SET007 Axioms: SET007+586.ax


%------------------------------------------------------------------------------
% File     : SET007+586 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : The Sequential Closure Operator in 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    : frechet2 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   65 (   1 unit)
%            Number of atoms       :  627 (  45 equality)
%            Maximal formula depth :   19 (  10 average)
%            Number of connectives :  644 (  82 ~  ;   1  |; 311  &)
%                                         (  13 <=>; 237 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   46 (   1 propositional; 0-3 arity)
%            Number of functors    :   40 (  13 constant; 0-5 arity)
%            Number of variables   :  225 (   0 singleton; 205 !;  20 ?)
%            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(t1_frechet2,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)) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,k1_numbers)
                & v1_seqm_3(C)
                & v7_seqm_3(C)
                & m2_relset_1(C,k5_numbers,k1_numbers) )
             => ( v1_funct_1(k5_relat_1(C,B))
                & v1_funct_2(k5_relat_1(C,B),k5_numbers,u1_struct_0(A))
                & m2_relset_1(k5_relat_1(C,B),k5_numbers,u1_struct_0(A)) ) ) ) ) )).

fof(t2_frechet2,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & m2_relset_1(A,k5_numbers,k1_numbers) )
     => ( A = k6_relat_1(k5_numbers)
       => ( v1_funct_1(A)
          & v1_funct_2(A,k5_numbers,k1_numbers)
          & v1_seqm_3(A)
          & v7_seqm_3(A)
          & m2_relset_1(A,k5_numbers,k1_numbers) ) ) ) )).

fof(t3_frechet2,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_bhsp_3(B,A,B) ) ) )).

fof(t4_frechet2,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)) )
         => ! [C] :
              ( m1_bhsp_3(C,A,B)
             => r1_tarski(k1_frechet(A,C),k1_frechet(A,B)) ) ) ) )).

fof(t5_frechet2,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)) )
         => ! [C] :
              ( m1_bhsp_3(C,A,B)
             => ! [D] :
                  ( m1_bhsp_3(D,A,C)
                 => m1_bhsp_3(D,A,B) ) ) ) ) )).

fof(t6_frechet2,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)) )
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
             => ~ ( ! [D] :
                      ( m1_bhsp_3(D,A,B)
                     => ~ r1_tarski(k1_frechet(A,D),C) )
                  & ! [D] :
                      ( m2_subset_1(D,k1_numbers,k5_numbers)
                     => ? [E] :
                          ( m2_subset_1(E,k1_numbers,k5_numbers)
                          & r1_xreal_0(D,E)
                          & r2_hidden(k2_normsp_1(A,B,E),C) ) ) ) ) ) ) )).

fof(t7_frechet2,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)) )
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
             => ! [D] :
                  ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
                 => ~ ( r1_tarski(k1_frechet(A,B),k4_subset_1(u1_struct_0(A),C,D))
                      & ! [E] :
                          ( m1_bhsp_3(E,A,B)
                         => ( ~ r1_tarski(k1_frechet(A,E),C)
                            & ~ r1_tarski(k1_frechet(A,E),D) ) ) ) ) ) ) ) )).

fof(t8_frechet2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(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))
               => ! [D] :
                    ( m1_subset_1(D,u1_struct_0(A))
                   => ( ( r2_hidden(C,k2_frechet(A,B))
                        & r2_hidden(D,k2_frechet(A,B)) )
                     => C = D ) ) ) )
       => v1_urysohn1(A) ) ) )).

fof(t9_frechet2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ( v3_compts_1(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))
               => ! [D] :
                    ( m1_subset_1(D,u1_struct_0(A))
                   => ( ( r2_hidden(C,k2_frechet(A,B))
                        & r2_hidden(D,k2_frechet(A,B)) )
                     => C = D ) ) ) ) ) ) )).

fof(t10_frechet2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ( v1_frechet(A)
       => ( v3_compts_1(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))
                 => ! [D] :
                      ( m1_subset_1(D,u1_struct_0(A))
                     => ( ( r2_hidden(C,k2_frechet(A,B))
                          & r2_hidden(D,k2_frechet(A,B)) )
                       => C = D ) ) ) ) ) ) ) )).

fof(t11_frechet2,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)) )
         => ( ~ v2_frechet(B,A)
           => k2_frechet(A,B) = k1_xboole_0 ) ) ) )).

fof(t12_frechet2,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)) )
               => ( r1_tarski(k1_frechet(A,C),B)
                 => r1_tarski(k2_frechet(A,C),B) ) ) ) ) ) )).

