SET007 Axioms: SET007+587.ax


%------------------------------------------------------------------------------
% File     : SET007+587 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Properties of the Product of Compact Topological Spaces
% 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_3 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   44 (   0 unit)
%            Number of atoms       :  422 (  20 equality)
%            Maximal formula depth :   23 (  10 average)
%            Number of connectives :  441 (  63 ~  ;   0  |; 247  &)
%                                         (   5 <=>; 126 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   30 (   0 propositional; 1-3 arity)
%            Number of functors    :   24 (   0 constant; 1-5 arity)
%            Number of variables   :  140 (   0 singleton; 132 !;   8 ?)
%            Maximal term depth    :    5 (   2 average)
% 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_3,axiom,(
    ! [A,B] :
      ( v1_xboole_0(B)
     => v1_xboole_0(k2_zfmisc_1(A,B)) ) )).

fof(fc2_borsuk_3,axiom,(
    ! [A,B] :
      ( v1_xboole_0(A)
     => v1_xboole_0(k2_zfmisc_1(A,B)) ) )).

fof(fc3_borsuk_3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_pre_topc(A) )
     => ( v1_relat_1(k7_grcat_1(A))
        & v1_funct_1(k7_grcat_1(A))
        & v2_funct_1(k7_grcat_1(A))
        & ~ v1_xboole_0(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)
        & v3_tops_2(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(fc4_borsuk_3,axiom,(
    ! [A,B] :
      ( ( l1_pre_topc(A)
        & v1_xboole_0(B)
        & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
     => ( v3_struct_0(k3_pre_topc(A,B))
        & v1_pre_topc(k3_pre_topc(A,B))
        & v2_t_0topsp(k3_pre_topc(A,B)) ) ) )).

fof(rc1_borsuk_3,axiom,(
    ? [A] :
      ( l1_pre_topc(A)
      & v3_struct_0(A)
      & v1_pre_topc(A)
      & v2_pre_topc(A)
      & v2_t_0topsp(A) ) )).

fof(cc1_borsuk_3,axiom,(
    ! [A] :
      ( l1_pre_topc(A)
     => ( ( v3_struct_0(A)
          & v2_pre_topc(A) )
       => ( v2_pre_topc(A)
          & v2_compts_1(A) ) ) ) )).

fof(fc5_borsuk_3,axiom,(
    ! [A,B] :
      ( ( v2_pre_topc(A)
        & l1_pre_topc(A)
        & v3_struct_0(B)
        & v2_pre_topc(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))
        & v2_compts_1(k6_borsuk_1(A,B)) ) ) )).

fof(fc6_borsuk_3,axiom,(
    ! [A,B] :
      ( ( v2_pre_topc(A)
        & v2_compts_1(A)
        & l1_pre_topc(A)
        & v2_pre_topc(B)
        & v2_compts_1(B)
        & l1_pre_topc(B) )
     => ( v1_pre_topc(k6_borsuk_1(A,B))
        & v2_pre_topc(k6_borsuk_1(A,B))
        & v2_compts_1(k6_borsuk_1(A,B)) ) ) )).

fof(rc2_borsuk_3,axiom,(
    ! [A] :
      ( l1_pre_topc(A)
     => ? [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
          & v6_compts_1(B,A) ) ) )).

fof(fc7_borsuk_3,axiom,(
    ! [A,B] :
      ( ( v2_pre_topc(A)
        & l1_pre_topc(A)
        & v6_compts_1(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))
        & v2_compts_1(k3_pre_topc(A,B)) ) ) )).

fof(t1_borsuk_3,axiom,(
    ! [A] :
      ( ( v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ! [B] :
          ( ( v2_pre_topc(B)
            & l1_pre_topc(B) )
         => k2_pre_topc(k6_borsuk_1(A,B)) = k7_borsuk_1(A,B,k2_pre_topc(A),k2_pre_topc(B)) ) ) )).

