SET007 Axioms: SET007+521.ax
%------------------------------------------------------------------------------
% File : SET007+521 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Introduction to the Homotopy Theory
% Version : [Urb08] axioms.
% English :
% Refs : [Mat90] Matuszewski (1990), Formalized Mathematics
% : [Urb07] Urban (2007), MPTP 0.2: Design, Implementation, and In
% : [Urb08] Urban (2006), Email to G. Sutcliffe
% Source : [Urb08]
% Names : borsuk_2 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 51 ( 3 unt; 0 def)
% Number of atoms : 518 ( 26 equ)
% Maximal formula atoms : 33 ( 10 avg)
% Number of connectives : 541 ( 74 ~; 0 |; 326 &)
% ( 10 <=>; 131 =>; 0 <=; 0 <~>)
% Maximal formula depth : 27 ( 11 avg)
% Maximal term depth : 4 ( 1 avg)
% Number of predicates : 39 ( 37 usr; 1 prp; 0-5 aty)
% Number of functors : 34 ( 34 usr; 8 con; 0-6 aty)
% Number of variables : 179 ( 170 !; 9 ?)
% 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(fc1_borsuk_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_pre_topc(A) )
=> ( ~ v1_xboole_0(k7_grcat_1(A))
& v1_relat_1(k7_grcat_1(A))
& v1_funct_1(k7_grcat_1(A))
& v2_funct_1(k7_grcat_1(A))
& v1_funct_2(k7_grcat_1(A),u1_struct_0(A),u1_struct_0(A))
& v5_pre_topc(k7_grcat_1(A),A,A)
& v1_partfun1(k7_grcat_1(A),u1_struct_0(A),u1_struct_0(A))
& v1_t_0topsp(k7_grcat_1(A),A,A) ) ) ).
fof(rc1_borsuk_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_pre_topc(A) )
=> ? [B] :
( m1_relset_1(B,u1_struct_0(A),u1_struct_0(A))
& ~ v1_xboole_0(B)
& v1_relat_1(B)
& v1_funct_1(B)
& v2_funct_1(B)
& v1_funct_2(B,u1_struct_0(A),u1_struct_0(A))
& v5_pre_topc(B,A,A)
& v1_partfun1(B,u1_struct_0(A),u1_struct_0(A)) ) ) ).
fof(cc1_borsuk_2,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k5_topmetr))
=> ( v1_xreal_0(A)
& v1_xcmplx_0(A) ) ) ).
fof(rc2_borsuk_2,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_xboole_0(C)
& v1_relat_1(C)
& v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(k5_topmetr),u1_struct_0(A))
& v5_pre_topc(C,k5_topmetr,A)
& v1_partfun1(C,u1_struct_0(k5_topmetr),u1_struct_0(A)) ) ) ).
fof(rc3_borsuk_2,axiom,
? [A] :
( l1_pre_topc(A)
& ~ v3_struct_0(A)
& v2_pre_topc(A)
& v1_borsuk_2(A) ) ).
fof(cc2_borsuk_2,axiom,
! [A,B,C] :
( ( v1_borsuk_2(A)
& l1_pre_topc(A)
& m1_subset_1(B,u1_struct_0(A))
& m1_subset_1(C,u1_struct_0(A)) )
=> ! [D] :
( m1_borsuk_2(D,A,B,C)
=> ( v1_relat_1(D)
& v5_pre_topc(D,k5_topmetr,A) ) ) ) ).
fof(cc3_borsuk_2,axiom,
! [A] :
( l1_pre_topc(A)
=> ( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& v1_borsuk_2(A) )
=> ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& v1_connsp_1(A) ) ) ) ).
fof(rc4_borsuk_2,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_xboole_0(C)
& v1_relat_1(C)
& v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(k5_topmetr),u1_struct_0(A))
& v1_partfun1(C,u1_struct_0(k5_topmetr),u1_struct_0(A))
& v5_seqm_3(C) ) ) ).
