SET007 Axioms: SET007+548.ax


%------------------------------------------------------------------------------
% File     : SET007+548 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : First-countable, Sequential, and Frechet 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    : frechet [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   56 (  10 unit)
%            Number of atoms       :  339 (  32 equality)
%            Maximal formula depth :   25 (   7 average)
%            Number of connectives :  336 (  53 ~  ;   5  |; 149  &)
%                                         (  14 <=>; 115 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   33 (   1 propositional; 0-3 arity)
%            Number of functors    :   41 (   9 constant; 0-4 arity)
%            Number of variables   :  122 (   0 singleton; 114 !;   8 ?)
%            Maximal term depth    :    5 (   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(cc1_frechet,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_yellow_8(C,A,B)
         => ~ v1_xboole_0(C) ) ) )).

fof(fc1_frechet,axiom,
    ( ~ v3_struct_0(k3_topmetr)
    & v1_pre_topc(k3_topmetr)
    & v2_pre_topc(k3_topmetr)
    & v1_frechet(k3_topmetr) )).

fof(cc2_frechet,axiom,(
    ! [A] :
      ( l1_pre_topc(A)
     => ( ( ~ v3_struct_0(A)
          & v2_pre_topc(A)
          & v1_frechet(A) )
       => ( ~ v3_struct_0(A)
          & v2_pre_topc(A)
          & v3_frechet(A) ) ) ) )).

fof(cc3_frechet,axiom,(
    ! [A] :
      ( l1_pre_topc(A)
     => ( ( ~ v3_struct_0(A)
          & v2_pre_topc(A)
          & v3_frechet(A) )
       => ( ~ v3_struct_0(A)
          & v2_pre_topc(A)
          & v4_frechet(A) ) ) ) )).

fof(t1_frechet,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_struct_0(A) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,u1_struct_0(A))
            & m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
         => m1_subset_1(k2_relat_1(B),k1_zfmisc_1(u1_struct_0(A))) ) ) )).

fof(t2_frechet,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_struct_0(A) )
     => ! [B] :
          ( l1_struct_0(B)
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,u1_struct_0(A))
                & m2_relset_1(C,k5_numbers,u1_struct_0(A)) )
             => ( r1_tarski(k1_frechet(A,C),u1_struct_0(B))
               => ( v1_funct_1(C)
                  & v1_funct_2(C,k5_numbers,u1_struct_0(B))
                  & m2_relset_1(C,k5_numbers,u1_struct_0(B)) ) ) ) ) ) )).

fof(t3_frechet,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_yellow_8(C,A,B)
             => C != k1_xboole_0 ) ) ) )).

fof(t4_frechet,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)))
             => ( ( v3_pre_topc(B,A)
                  & v4_pre_topc(C,A) )
               => v3_pre_topc(k6_subset_1(u1_struct_0(A),B,C),A) ) ) ) ) )).

fof(t5_frechet,axiom,(
    ! [A] :
      ( l1_pre_topc(A)
     => ( ( v4_pre_topc(k1_pre_topc(A),A)
          & v4_pre_topc(k2_pre_topc(A),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)))
                 => ( ( v4_pre_topc(B,A)
                      & v4_pre_topc(C,A) )
                   => v4_pre_topc(k4_subset_1(u1_struct_0(A),B,C),A) ) ) )
          & ! [B] :
              ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
             => ( v2_tops_2(B,A)
               => v4_pre_topc(k6_setfam_1(u1_struct_0(A),B),A) ) ) )
       => ( v2_pre_topc(A)
          & l1_pre_topc(A) ) ) ) )).

fof(t6_frechet,axiom,(
    ! [A] :
      ( ( v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ! [B] :
          ( ( ~ v3_struct_0(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)))
                   => ( v4_pre_topc(D,B)
                    <=> v4_pre_topc(k5_pre_topc(A,B,C,D),A) ) )
               => ( v2_pre_topc(B)
                  & l1_pre_topc(B) ) ) ) ) ) )).