fof(t2_borsuk_3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & v2_pre_topc(B)
            & l1_pre_topc(B) )
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ( v1_funct_1(k3_borsuk_1(B,A,C))
                & v1_funct_2(k3_borsuk_1(B,A,C),u1_struct_0(B),u1_struct_0(k3_pre_topc(A,k1_struct_0(A,C))))
                & v5_pre_topc(k3_borsuk_1(B,A,C),B,k3_pre_topc(A,k1_struct_0(A,C)))
                & m2_relset_1(k3_borsuk_1(B,A,C),u1_struct_0(B),u1_struct_0(k3_pre_topc(A,k1_struct_0(A,C)))) ) ) ) ) )).

fof(t3_borsuk_3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & v2_pre_topc(B)
            & l1_pre_topc(B) )
         => ! [C] :
              ( ( ~ v3_struct_0(C)
                & v2_pre_topc(C)
                & l1_pre_topc(C) )
             => ( ( r1_borsuk_3(A,B)
                  & r1_borsuk_3(B,C) )
               => r1_borsuk_3(A,C) ) ) ) ) )).

fof(t4_borsuk_3,axiom,(
    ! [A] :
      ( ( v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ! [B] :
          ( ( v3_struct_0(B)
            & v2_pre_topc(B)
            & l1_pre_topc(B) )
         => ( v3_struct_0(k6_borsuk_1(A,B))
            & v3_struct_0(k6_borsuk_1(B,A)) ) ) ) )).

fof(t5_borsuk_3,axiom,(
    ! [A] :
      ( ( v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => v2_compts_1(A) ) )).

fof(t6_borsuk_3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & v2_pre_topc(B)
            & l1_pre_topc(B) )
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ! [D] :
                  ( ( v1_funct_1(D)
                    & v1_funct_2(D,u1_struct_0(k6_borsuk_1(B,k3_pre_topc(A,k1_struct_0(A,C)))),u1_struct_0(B))
                    & m2_relset_1(D,u1_struct_0(k6_borsuk_1(B,k3_pre_topc(A,k1_struct_0(A,C)))),u1_struct_0(B)) )
                 => ( D = k9_funct_3(u1_struct_0(B),k1_struct_0(A,C))
                   => v2_funct_1(D) ) ) ) ) ) )).

fof(t7_borsuk_3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & v2_pre_topc(B)
            & l1_pre_topc(B) )
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ! [D] :
                  ( ( v1_funct_1(D)
                    & v1_funct_2(D,u1_struct_0(k6_borsuk_1(k3_pre_topc(A,k1_struct_0(A,C)),B)),u1_struct_0(B))
                    & m2_relset_1(D,u1_struct_0(k6_borsuk_1(k3_pre_topc(A,k1_struct_0(A,C)),B)),u1_struct_0(B)) )
                 => ( D = k10_funct_3(k1_struct_0(A,C),u1_struct_0(B))
                   => v2_funct_1(D) ) ) ) ) ) )).

fof(t8_borsuk_3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & v2_pre_topc(B)
            & l1_pre_topc(B) )
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ! [D] :
                  ( ( v1_funct_1(D)
                    & v1_funct_2(D,u1_struct_0(k6_borsuk_1(B,k3_pre_topc(A,k1_struct_0(A,C)))),u1_struct_0(B))
                    & m2_relset_1(D,u1_struct_0(k6_borsuk_1(B,k3_pre_topc(A,k1_struct_0(A,C)))),u1_struct_0(B)) )
                 => ( D = k9_funct_3(u1_struct_0(B),k1_struct_0(A,C))
                   => k2_tops_2(k6_borsuk_1(B,k3_pre_topc(A,k1_struct_0(A,C))),B,D) = k14_funct_3(u1_struct_0(B),u1_struct_0(B),u1_struct_0(A),k7_grcat_1(B),k3_borsuk_1(B,A,C)) ) ) ) ) ) )).