fof(fc2_borsuk_2,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A)
& m1_subset_1(B,u1_struct_0(A))
& v5_seqm_3(C)
& m1_borsuk_2(C,A,B,B) )
=> ( ~ v1_xboole_0(k1_borsuk_2(A,B,B,B,C,C))
& v1_relat_1(k1_borsuk_2(A,B,B,B,C,C))
& v1_funct_1(k1_borsuk_2(A,B,B,B,C,C))
& v1_funct_2(k1_borsuk_2(A,B,B,B,C,C),u1_struct_0(k5_topmetr),u1_struct_0(A))
& v1_partfun1(k1_borsuk_2(A,B,B,B,C,C),u1_struct_0(k5_topmetr),u1_struct_0(A))
& v5_seqm_3(k1_borsuk_2(A,B,B,B,C,C)) ) ) ).
fof(fc3_borsuk_2,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A)
& m1_subset_1(B,u1_struct_0(A))
& v5_seqm_3(C)
& m1_borsuk_2(C,A,B,B) )
=> ( ~ v1_xboole_0(k2_borsuk_2(A,B,B,C))
& v1_relat_1(k2_borsuk_2(A,B,B,C))
& v1_funct_1(k2_borsuk_2(A,B,B,C))
& v1_funct_2(k2_borsuk_2(A,B,B,C),u1_struct_0(k5_topmetr),u1_struct_0(A))
& v1_partfun1(k2_borsuk_2(A,B,B,C),u1_struct_0(k5_topmetr),u1_struct_0(A))
& v5_seqm_3(k2_borsuk_2(A,B,B,C)) ) ) ).
fof(cc4_borsuk_2,axiom,
! [A] :
( l1_pre_topc(A)
=> ( v3_struct_0(A)
=> v2_t_0topsp(A) ) ) ).
fof(fc4_borsuk_2,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& v2_t_0topsp(A)
& l1_pre_topc(A)
& ~ v3_struct_0(B)
& v2_pre_topc(B)
& v2_t_0topsp(B)
& l1_pre_topc(B) )
=> ( ~ v3_struct_0(k6_borsuk_1(A,B))
& v1_pre_topc(k6_borsuk_1(A,B))
& v2_pre_topc(k6_borsuk_1(A,B))
& v2_t_0topsp(k6_borsuk_1(A,B)) ) ) ).
fof(fc5_borsuk_2,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& v1_urysohn1(A)
& l1_pre_topc(A)
& ~ v3_struct_0(B)
& v2_pre_topc(B)
& v1_urysohn1(B)
& l1_pre_topc(B) )
=> ( ~ v3_struct_0(k6_borsuk_1(A,B))
& v1_pre_topc(k6_borsuk_1(A,B))
& v2_pre_topc(k6_borsuk_1(A,B))
& v1_urysohn1(k6_borsuk_1(A,B)) ) ) ).
fof(fc6_borsuk_2,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& v3_compts_1(A)
& l1_pre_topc(A)
& ~ v3_struct_0(B)
& v2_pre_topc(B)
& v3_compts_1(B)
& l1_pre_topc(B) )
=> ( ~ v3_struct_0(k6_borsuk_1(A,B))
& v1_pre_topc(k6_borsuk_1(A,B))
& v2_pre_topc(k6_borsuk_1(A,B))
& v2_t_0topsp(k6_borsuk_1(A,B))
& v3_compts_1(k6_borsuk_1(A,B))
& v3_yellow_8(k6_borsuk_1(A,B)) ) ) ).
fof(fc7_borsuk_2,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)
& v3_yellow_8(k22_borsuk_1) ) ).
fof(t1_borsuk_2,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] :
( ( 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)) )
=> ! [F] :
( ( v1_funct_1(F)
& v1_funct_2(F,u1_struct_0(C),u1_struct_0(B))
& m2_relset_1(F,u1_struct_0(C),u1_struct_0(B)) )
=> ~ ( m1_pre_topc(A,D)
& m1_pre_topc(C,D)
& k2_xboole_0(k2_pre_topc(A),k2_pre_topc(C)) = k2_pre_topc(D)
& v2_compts_1(A)
& v2_compts_1(C)
& v3_compts_1(D)
& v5_pre_topc(E,A,B)
& v5_pre_topc(F,C,B)
& ! [G] :
( r2_hidden(G,k3_xboole_0(k2_pre_topc(A),k2_pre_topc(C)))
=> k1_funct_1(E,G) = k1_funct_1(F,G) )
& ! [G] :
( ( v1_funct_1(G)
& v1_funct_2(G,u1_struct_0(D),u1_struct_0(B))
& m2_relset_1(G,u1_struct_0(D),u1_struct_0(B)) )
=> ~ ( G = k1_funct_4(E,F)
& v5_pre_topc(G,D,B) ) ) ) ) ) ) ) ) ) ).
