SET007 Axioms: SET007+317.ax


%------------------------------------------------------------------------------
% File     : SET007+317 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Metric Spaces as Topological Spaces - Fundamental Concepts
% 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    : topmetr [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   49 (   7 unit)
%            Number of atoms       :  353 (  33 equality)
%            Maximal formula depth :   16 (   8 average)
%            Number of connectives :  341 (  37 ~  ;   0  |; 166  &)
%                                         (   9 <=>; 129 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   36 (   1 propositional; 0-3 arity)
%            Number of functors    :   35 (   8 constant; 0-5 arity)
%            Number of variables   :  119 (   0 singleton; 114 !;   5 ?)
%            Maximal term depth    :    4 (   1 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(rc1_topmetr,axiom,(
    ! [A] :
      ( ( v6_metric_1(A)
        & v7_metric_1(A)
        & v8_metric_1(A)
        & v9_metric_1(A)
        & l1_metric_1(A) )
     => ? [B] :
          ( m1_topmetr(B,A)
          & v1_metric_1(B)
          & v6_metric_1(B)
          & v7_metric_1(B)
          & v8_metric_1(B)
          & v9_metric_1(B) ) ) )).

fof(rc2_topmetr,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v6_metric_1(A)
        & v7_metric_1(A)
        & v8_metric_1(A)
        & v9_metric_1(A)
        & l1_metric_1(A) )
     => ? [B] :
          ( m1_topmetr(B,A)
          & ~ v3_struct_0(B)
          & v1_metric_1(B)
          & v6_metric_1(B)
          & v7_metric_1(B)
          & v8_metric_1(B)
          & v9_metric_1(B) ) ) )).

fof(fc1_topmetr,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(A)
        & v6_metric_1(A)
        & v7_metric_1(A)
        & v8_metric_1(A)
        & v9_metric_1(A)
        & l1_metric_1(A)
        & ~ v1_xboole_0(B)
        & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
     => ( ~ v3_struct_0(k1_topmetr(A,B))
        & v1_metric_1(k1_topmetr(A,B))
        & v6_metric_1(k1_topmetr(A,B))
        & v7_metric_1(k1_topmetr(A,B))
        & v8_metric_1(k1_topmetr(A,B))
        & v9_metric_1(k1_topmetr(A,B)) ) ) )).

fof(fc2_topmetr,axiom,
    ( ~ v3_struct_0(k3_topmetr)
    & v1_pre_topc(k3_topmetr)
    & v2_pre_topc(k3_topmetr) )).

fof(fc3_topmetr,axiom,(
    ! [A,B,C] :
      ( ( v1_funct_1(B)
        & m1_relset_1(B,A,u1_struct_0(k3_topmetr)) )
     => ( v1_xcmplx_0(k1_funct_1(B,C))
        & v1_xreal_0(k1_funct_1(B,C)) ) ) )).

fof(t1_topmetr,axiom,(
    ! [A] :
      ( l1_pre_topc(A)
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
         => ( r1_pre_topc(A,B)
          <=> r1_tarski(u1_struct_0(A),k5_setfam_1(u1_struct_0(A),B)) ) ) ) )).

fof(t2_topmetr,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] :
              ( m1_subset_1(C,u1_struct_0(B))
             => m1_subset_1(C,u1_struct_0(A)) ) ) ) )).

fof(t3_topmetr,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & m1_pre_topc(B,A) )
         => ( v3_compts_1(A)
           => v3_compts_1(B) ) ) ) )).

fof(t4_topmetr,axiom,(
    ! [A] :
      ( ( v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ! [B] :
          ( m1_pre_topc(B,A)
         => ! [C] :
              ( m1_pre_topc(C,A)
             => ( r1_tarski(u1_struct_0(B),u1_struct_0(C))
               => m1_pre_topc(B,C) ) ) ) ) )).

fof(t5_topmetr,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(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)))
             => ( m1_pre_topc(k3_pre_topc(A,B),k3_pre_topc(A,k4_subset_1(u1_struct_0(A),B,C)))
                & m1_pre_topc(k3_pre_topc(A,C),k3_pre_topc(A,k4_subset_1(u1_struct_0(A),B,C))) ) ) ) ) )).

fof(t6_topmetr,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_xboole_0(C)
                & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
             => ( r2_hidden(B,C)
               => ! [D] :
                    ( m1_connsp_2(D,A,B)
                   => ! [E] :
                        ( m1_subset_1(E,u1_struct_0(k3_pre_topc(A,C)))
                       => ! [F] :
                            ( m1_subset_1(F,k1_zfmisc_1(u1_struct_0(k3_pre_topc(A,C))))
                           => ( ( F = k5_subset_1(u1_struct_0(A),D,C)
                                & E = B )
                             => m1_connsp_2(F,k3_pre_topc(A,C),E) ) ) ) ) ) ) ) ) )).