fof(t9_borsuk_3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & v2_pre_topc(B)
            & l1_pre_topc(B) )
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ! [D] :
                  ( ( v1_funct_1(D)
                    & v1_funct_2(D,u1_struct_0(k6_borsuk_1(k3_pre_topc(A,k1_struct_0(A,C)),B)),u1_struct_0(B))
                    & m2_relset_1(D,u1_struct_0(k6_borsuk_1(k3_pre_topc(A,k1_struct_0(A,C)),B)),u1_struct_0(B)) )
                 => ( D = k10_funct_3(k1_struct_0(A,C),u1_struct_0(B))
                   => k2_tops_2(k6_borsuk_1(k3_pre_topc(A,k1_struct_0(A,C)),B),B,D) = k14_funct_3(u1_struct_0(B),u1_struct_0(A),u1_struct_0(B),k3_borsuk_1(B,A,C),k7_grcat_1(B)) ) ) ) ) ) )).

fof(t10_borsuk_3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & v2_pre_topc(B)
            & l1_pre_topc(B) )
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ! [D] :
                  ( ( v1_funct_1(D)
                    & v1_funct_2(D,u1_struct_0(k6_borsuk_1(B,k3_pre_topc(A,k1_struct_0(A,C)))),u1_struct_0(B))
                    & m2_relset_1(D,u1_struct_0(k6_borsuk_1(B,k3_pre_topc(A,k1_struct_0(A,C)))),u1_struct_0(B)) )
                 => ( D = k9_funct_3(u1_struct_0(B),k1_struct_0(A,C))
                   => v3_tops_2(D,k6_borsuk_1(B,k3_pre_topc(A,k1_struct_0(A,C))),B) ) ) ) ) ) )).

fof(t11_borsuk_3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & v2_pre_topc(B)
            & l1_pre_topc(B) )
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ! [D] :
                  ( ( v1_funct_1(D)
                    & v1_funct_2(D,u1_struct_0(k6_borsuk_1(k3_pre_topc(A,k1_struct_0(A,C)),B)),u1_struct_0(B))
                    & m2_relset_1(D,u1_struct_0(k6_borsuk_1(k3_pre_topc(A,k1_struct_0(A,C)),B)),u1_struct_0(B)) )
                 => ( D = k10_funct_3(k1_struct_0(A,C),u1_struct_0(B))
                   => v3_tops_2(D,k6_borsuk_1(k3_pre_topc(A,k1_struct_0(A,C)),B),B) ) ) ) ) ) )).

fof(t15_borsuk_3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & v2_pre_topc(B)
            & l1_pre_topc(B) )
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => r1_borsuk_3(k6_borsuk_1(k3_pre_topc(A,k1_struct_0(A,C)),B),B) ) ) ) )).

fof(t16_borsuk_3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & v2_pre_topc(B)
            & l1_pre_topc(B) )
         => ( ( r1_borsuk_3(A,B)
              & v2_compts_1(A) )
           => v2_compts_1(B) ) ) ) )).

fof(t17_borsuk_3,axiom,(
    ! [A] :
      ( ( v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ! [B] :
          ( ( v2_pre_topc(B)
            & l1_pre_topc(B) )
         => ! [C] :
              ( m1_pre_topc(C,A)
             => m1_pre_topc(k6_borsuk_1(B,C),k6_borsuk_1(B,A)) ) ) ) )).

fof(t18_borsuk_3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & v2_pre_topc(B)
            & v2_compts_1(B)
            & l1_pre_topc(B) )
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ! [D] :
                  ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k6_borsuk_1(B,A))))
                 => ( D = k7_borsuk_1(B,A,k2_pre_topc(B),k1_struct_0(A,C))
                   => v6_compts_1(D,k6_borsuk_1(B,A)) ) ) ) ) ) )).

fof(t19_borsuk_3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & v2_pre_topc(B)
            & v2_compts_1(B)
            & l1_pre_topc(B) )
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => v2_compts_1(k6_borsuk_1(B,k3_pre_topc(A,k1_struct_0(A,C)))) ) ) ) )).

