1_struct_0(C))
& m2_relset_1(k4_topalg_5(A,B,C,D,E),u1_struct_0(B),u1_struct_0(C)) ) ) ).
fof(dt_k5_topalg_5,axiom,
! [A] :
( v1_xreal_0(A)
=> ( v1_funct_1(k5_topalg_5(A))
& v1_funct_2(k5_topalg_5(A),u1_struct_0(k5_topmetr),u1_struct_0(k8_toprealb(np__2)))
& m2_relset_1(k5_topalg_5(A),u1_struct_0(k5_topmetr),u1_struct_0(k8_toprealb(np__2))) ) ) ).
fof(dt_k6_topalg_5,axiom,
! [A] :
( v1_int_1(A)
=> m1_borsuk_2(k6_topalg_5(A),k8_toprealb(np__2),k9_toprealb,k9_toprealb) ) ).
fof(redefinition_k6_topalg_5,axiom,
! [A] :
( v1_int_1(A)
=> k6_topalg_5(A) = k5_topalg_5(A) ) ).
fof(dt_k7_topalg_5,axiom,
( v1_funct_1(k7_topalg_5)
& v1_funct_2(k7_topalg_5,u1_struct_0(k3_gr_cy_1),u1_struct_0(k3_topalg_1(k8_toprealb(np__2),k9_toprealb)))
& m2_relset_1(k7_topalg_5,u1_struct_0(k3_gr_cy_1),u1_struct_0(k3_topalg_1(k8_toprealb(np__2),k9_toprealb))) ) ).
fof(dt_k8_topalg_5,axiom,
( v1_funct_1(k8_topalg_5)
& v1_funct_2(k8_topalg_5,u1_struct_0(k3_gr_cy_1),u1_struct_0(k3_topalg_1(k8_toprealb(np__2),k9_toprealb)))
& v1_group_6(k8_topalg_5,k3_gr_cy_1,k3_topalg_1(k8_toprealb(np__2),k9_toprealb))
& m2_relset_1(k8_topalg_5,u1_struct_0(k3_gr_cy_1),u1_struct_0(k3_topalg_1(k8_toprealb(np__2),k9_toprealb))) ) ).
fof(redefinition_k8_topalg_5,axiom,
k8_topalg_5 = k7_topalg_5 ).
fof(t2_topalg_5,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ~ ( r1_xreal_0(A,B)
& ! [C] :
( m1_cantor_1(C,k4_topmetr(A,B))
=> ~ ( ? [D] :
( m1_pboole(D,u1_struct_0(k4_topmetr(A,B)))
& ! [E] :
( m1_subset_1(E,u1_struct_0(k2_topmetr(A,B)))
=> ( k1_funct_1(D,E) = a_3_0_topalg_5(A,B,E)
& C = k3_card_3(D) ) ) )
& ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k4_topmetr(A,B))))
=> ( r2_hidden(D,C)
=> v2_connsp_1(D,k4_topmetr(A,B)) ) ) ) ) ) ) ) ).
fof(fraenkel_a_3_0_topalg_5,axiom,
! [A,B,C,D] :
( ( v1_xreal_0(B)
& v1_xreal_0(C)
& m1_subset_1(D,u1_struct_0(k2_topmetr(B,C))) )
=> ( r2_hidden(A,a_3_0_topalg_5(B,C,D))
<=> ? [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
& A = k9_metric_1(k2_topmetr(B,C),D,k6_real_1(np__1,E))
& E != np__0 ) ) ) ).
%------------------------------------------------------------------------------