fof(t7_frechet,axiom,(
    ! [A] :
      ( m1_subset_1(A,u1_struct_0(k8_metric_1))
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ! [C] :
              ( m1_subset_1(C,k1_numbers)
             => ( B = A
               => ( r1_xreal_0(C,np__0)
                  | k9_metric_1(k8_metric_1,A,C) = k2_rcomp_1(k5_real_1(B,C),k3_real_1(B,C)) ) ) ) ) ) )).

fof(t8_frechet,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k3_topmetr)))
     => ( v3_pre_topc(A,k3_topmetr)
      <=> ! [B] :
            ( m1_subset_1(B,k1_numbers)
           => ~ ( r2_hidden(B,A)
                & ! [C] :
                    ( m1_subset_1(C,k1_numbers)
                   => ~ ( ~ r1_xreal_0(C,np__0)
                        & r1_tarski(k2_rcomp_1(k5_real_1(B,C),k3_real_1(B,C)),A) ) ) ) ) ) ) )).

fof(t9_frechet,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,u1_struct_0(k3_topmetr))
        & m2_relset_1(A,k5_numbers,u1_struct_0(k3_topmetr)) )
     => ( ! [B] :
            ( m2_subset_1(B,k1_numbers,k5_numbers)
           => r2_hidden(k8_funct_2(k5_numbers,u1_struct_0(k3_topmetr),A,B),k2_rcomp_1(k5_real_1(B,k6_real_1(np__1,np__4)),k3_real_1(B,k6_real_1(np__1,np__4)))) )
       => v4_pre_topc(k1_frechet(k3_topmetr,A),k3_topmetr) ) ) )).

fof(t10_frechet,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k3_topmetr)))
     => ( A = k5_numbers
       => v4_pre_topc(A,k3_topmetr) ) ) )).

fof(t12_frechet,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B] :
          ( ( v1_relat_1(B)
            & v1_funct_1(B) )
         => k2_relat_1(k1_funct_4(A,B)) = k2_xboole_0(k9_relat_1(A,k4_xboole_0(k1_relat_1(A),k1_relat_1(B))),k2_relat_1(B)) ) ) )).

fof(t13_frechet,axiom,(
    ! [A,B] :
      ( r1_tarski(B,A)
     => k9_relat_1(k6_relat_1(A),B) = B ) )).

fof(t14_frechet,axiom,(
    $true )).

fof(t15_frechet,axiom,(
    ! [A,B,C] : k1_relat_1(k1_funct_4(k6_relat_1(A),k2_funcop_1(B,C))) = k2_xboole_0(A,B) )).

fof(t16_frechet,axiom,(
    ! [A,B,C] :
      ( B != k1_xboole_0
     => k2_relat_1(k1_funct_4(k6_relat_1(A),k2_funcop_1(B,C))) = k2_xboole_0(k4_xboole_0(A,B),k1_tarski(C)) ) )).

fof(t17_frechet,axiom,(
    ! [A,B,C,D] :
      ( r1_tarski(C,A)
     => k10_relat_1(k1_funct_4(k6_relat_1(A),k2_funcop_1(B,D)),k4_xboole_0(C,k1_tarski(D))) = k4_xboole_0(k4_xboole_0(C,B),k1_tarski(D)) ) )).

fof(t18_frechet,axiom,(
    ! [A,B,C] :
      ( ~ r2_hidden(C,A)
     => k10_relat_1(k1_funct_4(k6_relat_1(A),k2_funcop_1(B,C)),k1_tarski(C)) = B ) )).

fof(t19_frechet,axiom,(
    ! [A,B,C,D] :
      ( r1_tarski(C,A)
     => ( r2_hidden(D,A)
        | k10_relat_1(k1_funct_4(k6_relat_1(A),k2_funcop_1(B,D)),k2_xboole_0(C,k1_tarski(D))) = k2_xboole_0(C,B) ) ) )).