fof(t23_borsuk_3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & v2_compts_1(A)
        & l1_pre_topc(A) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & v2_pre_topc(B)
            & v2_compts_1(B)
            & l1_pre_topc(B) )
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(k6_borsuk_1(B,A)))))
             => ~ ( r1_pre_topc(k6_borsuk_1(B,A),C)
                  & v1_tops_2(C,k6_borsuk_1(B,A))
                  & ! [D] :
                      ( m1_subset_1(D,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(k6_borsuk_1(B,A)))))
                     => ~ ( r1_tarski(D,C)
                          & r1_pre_topc(k6_borsuk_1(B,A),D)
                          & v1_finset_1(D) ) ) ) ) ) ) )).

fof(t24_borsuk_3,axiom,(
    ! [A] :
      ( ( v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ! [B] :
          ( ( v2_pre_topc(B)
            & l1_pre_topc(B) )
         => ( ( v2_compts_1(A)
              & v2_compts_1(B) )
           => v2_compts_1(k6_borsuk_1(A,B)) ) ) ) )).

fof(t25_borsuk_3,axiom,(
    ! [A] :
      ( ( v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ! [B] :
          ( ( v2_pre_topc(B)
            & l1_pre_topc(B) )
         => ! [C] :
              ( m1_pre_topc(C,A)
             => ! [D] :
                  ( m1_pre_topc(D,B)
                 => m1_pre_topc(k6_borsuk_1(C,D),k6_borsuk_1(A,B)) ) ) ) ) )).

fof(t26_borsuk_3,axiom,(
    ! [A] :
      ( ( v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ! [B] :
          ( ( v2_pre_topc(B)
            & l1_pre_topc(B) )
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k6_borsuk_1(B,A))))
             => ! [D] :
                  ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
                 => ! [E] :
                      ( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(B)))
                     => ( C = k7_borsuk_1(B,A,E,D)
                       => g1_pre_topc(u1_struct_0(k6_borsuk_1(k3_pre_topc(B,E),k3_pre_topc(A,D))),u1_pre_topc(k6_borsuk_1(k3_pre_topc(B,E),k3_pre_topc(A,D)))) = g1_pre_topc(u1_struct_0(k3_pre_topc(k6_borsuk_1(B,A),C)),u1_pre_topc(k3_pre_topc(k6_borsuk_1(B,A),C))) ) ) ) ) ) ) )).

fof(t27_borsuk_3,axiom,(
    ! [A] :
      ( ( v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ! [B] :
          ( ( v2_pre_topc(B)
            & l1_pre_topc(B) )
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
             => ! [D] :
                  ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(B)))
                 => ( ( v6_compts_1(C,A)
                      & v6_compts_1(D,B) )
                   => ( v6_compts_1(k7_borsuk_1(A,B,C,D),k6_borsuk_1(A,B))
                      & m1_subset_1(k7_borsuk_1(A,B,C,D),k1_zfmisc_1(u1_struct_0(k6_borsuk_1(A,B)))) ) ) ) ) ) ) )).

fof(symmetry_r1_borsuk_3,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(A)
        & l1_pre_topc(A)
        & ~ v3_struct_0(B)
        & l1_pre_topc(B) )
     => ( r1_borsuk_3(A,B)
       => r1_borsuk_3(B,A) ) ) )).

fof(reflexivity_r1_borsuk_3,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(A)
        & l1_pre_topc(A)
        & ~ v3_struct_0(B)
        & l1_pre_topc(B) )
     => r1_borsuk_3(A,A) ) )).

fof(redefinition_r1_borsuk_3,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(A)
        & l1_pre_topc(A)
        & ~ v3_struct_0(B)
        & l1_pre_topc(B) )
     => ( r1_borsuk_3(A,B)
      <=> r1_t_0topsp(A,B) ) ) )).