fof(t2_borsuk_2,axiom,
$true ).
fof(t3_borsuk_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v2_pre_topc(B)
& l1_pre_topc(B) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
& m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
=> ( v3_tops_2(C,A,B)
=> v1_t_0topsp(k2_tops_2(A,B,C),B,A) ) ) ) ) ).
fof(t4_borsuk_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ? [C] :
( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(k5_topmetr),u1_struct_0(A))
& m2_relset_1(C,u1_struct_0(k5_topmetr),u1_struct_0(A))
& v5_pre_topc(C,k5_topmetr,A)
& k1_funct_1(C,np__0) = B
& k1_funct_1(C,np__1) = B ) ) ) ).
fof(d1_borsuk_2,axiom,
! [A] :
( l1_pre_topc(A)
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( r1_borsuk_2(A,B,C)
<=> ? [D] :
( v1_funct_1(D)
& v1_funct_2(D,u1_struct_0(k5_topmetr),u1_struct_0(A))
& m2_relset_1(D,u1_struct_0(k5_topmetr),u1_struct_0(A))
& v5_pre_topc(D,k5_topmetr,A)
& k1_funct_1(D,np__0) = B
& k1_funct_1(D,np__1) = C ) ) ) ) ) ).
fof(d2_borsuk_2,axiom,
! [A] :
( l1_pre_topc(A)
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( r1_borsuk_2(A,B,C)
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,u1_struct_0(k5_topmetr),u1_struct_0(A))
& m2_relset_1(D,u1_struct_0(k5_topmetr),u1_struct_0(A)) )
=> ( m1_borsuk_2(D,A,B,C)
<=> ( v5_pre_topc(D,k5_topmetr,A)
& k1_funct_1(D,np__0) = B
& k1_funct_1(D,np__1) = C ) ) ) ) ) ) ) ).
fof(d3_borsuk_2,axiom,
! [A] :
( l1_pre_topc(A)
=> ( v1_borsuk_2(A)
<=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> r1_borsuk_2(A,B,C) ) ) ) ) ).
fof(d4_borsuk_2,axiom,
! [A] :
( ( v1_borsuk_2(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,u1_struct_0(k5_topmetr),u1_struct_0(A))
& m2_relset_1(D,u1_struct_0(k5_topmetr),u1_struct_0(A)) )
=> ( m1_borsuk_2(D,A,B,C)
<=> ( v5_pre_topc(D,k5_topmetr,A)
& k1_funct_1(D,np__0) = B
& k1_funct_1(D,np__1) = C ) ) ) ) ) ) ).
fof(t5_borsuk_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ( v1_borsuk_2(A)
=> v1_connsp_1(A) ) ) ).
fof(d5_borsuk_2,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_subset_1(D,u1_struct_0(A))
=> ! [E] :
( m1_borsuk_2(E,A,B,C)
=> ! [F] :
( m1_borsuk_2(F,A,C,D)
=> ( ( r2_borsuk_2(A,B,C)
& r2_borsuk_2(A,C,D) )
=> ! [G] :
( m1_borsuk_2(G,A,B,D)
=> ( G = k1_borsuk_2(A,B,C,D,E,F)
<=> ! [H] :
( m1_subset_1(H,u1_struct_0(k5_topmetr))
=> ( ( r1_xreal_0(H,k6_real_1(np__1,np__2))
=> k8_funct_2(u1_struct_0(k5_topmetr),u1_struct_0(A),G,H) = k1_funct_1(E,k3_xcmplx_0(np__2,H)) )
& ( r1_xreal_0(k6_real_1(np__1,np__2),H)
=> k8_funct_2(u1_struct_0(k5_topmetr),u1_struct_0(A),G,H) = k1_funct_1(F,k6_xcmplx_0(k3_xcmplx_0(np__2,H),np__1)) ) ) ) ) ) ) ) ) ) ) ) ) ).
fof(t6_borsuk_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( ( v5_seqm_3(C)
& m1_borsuk_2(C,A,B,B) )
=> C = k3_borsuk_1(k5_topmetr,A,B) ) ) ) ).