fof(t20_frechet,axiom,(
    ! [A,B,C,D] :
      ( r1_tarski(C,A)
     => ( r2_hidden(D,A)
        | k10_relat_1(k1_funct_4(k6_relat_1(A),k2_funcop_1(B,D)),k4_xboole_0(C,k1_tarski(D))) = k4_xboole_0(C,B) ) ) )).

fof(d1_frechet,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_pre_topc(A) )
     => ( v1_frechet(A)
      <=> ! [B] :
            ( m1_subset_1(B,u1_struct_0(A))
           => ? [C] :
                ( m1_yellow_8(C,A,B)
                & v1_card_4(C) ) ) ) ) )).

fof(t21_frechet,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) )
     => v1_frechet(k5_pcomps_1(A)) ) )).

fof(t22_frechet,axiom,(
    v1_frechet(k3_topmetr) )).

fof(d2_frechet,axiom,(
    ! [A] :
      ( l1_pre_topc(A)
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,u1_struct_0(A))
            & m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ( r1_frechet(A,B,C)
              <=> ! [D] :
                    ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
                   => ~ ( v3_pre_topc(D,A)
                        & r2_hidden(C,D)
                        & ! [E] :
                            ( m2_subset_1(E,k1_numbers,k5_numbers)
                           => ? [F] :
                                ( m2_subset_1(F,k1_numbers,k5_numbers)
                                & r1_xreal_0(E,F)
                                & ~ r2_hidden(k8_funct_2(k5_numbers,u1_struct_0(A),B,F),D) ) ) ) ) ) ) ) ) )).

fof(t23_frechet,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_pre_topc(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,u1_struct_0(A))
                & m2_relset_1(C,k5_numbers,u1_struct_0(A)) )
             => ( C = k2_funcop_1(k5_numbers,B)
               => r1_frechet(A,C,B) ) ) ) ) )).

fof(d3_frechet,axiom,(
    ! [A] :
      ( l1_pre_topc(A)
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,u1_struct_0(A))
            & m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
         => ( v2_frechet(B,A)
          <=> ? [C] :
                ( m1_subset_1(C,u1_struct_0(A))
                & r1_frechet(A,B,C) ) ) ) ) )).

fof(d4_frechet,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_pre_topc(A) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,u1_struct_0(A))
            & m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
             => ( C = k2_frechet(A,B)
              <=> ! [D] :
                    ( m1_subset_1(D,u1_struct_0(A))
                   => ( r2_hidden(D,C)
                    <=> r1_frechet(A,B,D) ) ) ) ) ) ) )).

fof(d5_frechet,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_pre_topc(A) )
     => ( v3_frechet(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,k6_pre_topc(A,B))
                    & ! [D] :
                        ( ( v1_funct_1(D)
                          & v1_funct_2(D,k5_numbers,u1_struct_0(A))
                          & m2_relset_1(D,k5_numbers,u1_struct_0(A)) )
                       => ~ ( r1_tarski(k1_frechet(A,D),B)
                            & r2_hidden(C,k2_frechet(A,D)) ) ) ) ) ) ) ) )).

fof(d6_frechet,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_pre_topc(A) )
     => ( v4_frechet(A)
      <=> ! [B] :
            ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
           => ( v4_pre_topc(B,A)
            <=> ! [C] :
                  ( ( v1_funct_1(C)
                    & v1_funct_2(C,k5_numbers,u1_struct_0(A))
                    & m2_relset_1(C,k5_numbers,u1_struct_0(A)) )
                 => ( ( v2_frechet(C,A)
                      & r1_tarski(k1_frechet(A,C),B) )
                   => r1_tarski(k2_frechet(A,C),B) ) ) ) ) ) ) )).

fof(t24_frechet,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ( v1_frechet(A)
       => v3_frechet(A) ) ) )).