fof(t12_borsuk_3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & v2_pre_topc(B)
            & v2_compts_1(B)
            & l1_pre_topc(B) )
         => ! [C] :
              ( ( v3_pre_topc(C,k6_borsuk_1(A,B))
                & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k6_borsuk_1(A,B)))) )
             => ! [D] :
                  ~ ( r2_hidden(D,a_3_0_borsuk_3(A,B,C))
                    & ! [E] :
                        ( m1_pboole(E,u1_struct_0(B))
                       => ? [F] :
                            ( r2_hidden(F,u1_struct_0(B))
                            & ! [G] :
                                ( m1_subset_1(G,k1_zfmisc_1(u1_struct_0(A)))
                               => ! [H] :
                                    ( m1_subset_1(H,k1_zfmisc_1(u1_struct_0(B)))
                                   => ~ ( k1_funct_1(E,F) = k4_tarski(G,H)
                                        & r2_hidden(k4_tarski(D,F),k7_borsuk_1(A,B,G,H))
                                        & v3_pre_topc(G,A)
                                        & v3_pre_topc(H,B)
                                        & r1_tarski(k7_borsuk_1(A,B,G,H),C) ) ) ) ) ) ) ) ) ) )).

fof(t13_borsuk_3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & v2_pre_topc(B)
            & v2_compts_1(B)
            & l1_pre_topc(B) )
         => ! [C] :
              ( ( v3_pre_topc(C,k6_borsuk_1(B,A))
                & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k6_borsuk_1(B,A)))) )
             => ! [D] :
                  ~ ( r2_hidden(D,a_3_1_borsuk_3(A,B,C))
                    & ! [E] :
                        ( ( v3_pre_topc(E,A)
                          & m1_subset_1(E,k1_zfmisc_1(u1_struct_0(A))) )
                       => ~ ( r2_hidden(D,E)
                            & r1_tarski(E,a_3_1_borsuk_3(A,B,C)) ) ) ) ) ) ) )).

fof(t14_borsuk_3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & v2_pre_topc(B)
            & v2_compts_1(B)
            & l1_pre_topc(B) )
         => ! [C] :
              ( ( v3_pre_topc(C,k6_borsuk_1(B,A))
                & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k6_borsuk_1(B,A)))) )
             => r2_hidden(a_3_1_borsuk_3(A,B,C),u1_pre_topc(A)) ) ) ) )).

fof(t20_borsuk_3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & v2_compts_1(A)
        & l1_pre_topc(A) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & v2_pre_topc(B)
            & v2_compts_1(B)
            & l1_pre_topc(B) )
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
             => ( C = a_2_0_borsuk_3(A,B)
               => ( v1_tops_2(C,A)
                  & r1_compts_1(A,C,k2_pre_topc(A)) ) ) ) ) ) )).

fof(t21_borsuk_3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & v2_compts_1(A)
        & l1_pre_topc(A) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & v2_pre_topc(B)
            & v2_compts_1(B)
            & l1_pre_topc(B) )
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
             => ! [D] :
                  ( m1_subset_1(D,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(k6_borsuk_1(B,A)))))
                 => ( ( r1_pre_topc(k6_borsuk_1(B,A),D)
                      & v1_tops_2(D,k6_borsuk_1(B,A))
                      & C = a_3_2_borsuk_3(A,B,D) )
                   => ( v1_tops_2(C,A)
                      & r1_pre_topc(A,C) ) ) ) ) ) ) )).

fof(t22_borsuk_3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & v2_compts_1(A)
        & l1_pre_topc(A) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & v2_pre_topc(B)
            & v2_compts_1(B)
            & l1_pre_topc(B) )
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
             => ! [D] :
                  ( m1_subset_1(D,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(k6_borsuk_1(B,A)))))
                 => ~ ( r1_pre_topc(k6_borsuk_1(B,A),D)
                      & v1_tops_2(D,k6_borsuk_1(B,A))
                      & C = a_3_2_borsuk_3(A,B,D)
                      & ! [E] :
                          ( m1_subset_1(E,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
                         => ~ ( r1_tarski(E,C)
                              & v1_finset_1(E)
                              & r1_pre_topc(A,E) ) ) ) ) ) ) ) )).