fof(t7_borsuk_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( ( v5_seqm_3(C)
& m1_borsuk_2(C,A,B,B) )
=> k1_borsuk_2(A,B,B,B,C,C) = C ) ) ) ).
fof(d6_borsuk_2,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)
=> ( r2_borsuk_2(A,B,C)
=> ! [E] :
( m1_borsuk_2(E,A,C,B)
=> ( E = k2_borsuk_2(A,B,C,D)
<=> ! [F] :
( m1_subset_1(F,u1_struct_0(k5_topmetr))
=> k8_funct_2(u1_struct_0(k5_topmetr),u1_struct_0(A),E,F) = k1_funct_1(D,k6_xcmplx_0(np__1,F)) ) ) ) ) ) ) ) ) ).
fof(t8_borsuk_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( ( v5_seqm_3(C)
& m1_borsuk_2(C,A,B,B) )
=> k2_borsuk_2(A,B,B,C) = C ) ) ) ).
fof(t9_borsuk_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v2_pre_topc(B)
& l1_pre_topc(B) )
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(B))))
=> ! [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)) )
=> k5_pre_topc(A,B,D,k5_setfam_1(u1_struct_0(B),C)) = k5_setfam_1(u1_struct_0(A),k1_weierstr(A,B,D,C)) ) ) ) ) ).
fof(t10_borsuk_2,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] :
( ( v1_funct_1(E)
& v1_funct_2(E,u1_struct_0(A),u1_struct_0(C))
& v5_pre_topc(E,A,C)
& m2_relset_1(E,u1_struct_0(A),u1_struct_0(C)) )
=> ! [F] :
( ( v1_funct_1(F)
& v1_funct_2(F,u1_struct_0(B),u1_struct_0(D))
& v5_pre_topc(F,B,D)
& m2_relset_1(F,u1_struct_0(B),u1_struct_0(D)) )
=> ! [G] :
( m1_subset_1(G,k1_zfmisc_1(u1_struct_0(k6_borsuk_1(C,D))))
=> ! [H] :
( m1_subset_1(H,k1_zfmisc_1(u1_struct_0(k6_borsuk_1(C,D))))
=> ( r2_hidden(H,k11_borsuk_1(C,D,G))
=> v3_pre_topc(k5_pre_topc(k6_borsuk_1(A,B),k6_borsuk_1(C,D),k3_borsuk_2(A,C,B,D,E,F),H),k6_borsuk_1(A,B)) ) ) ) ) ) ) ) ) ) ).
fof(t11_borsuk_2,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] :
( ( v1_funct_1(E)
& v1_funct_2(E,u1_struct_0(A),u1_struct_0(C))
& v5_pre_topc(E,A,C)
& m2_relset_1(E,u1_struct_0(A),u1_struct_0(C)) )
=> ! [F] :
( ( v1_funct_1(F)
& v1_funct_2(F,u1_struct_0(B),u1_struct_0(D))
& v5_pre_topc(F,B,D)
& m2_relset_1(F,u1_struct_0(B),u1_struct_0(D)) )
=> ! [G] :
( m1_subset_1(G,k1_zfmisc_1(u1_struct_0(k6_borsuk_1(C,D))))
=> ( v3_pre_topc(G,k6_borsuk_1(C,D))
=> v3_pre_topc(k5_pre_topc(k6_borsuk_1(A,B),k6_borsuk_1(C,D),k3_borsuk_2(A,C,B,D,E,F),G),k6_borsuk_1(A,B)) ) ) ) ) ) ) ) ) ).
fof(t12_borsuk_2,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] :
( ( v1_funct_1(E)
& v1_funct_2(E,u1_struct_0(A),u1_struct_0(C))
& v5_pre_topc(E,A,C)
& m2_relset_1(E,u1_struct_0(A),u1_struct_0(C)) )
=> ! [F] :
( ( v1_funct_1(F)
& v1_funct_2(F,u1_struct_0(B),u1_struct_0(D))
& v5_pre_topc(F,B,D)
& m2_relset_1(F,u1_struct_0(B),u1_struct_0(D)) )
=> v5_pre_topc(k3_borsuk_2(A,C,B,D,E,F),k6_borsuk_1(A,B),k6_borsuk_1(C,D)) ) ) ) ) ) ) ).
fof(t13_borsuk_2,axiom,
$true ).