fof(t7_topmetr,axiom,(
    ! [A] :
      ( ( v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ! [B] :
          ( ( v2_pre_topc(B)
            & l1_pre_topc(B) )
         => ! [C] :
              ( ( v2_pre_topc(C)
                & l1_pre_topc(C) )
             => ! [D] :
                  ( ( v1_funct_1(D)
                    & v1_funct_2(D,u1_struct_0(A),u1_struct_0(C))
                    & m2_relset_1(D,u1_struct_0(A),u1_struct_0(C)) )
                 => ( ( v5_pre_topc(D,A,C)
                      & m1_pre_topc(C,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)) )
                       => ( E = D
                         => v5_pre_topc(E,A,B) ) ) ) ) ) ) ) )).

fof(t8_topmetr,axiom,(
    ! [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)) )
             => ! [D] :
                  ( m1_pre_topc(D,B)
                 => ( v5_pre_topc(C,A,B)
                   => ! [E] :
                        ( ( v1_funct_1(E)
                          & v1_funct_2(E,u1_struct_0(A),u1_struct_0(D))
                          & m2_relset_1(E,u1_struct_0(A),u1_struct_0(D)) )
                       => ( E = C
                         => v5_pre_topc(E,A,D) ) ) ) ) ) ) ) )).

fof(t9_topmetr,axiom,(
    ! [A] :
      ( ( v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ! [B] :
          ( ( v2_pre_topc(B)
            & l1_pre_topc(B) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
                & m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
             => ! [D] :
                  ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(B)))
                 => ( v5_pre_topc(C,A,B)
                   => ! [E] :
                        ( ( v1_funct_1(E)
                          & v1_funct_2(E,u1_struct_0(A),u1_struct_0(k3_pre_topc(B,D)))
                          & m2_relset_1(E,u1_struct_0(A),u1_struct_0(k3_pre_topc(B,D))) )
                       => ( E = C
                         => v5_pre_topc(E,A,k3_pre_topc(B,D)) ) ) ) ) ) ) ) )).

fof(t10_topmetr,axiom,(
    ! [A] :
      ( l1_pre_topc(A)
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & l1_pre_topc(B) )
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
             => ! [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)) )
                 => ! [E] :
                      ( ( v1_funct_1(E)
                        & v1_funct_2(E,u1_struct_0(k3_pre_topc(A,C)),u1_struct_0(B))
                        & m2_relset_1(E,u1_struct_0(k3_pre_topc(A,C)),u1_struct_0(B)) )
                     => ( ( v5_pre_topc(D,A,B)
                          & E = k2_partfun1(u1_struct_0(A),u1_struct_0(B),D,C) )
                       => v5_pre_topc(E,k3_pre_topc(A,C),B) ) ) ) ) ) ) )).

fof(d1_topmetr,axiom,(
    ! [A] :
      ( ( v6_metric_1(A)
        & v7_metric_1(A)
        & v8_metric_1(A)
        & v9_metric_1(A)
        & l1_metric_1(A) )
     => ! [B] :
          ( ( v6_metric_1(B)
            & v7_metric_1(B)
            & v8_metric_1(B)
            & v9_metric_1(B)
            & l1_metric_1(B) )
         => ( m1_topmetr(B,A)
          <=> ( r1_tarski(u1_struct_0(B),u1_struct_0(A))
              & ! [C] :
                  ( m1_subset_1(C,u1_struct_0(B))
                 => ! [D] :
                      ( m1_subset_1(D,u1_struct_0(B))
                     => k1_metric_1(u1_struct_0(B),u1_struct_0(B),u1_metric_1(B),C,D) = k1_binop_1(u1_metric_1(A),C,D) ) ) ) ) ) ) )).

fof(t11_topmetr,axiom,(
    $true )).

fof(t12_topmetr,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v6_metric_1(A)
        & v7_metric_1(A)
        & v8_metric_1(A)
        & v9_metric_1(A)
        & l1_metric_1(A) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & m1_topmetr(B,A) )
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(B))
             => m1_subset_1(C,u1_struct_0(A)) ) ) ) )).

fof(t13_topmetr,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( ( v6_metric_1(B)
            & v7_metric_1(B)
            & v8_metric_1(B)
            & v9_metric_1(B)
            & l1_metric_1(B) )
         => ! [C] :
              ( m1_topmetr(C,B)
             => ! [D] :
                  ( m1_subset_1(D,u1_struct_0(B))
                 => ! [E] :
                      ( m1_subset_1(E,u1_struct_0(C))
                     => ( D = E
                       => k9_metric_1(C,E,A) = k3_xboole_0(k9_metric_1(B,D,A),u1_struct_0(C)) ) ) ) ) ) ) )).