fof(fraenkel_a_3_0_borsuk_3,axiom,(
    ! [A,B,C,D] :
      ( ( ~ v3_struct_0(B)
        & v2_pre_topc(B)
        & l1_pre_topc(B)
        & ~ v3_struct_0(C)
        & v2_pre_topc(C)
        & v2_compts_1(C)
        & l1_pre_topc(C)
        & v3_pre_topc(D,k6_borsuk_1(B,C))
        & m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k6_borsuk_1(B,C)))) )
     => ( r2_hidden(A,a_3_0_borsuk_3(B,C,D))
      <=> ? [E] :
            ( m1_subset_1(E,u1_struct_0(B))
            & A = E
            & r1_tarski(k2_zfmisc_1(k1_struct_0(B,E),u1_struct_0(C)),D) ) ) ) )).

fof(fraenkel_a_3_1_borsuk_3,axiom,(
    ! [A,B,C,D] :
      ( ( ~ v3_struct_0(B)
        & v2_pre_topc(B)
        & l1_pre_topc(B)
        & ~ v3_struct_0(C)
        & v2_pre_topc(C)
        & v2_compts_1(C)
        & l1_pre_topc(C)
        & v3_pre_topc(D,k6_borsuk_1(C,B))
        & m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k6_borsuk_1(C,B)))) )
     => ( r2_hidden(A,a_3_1_borsuk_3(B,C,D))
      <=> ? [E] :
            ( m1_subset_1(E,u1_struct_0(B))
            & A = E
            & r1_tarski(k7_borsuk_1(C,B,k2_pre_topc(C),k1_struct_0(B,E)),D) ) ) ) )).

fof(fraenkel_a_2_0_borsuk_3,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(B)
        & v2_pre_topc(B)
        & v2_compts_1(B)
        & l1_pre_topc(B)
        & ~ v3_struct_0(C)
        & v2_pre_topc(C)
        & v2_compts_1(C)
        & l1_pre_topc(C) )
     => ( r2_hidden(A,a_2_0_borsuk_3(B,C))
      <=> ? [D] :
            ( v3_pre_topc(D,B)
            & m1_subset_1(D,k1_zfmisc_1(u1_struct_0(B)))
            & A = D
            & r1_tarski(k7_borsuk_1(C,B,k2_pre_topc(C),D),k5_setfam_1(u1_struct_0(k6_borsuk_1(C,B)),k11_borsuk_1(C,B,k2_pre_topc(k6_borsuk_1(C,B))))) ) ) ) )).

fof(fraenkel_a_3_2_borsuk_3,axiom,(
    ! [A,B,C,D] :
      ( ( ~ v3_struct_0(B)
        & v2_pre_topc(B)
        & v2_compts_1(B)
        & l1_pre_topc(B)
        & ~ v3_struct_0(C)
        & v2_pre_topc(C)
        & v2_compts_1(C)
        & l1_pre_topc(C)
        & m1_subset_1(D,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(k6_borsuk_1(C,B))))) )
     => ( r2_hidden(A,a_3_2_borsuk_3(B,C,D))
      <=> ? [E] :
            ( v3_pre_topc(E,B)
            & m1_subset_1(E,k1_zfmisc_1(u1_struct_0(B)))
            & A = E
            & ? [F] :
                ( m1_subset_1(F,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(k6_borsuk_1(C,B)))))
                & r1_tarski(F,D)
                & v1_finset_1(F)
                & r1_tarski(k7_borsuk_1(C,B,k2_pre_topc(C),E),k5_setfam_1(u1_struct_0(k6_borsuk_1(C,B)),F)) ) ) ) ) )).
%------------------------------------------------------------------------------