fof(t14_borsuk_2,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) )
=> ( ( v1_urysohn1(A)
& v1_urysohn1(B) )
=> v1_urysohn1(k6_borsuk_1(A,B)) ) ) ) ).
fof(d7_borsuk_2,axiom,
! [A] :
( ( ~ v3_struct_0(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)
=> ( r3_borsuk_2(A,B,C,D,E)
<=> ? [F] :
( v1_funct_1(F)
& v1_funct_2(F,u1_struct_0(k6_borsuk_1(k5_topmetr,k5_topmetr)),u1_struct_0(A))
& m2_relset_1(F,u1_struct_0(k6_borsuk_1(k5_topmetr,k5_topmetr)),u1_struct_0(A))
& v5_pre_topc(F,k6_borsuk_1(k5_topmetr,k5_topmetr),A)
& ! [G] :
( m1_subset_1(G,u1_struct_0(k5_topmetr))
=> ( k1_binop_1(F,G,np__0) = k8_funct_2(u1_struct_0(k5_topmetr),u1_struct_0(A),D,G)
& k1_binop_1(F,G,np__1) = k8_funct_2(u1_struct_0(k5_topmetr),u1_struct_0(A),E,G)
& ! [H] :
( m1_subset_1(H,u1_struct_0(k5_topmetr))
=> ( k1_binop_1(F,np__0,H) = B
& k1_binop_1(F,np__1,H) = C ) ) ) ) ) ) ) ) ) ) ) ).
fof(t15_borsuk_2,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)
=> ( r2_borsuk_2(A,B,C)
=> r3_borsuk_2(A,B,C,D,D) ) ) ) ) ) ).
fof(dt_m1_borsuk_2,axiom,
! [A,B,C] :
( ( l1_pre_topc(A)
& m1_subset_1(B,u1_struct_0(A))
& m1_subset_1(C,u1_struct_0(A)) )
=> ! [D] :
( m1_borsuk_2(D,A,B,C)
=> ( v1_funct_1(D)
& v1_funct_2(D,u1_struct_0(k5_topmetr),u1_struct_0(A))
& m2_relset_1(D,u1_struct_0(k5_topmetr),u1_struct_0(A)) ) ) ) ).
fof(existence_m1_borsuk_2,axiom,
! [A,B,C] :
( ( l1_pre_topc(A)
& m1_subset_1(B,u1_struct_0(A))
& m1_subset_1(C,u1_struct_0(A)) )
=> ? [D] : m1_borsuk_2(D,A,B,C) ) ).
fof(reflexivity_r2_borsuk_2,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A)
& m1_subset_1(B,u1_struct_0(A))
& m1_subset_1(C,u1_struct_0(A)) )
=> r2_borsuk_2(A,B,B) ) ).
fof(redefinition_r2_borsuk_2,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A)
& m1_subset_1(B,u1_struct_0(A))
& m1_subset_1(C,u1_struct_0(A)) )
=> ( r2_borsuk_2(A,B,C)
<=> r1_borsuk_2(A,B,C) ) ) ).
fof(symmetry_r3_borsuk_2,axiom,
! [A,B,C,D,E] :
( ( ~ v3_struct_0(A)
& l1_pre_topc(A)
& m1_subset_1(B,u1_struct_0(A))
& m1_subset_1(C,u1_struct_0(A))
& m1_borsuk_2(D,A,B,C)
& m1_borsuk_2(E,A,B,C) )
=> ( r3_borsuk_2(A,B,C,D,E)
=> r3_borsuk_2(A,B,C,E,D) ) ) ).
fof(symmetry_r4_borsuk_2,axiom,
! [A,B,C,D,E] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& v1_borsuk_2(A)
& l1_pre_topc(A)
& m1_subset_1(B,u1_struct_0(A))
& m1_subset_1(C,u1_struct_0(A))
& m1_borsuk_2(D,A,B,C)
& m1_borsuk_2(E,A,B,C) )
=> ( r4_borsuk_2(A,B,C,D,E)
=> r4_borsuk_2(A,B,C,E,D) ) ) ).
fof(reflexivity_r4_borsuk_2,axiom,
! [A,B,C,D,E] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& v1_borsuk_2(A)
& l1_pre_topc(A)
& m1_subset_1(B,u1_struct_0(A))
& m1_subset_1(C,u1_struct_0(A))
& m1_borsuk_2(D,A,B,C)
& m1_borsuk_2(E,A,B,C) )
=> r4_borsuk_2(A,B,C,D,D) ) ).