fof(t25_frechet,axiom,(
    $true )).

fof(t26_frechet,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)))
         => ( v4_pre_topc(B,A)
           => ! [C] :
                ( ( v1_funct_1(C)
                  & v1_funct_2(C,k5_numbers,u1_struct_0(A))
                  & m2_relset_1(C,k5_numbers,u1_struct_0(A)) )
               => ( ( v2_frechet(C,A)
                    & r1_tarski(k1_frechet(A,C),B) )
                 => r1_tarski(k2_frechet(A,C),B) ) ) ) ) ) )).

fof(t27_frechet,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] :
                  ( ( v1_funct_1(C)
                    & v1_funct_2(C,k5_numbers,u1_struct_0(A))
                    & m2_relset_1(C,k5_numbers,u1_struct_0(A)) )
                 => ( ( v2_frechet(C,A)
                      & r1_tarski(k1_frechet(A,C),B) )
                   => r1_tarski(k2_frechet(A,C),B) ) )
             => v4_pre_topc(B,A) ) )
       => v4_frechet(A) ) ) )).

fof(t28_frechet,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ( v3_frechet(A)
       => v4_frechet(A) ) ) )).

fof(d7_frechet,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v1_pre_topc(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ( A = k3_frechet
      <=> ( u1_struct_0(A) = k2_xboole_0(k4_xboole_0(k1_numbers,k5_numbers),k1_tarski(k1_numbers))
          & ? [B] :
              ( v1_funct_1(B)
              & v1_funct_2(B,u1_struct_0(k3_topmetr),u1_struct_0(A))
              & m2_relset_1(B,u1_struct_0(k3_topmetr),u1_struct_0(A))
              & B = k1_funct_4(k6_relat_1(k1_numbers),k2_funcop_1(k5_numbers,k1_numbers))
              & ! [C] :
                  ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
                 => ( v4_pre_topc(C,A)
                  <=> v4_pre_topc(k5_pre_topc(k3_topmetr,A,B,C),k3_topmetr) ) ) ) ) ) ) )).

fof(t29_frechet,axiom,(
    $true )).

fof(t30_frechet,axiom,(
    m1_subset_1(k1_numbers,u1_struct_0(k3_frechet)) )).

fof(t31_frechet,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k3_frechet)))
     => ( ( v3_pre_topc(A,k3_frechet)
          & r2_hidden(k1_numbers,A) )
      <=> ? [B] :
            ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k3_topmetr)))
            & v3_pre_topc(B,k3_topmetr)
            & r1_tarski(k5_numbers,B)
            & A = k2_xboole_0(k4_xboole_0(B,k5_numbers),k1_tarski(k1_numbers)) ) ) ) )).

fof(t32_frechet,axiom,(
    ! [A] :
      ( ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k3_frechet)))
       => ( r2_hidden(k1_numbers,A)
          | ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k3_topmetr)))
            & k3_xboole_0(k5_numbers,A) = k1_xboole_0 ) ) )
      & ( ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k3_topmetr)))
          & k3_xboole_0(k5_numbers,A) = k1_xboole_0 )
       => ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k3_frechet)))
          & ~ r2_hidden(k1_numbers,A) ) ) ) )).

fof(t33_frechet,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k3_topmetr)))
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k3_frechet)))
         => ( A = B
           => ( ( ( k3_xboole_0(k5_numbers,A) = k1_xboole_0
                  & v3_pre_topc(A,k3_topmetr) )
               => ( ~ r2_hidden(k1_numbers,B)
                  & v3_pre_topc(B,k3_frechet) ) )
              & ( v3_pre_topc(B,k3_frechet)
               => ( r2_hidden(k1_numbers,B)
                  | ( k3_xboole_0(k5_numbers,A) = k1_xboole_0
                    & v3_pre_topc(A,k3_topmetr) ) ) ) ) ) ) ) )).