fof(d2_topmetr,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v6_metric_1(A)
        & v7_metric_1(A)
        & v8_metric_1(A)
        & v9_metric_1(A)
        & l1_metric_1(A) )
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
         => ! [C] :
              ( ( v1_metric_1(C)
                & m1_topmetr(C,A) )
             => ( C = k1_topmetr(A,B)
              <=> u1_struct_0(C) = B ) ) ) ) )).

fof(d3_topmetr,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ( r1_xreal_0(A,B)
           => ! [C] :
                ( ( ~ v3_struct_0(C)
                  & v1_metric_1(C)
                  & m1_topmetr(C,k8_metric_1) )
               => ( C = k2_topmetr(A,B)
                <=> ! [D] :
                      ( ( ~ v1_xboole_0(D)
                        & m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k8_metric_1))) )
                     => ( D = k1_rcomp_1(A,B)
                       => C = k1_topmetr(k8_metric_1,D) ) ) ) ) ) ) ) )).

fof(t14_topmetr,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ( r1_xreal_0(A,B)
           => u1_struct_0(k2_topmetr(A,B)) = k1_rcomp_1(A,B) ) ) ) )).

fof(d4_topmetr,axiom,(
    ! [A] :
      ( l1_metric_1(A)
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
         => ( v1_topmetr(B,A)
          <=> ! [C] :
                ~ ( r2_hidden(C,B)
                  & ! [D] :
                      ( m1_subset_1(D,u1_struct_0(A))
                     => ! [E] :
                          ( m1_subset_1(E,k1_numbers)
                         => C != k9_metric_1(A,D,E) ) ) ) ) ) ) )).

fof(d5_topmetr,axiom,(
    ! [A] :
      ( l1_metric_1(A)
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
         => ( r1_topmetr(A,B)
          <=> r1_tarski(u1_struct_0(A),k5_setfam_1(u1_struct_0(A),B)) ) ) ) )).

fof(t15_topmetr,axiom,(
    ! [A] :
      ( m1_subset_1(A,u1_struct_0(k8_metric_1))
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(k8_metric_1))
         => ! [C] :
              ( v1_xreal_0(C)
             => ! [D] :
                  ( v1_xreal_0(D)
                 => ( ( C = A
                      & D = B )
                   => k4_metric_1(k8_metric_1,A,B) = k18_complex1(k6_xcmplx_0(C,D)) ) ) ) ) ) )).

fof(t16_topmetr,axiom,(
    ! [A] :
      ( l1_metric_1(A)
     => ( u1_struct_0(A) = u1_struct_0(k5_pcomps_1(A))
        & u1_pre_topc(k5_pcomps_1(A)) = k4_pcomps_1(A) ) ) )).

fof(t17_topmetr,axiom,(
    $true )).

fof(t18_topmetr,axiom,(
    $true )).

fof(t19_topmetr,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v6_metric_1(A)
        & v7_metric_1(A)
        & v8_metric_1(A)
        & v9_metric_1(A)
        & l1_metric_1(A) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & m1_topmetr(B,A) )
         => m1_pre_topc(k5_pcomps_1(B),k5_pcomps_1(A)) ) ) )).

fof(t20_topmetr,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
         => ! [C] :
              ( ( ~ v1_xboole_0(C)
                & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k14_euclid(A)))) )
             => ( B = C
               => k3_pre_topc(k15_euclid(A),B) = k5_pcomps_1(k1_topmetr(k14_euclid(A),C)) ) ) ) ) )).

fof(t21_topmetr,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( ( v9_metric_1(B)
            & l1_metric_1(B) )
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(B))
             => ! [D] :
                  ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k5_pcomps_1(B))))
                 => ( D = k9_metric_1(B,C,A)
                   => v3_pre_topc(D,k5_pcomps_1(B)) ) ) ) ) ) )).

fof(t22_topmetr,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v6_metric_1(A)
        & v7_metric_1(A)
        & v8_metric_1(A)
        & v9_metric_1(A)
        & l1_metric_1(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k5_pcomps_1(A))))
         => ( v3_pre_topc(B,k5_pcomps_1(A))
          <=> ! [C] :
                ( m1_subset_1(C,u1_struct_0(A))
               => ~ ( r2_hidden(C,B)
                    & ! [D] :
                        ( v1_xreal_0(D)
                       => ~ ( ~ r1_xreal_0(D,np__0)
                            & r1_tarski(k9_metric_1(A,C,D),B) ) ) ) ) ) ) ) )).

fof(d6_topmetr,axiom,(
    ! [A] :
      ( l1_metric_1(A)
     => ( v2_topmetr(A)
      <=> v2_compts_1(k5_pcomps_1(A)) ) ) )).