fof(redefinition_r4_borsuk_2,axiom,
! [A,B,C,D,E] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& v1_borsuk_2(A)
& l1_pre_topc(A)
& m1_subset_1(B,u1_struct_0(A))
& m1_subset_1(C,u1_struct_0(A))
& m1_borsuk_2(D,A,B,C)
& m1_borsuk_2(E,A,B,C) )
=> ( r4_borsuk_2(A,B,C,D,E)
<=> r3_borsuk_2(A,B,C,D,E) ) ) ).
fof(dt_k1_borsuk_2,axiom,
! [A,B,C,D,E,F] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A)
& m1_subset_1(B,u1_struct_0(A))
& m1_subset_1(C,u1_struct_0(A))
& m1_subset_1(D,u1_struct_0(A))
& m1_borsuk_2(E,A,B,C)
& m1_borsuk_2(F,A,C,D) )
=> m1_borsuk_2(k1_borsuk_2(A,B,C,D,E,F),A,B,D) ) ).
fof(dt_k2_borsuk_2,axiom,
! [A,B,C,D] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A)
& m1_subset_1(B,u1_struct_0(A))
& m1_subset_1(C,u1_struct_0(A))
& m1_borsuk_2(D,A,B,C) )
=> m1_borsuk_2(k2_borsuk_2(A,B,C,D),A,C,B) ) ).
fof(dt_k3_borsuk_2,axiom,
! [A,B,C,D,E,F] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A)
& ~ v3_struct_0(B)
& v2_pre_topc(B)
& l1_pre_topc(B)
& ~ v3_struct_0(C)
& v2_pre_topc(C)
& l1_pre_topc(C)
& ~ v3_struct_0(D)
& v2_pre_topc(D)
& l1_pre_topc(D)
& v1_funct_1(E)
& v1_funct_2(E,u1_struct_0(A),u1_struct_0(B))
& m1_relset_1(E,u1_struct_0(A),u1_struct_0(B))
& v1_funct_1(F)
& v1_funct_2(F,u1_struct_0(C),u1_struct_0(D))
& m1_relset_1(F,u1_struct_0(C),u1_struct_0(D)) )
=> ( v1_funct_1(k3_borsuk_2(A,B,C,D,E,F))
& v1_funct_2(k3_borsuk_2(A,B,C,D,E,F),u1_struct_0(k6_borsuk_1(A,C)),u1_struct_0(k6_borsuk_1(B,D)))
& m2_relset_1(k3_borsuk_2(A,B,C,D,E,F),u1_struct_0(k6_borsuk_1(A,C)),u1_struct_0(k6_borsuk_1(B,D))) ) ) ).
fof(redefinition_k3_borsuk_2,axiom,
! [A,B,C,D,E,F] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A)
& ~ v3_struct_0(B)
& v2_pre_topc(B)
& l1_pre_topc(B)
& ~ v3_struct_0(C)
& v2_pre_topc(C)
& l1_pre_topc(C)
& ~ v3_struct_0(D)
& v2_pre_topc(D)
& l1_pre_topc(D)
& v1_funct_1(E)
& v1_funct_2(E,u1_struct_0(A),u1_struct_0(B))
& m1_relset_1(E,u1_struct_0(A),u1_struct_0(B))
& v1_funct_1(F)
& v1_funct_2(F,u1_struct_0(C),u1_struct_0(D))
& m1_relset_1(F,u1_struct_0(C),u1_struct_0(D)) )
=> k3_borsuk_2(A,B,C,D,E,F) = k15_funct_3(E,F) ) ).
fof(s1_borsuk_2,axiom,
r1_tarski(k1_card_1(a_0_0_borsuk_2),k1_card_1(f2_s1_borsuk_2)) ).
fof(fraenkel_a_0_0_borsuk_2,axiom,
! [A] :
( r2_hidden(A,a_0_0_borsuk_2)
<=> ? [B] :
( m1_subset_1(B,f1_s1_borsuk_2)
& A = f3_s1_borsuk_2(B)
& r2_hidden(B,f2_s1_borsuk_2)
& p1_s1_borsuk_2(B) ) ) ).
%------------------------------------------------------------------------------