fof(t34_frechet,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k3_frechet)))
     => ( A = k1_tarski(k1_numbers)
       => v4_pre_topc(A,k3_frechet) ) ) )).

fof(t35_frechet,axiom,(
    ~ v1_frechet(k3_frechet) )).

fof(t36_frechet,axiom,(
    v3_frechet(k3_frechet) )).

fof(t37_frechet,axiom,(
    ? [A] :
      ( ~ v3_struct_0(A)
      & v2_pre_topc(A)
      & l1_pre_topc(A)
      & v3_frechet(A)
      & ~ v1_frechet(A) ) )).

fof(t38_frechet,axiom,(
    $true )).

fof(t39_frechet,axiom,(
    $true )).

fof(t40_frechet,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ~ ( ~ r1_xreal_0(A,np__0)
          & ! [B] :
              ( m2_subset_1(B,k1_numbers,k5_numbers)
             => ~ ( ~ r1_xreal_0(A,k6_real_1(np__1,B))
                  & ~ r1_xreal_0(B,np__0) ) ) ) ) )).

fof(dt_k1_frechet,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(A)
        & l1_struct_0(A)
        & v1_funct_1(B)
        & v1_funct_2(B,k5_numbers,u1_struct_0(A))
        & m1_relset_1(B,k5_numbers,u1_struct_0(A)) )
     => m1_subset_1(k1_frechet(A,B),k1_zfmisc_1(u1_struct_0(A))) ) )).

fof(redefinition_k1_frechet,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(A)
        & l1_struct_0(A)
        & v1_funct_1(B)
        & v1_funct_2(B,k5_numbers,u1_struct_0(A))
        & m1_relset_1(B,k5_numbers,u1_struct_0(A)) )
     => k1_frechet(A,B) = k2_relat_1(B) ) )).

fof(dt_k2_frechet,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(A)
        & l1_pre_topc(A)
        & v1_funct_1(B)
        & v1_funct_2(B,k5_numbers,u1_struct_0(A))
        & m1_relset_1(B,k5_numbers,u1_struct_0(A)) )
     => m1_subset_1(k2_frechet(A,B),k1_zfmisc_1(u1_struct_0(A))) ) )).

fof(dt_k3_frechet,axiom,
    ( ~ v3_struct_0(k3_frechet)
    & v1_pre_topc(k3_frechet)
    & v2_pre_topc(k3_frechet)
    & l1_pre_topc(k3_frechet) )).

fof(t11_frechet,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,u1_struct_0(k5_pcomps_1(A)))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ~ ( B = C
                  & ! [D] :
                      ( m1_yellow_8(D,k5_pcomps_1(A),B)
                     => ~ ( D = a_2_0_frechet(A,C)
                          & v1_card_4(D)
                          & ? [E] :
                              ( v1_funct_1(E)
                              & v1_funct_2(E,k5_numbers,D)
                              & m2_relset_1(E,k5_numbers,D)
                              & ! [F] :
                                  ~ ( r2_hidden(F,k5_numbers)
                                    & ! [G] :
                                        ( m2_subset_1(G,k1_numbers,k5_numbers)
                                       => ~ ( F = G
                                            & k1_funct_1(E,F) = k9_metric_1(A,C,k6_real_1(np__1,k1_nat_1(G,np__1))) ) ) ) ) ) ) ) ) ) ) )).

fof(fraenkel_a_2_0_frechet,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(B)
        & v6_metric_1(B)
        & v7_metric_1(B)
        & v8_metric_1(B)
        & v9_metric_1(B)
        & l1_metric_1(B)
        & m1_subset_1(C,u1_struct_0(B)) )
     => ( r2_hidden(A,a_2_0_frechet(B,C))
      <=> ? [D] :
            ( m2_subset_1(D,k1_numbers,k5_numbers)
            & A = k9_metric_1(B,C,k6_real_1(np__1,D))
            & D != np__0 ) ) ) )).
%------------------------------------------------------------------------------