fof(t23_topmetr,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v6_metric_1(A)
        & v7_metric_1(A)
        & v8_metric_1(A)
        & v9_metric_1(A)
        & l1_metric_1(A) )
     => ( v2_topmetr(A)
      <=> ! [B] :
            ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
           => ~ ( v1_topmetr(B,A)
                & r1_topmetr(A,B)
                & ! [C] :
                    ( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
                   => ~ ( r1_tarski(C,B)
                        & r1_topmetr(A,C)
                        & v1_finset_1(C) ) ) ) ) ) ) )).

fof(d7_topmetr,axiom,(
    k3_topmetr = k5_pcomps_1(k8_metric_1) )).

fof(t24_topmetr,axiom,(
    u1_struct_0(k3_topmetr) = k1_numbers )).

fof(d8_topmetr,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => k4_topmetr(A,B) = k5_pcomps_1(k2_topmetr(A,B)) ) ) )).

fof(t25_topmetr,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ( r1_xreal_0(A,B)
           => u1_struct_0(k4_topmetr(A,B)) = k1_rcomp_1(A,B) ) ) ) )).

fof(t26_topmetr,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ( r1_xreal_0(A,B)
           => ! [C] :
                ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k3_topmetr)))
               => ( C = k1_rcomp_1(A,B)
                 => k4_topmetr(A,B) = k3_pre_topc(k3_topmetr,C) ) ) ) ) ) )).

fof(t27_topmetr,axiom,(
    k4_topmetr(np__0,np__1) = k22_borsuk_1 )).

fof(t28_topmetr,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,u1_struct_0(k3_topmetr),u1_struct_0(k3_topmetr))
        & m2_relset_1(A,u1_struct_0(k3_topmetr),u1_struct_0(k3_topmetr)) )
     => ( ? [B] :
            ( m1_subset_1(B,k1_numbers)
            & ? [C] :
                ( m1_subset_1(C,k1_numbers)
                & ! [D] :
                    ( m1_subset_1(D,k1_numbers)
                   => k1_funct_1(A,D) = k3_real_1(k4_real_1(B,D),C) ) ) )
       => v5_pre_topc(A,k3_topmetr,k3_topmetr) ) ) )).

fof(dt_m1_topmetr,axiom,(
    ! [A] :
      ( ( v6_metric_1(A)
        & v7_metric_1(A)
        & v8_metric_1(A)
        & v9_metric_1(A)
        & l1_metric_1(A) )
     => ! [B] :
          ( m1_topmetr(B,A)
         => ( v6_metric_1(B)
            & v7_metric_1(B)
            & v8_metric_1(B)
            & v9_metric_1(B)
            & l1_metric_1(B) ) ) ) )).

fof(existence_m1_topmetr,axiom,(
    ! [A] :
      ( ( v6_metric_1(A)
        & v7_metric_1(A)
        & v8_metric_1(A)
        & v9_metric_1(A)
        & l1_metric_1(A) )
     => ? [B] : m1_topmetr(B,A) ) )).

fof(dt_k1_topmetr,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(A)
        & v6_metric_1(A)
        & v7_metric_1(A)
        & v8_metric_1(A)
        & v9_metric_1(A)
        & l1_metric_1(A)
        & ~ v1_xboole_0(B)
        & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
     => ( v1_metric_1(k1_topmetr(A,B))
        & m1_topmetr(k1_topmetr(A,B),A) ) ) )).

fof(dt_k2_topmetr,axiom,(
    ! [A,B] :
      ( ( v1_xreal_0(A)
        & v1_xreal_0(B) )
     => ( ~ v3_struct_0(k2_topmetr(A,B))
        & v1_metric_1(k2_topmetr(A,B))
        & m1_topmetr(k2_topmetr(A,B),k8_metric_1) ) ) )).

fof(dt_k3_topmetr,axiom,
    ( v1_pre_topc(k3_topmetr)
    & v2_pre_topc(k3_topmetr)
    & l1_pre_topc(k3_topmetr) )).

fof(dt_k4_topmetr,axiom,(
    ! [A,B] :
      ( ( v1_xreal_0(A)
        & v1_xreal_0(B) )
     => ( ~ v3_struct_0(k4_topmetr(A,B))
        & v1_pre_topc(k4_topmetr(A,B))
        & m1_pre_topc(k4_topmetr(A,B),k3_topmetr) ) ) )).

fof(dt_k5_topmetr,axiom,
    ( v1_pre_topc(k5_topmetr)
    & m1_pre_topc(k5_topmetr,k3_topmetr) )).

fof(redefinition_k5_topmetr,axiom,(
    k5_topmetr = k22_borsuk_1 )).
%------------------------------------------------------------------------------