fof(t13_frechet2,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,u1_struct_0(A))
             => ~ ( ~ r1_frechet(A,B,C)
                  & ! [D] :
                      ( m1_bhsp_3(D,A,B)
                     => ? [E] :
                          ( m1_bhsp_3(E,A,D)
                          & r1_frechet(A,E,C) ) ) ) ) ) ) )).

fof(t14_frechet2,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)) )
             => ( v5_pre_topc(C,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)) )
                   => ! [E] :
                        ( ( v1_funct_1(E)
                          & v1_funct_2(E,k5_numbers,u1_struct_0(B))
                          & m2_relset_1(E,k5_numbers,u1_struct_0(B)) )
                       => ( E = k7_funct_2(k5_numbers,u1_struct_0(A),u1_struct_0(B),D,C)
                         => r1_tarski(k4_pre_topc(A,B,C,k2_frechet(A,D)),k2_frechet(B,E)) ) ) ) ) ) ) ) )).

fof(t15_frechet2,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)) )
             => ( v4_frechet(A)
               => ( v5_pre_topc(C,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)) )
                     => ! [E] :
                          ( ( v1_funct_1(E)
                            & v1_funct_2(E,k5_numbers,u1_struct_0(B))
                            & m2_relset_1(E,k5_numbers,u1_struct_0(B)) )
                         => ( E = k7_funct_2(k5_numbers,u1_struct_0(A),u1_struct_0(B),D,C)
                           => r1_tarski(k4_pre_topc(A,B,C,k2_frechet(A,D)),k2_frechet(B,E)) ) ) ) ) ) ) ) ) )).

fof(d1_frechet2,axiom,(
    $true )).

fof(d2_frechet2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(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)))
             => ( C = k3_frechet2(A,B)
              <=> ! [D] :
                    ( m1_subset_1(D,u1_struct_0(A))
                   => ( r2_hidden(D,C)
                    <=> ? [E] :
                          ( v1_funct_1(E)
                          & v1_funct_2(E,k5_numbers,u1_struct_0(A))
                          & m2_relset_1(E,k5_numbers,u1_struct_0(A))
                          & r1_tarski(k1_frechet(A,E),B)
                          & r2_hidden(D,k2_frechet(A,E)) ) ) ) ) ) ) ) )).

fof(t16_frechet2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(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)) )
             => ! [D] :
                  ( m1_subset_1(D,u1_struct_0(A))
                 => ( ( r1_tarski(k1_frechet(A,C),B)
                      & r2_hidden(D,k2_frechet(A,C)) )
                   => r2_hidden(D,k6_pre_topc(A,B)) ) ) ) ) ) )).

fof(t17_frechet2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_pre_topc(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => r1_tarski(k3_frechet2(A,B),k6_pre_topc(A,B)) ) ) )).

fof(t18_frechet2,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_bhsp_3(C,A,B)
             => ! [D] :
                  ( m1_subset_1(D,u1_struct_0(A))
                 => ( r1_frechet(A,B,D)
                   => r1_frechet(A,C,D) ) ) ) ) ) )).

fof(t19_frechet2,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_bhsp_3(C,A,B)
             => r1_tarski(k2_frechet(A,B),k2_frechet(A,C)) ) ) ) )).

fof(t20_frechet2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_pre_topc(A) )
     => k3_frechet2(A,k1_pre_topc(A)) = k1_xboole_0 ) )).

fof(t21_frechet2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_pre_topc(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => r1_tarski(B,k3_frechet2(A,B)) ) ) )).

fof(t22_frechet2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(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)))
             => k4_subset_1(u1_struct_0(A),k3_frechet2(A,B),k3_frechet2(A,C)) = k3_frechet2(A,k4_subset_1(u1_struct_0(A),B,C)) ) ) ) )).

fof(t23_frechet2,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)))
           => k6_pre_topc(A,B) = k3_frechet2(A,B) ) ) ) )).

fof(t24_frechet2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(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,k1_zfmisc_1(u1_struct_0(A)))
               => ( k3_frechet2(A,k1_pre_topc(A)) = k1_xboole_0
                  & r1_tarski(B,k3_frechet2(A,B))
                  & k3_frechet2(A,k4_subset_1(u1_struct_0(A),B,C)) = k4_subset_1(u1_struct_0(A),k3_frechet2(A,B),k3_frechet2(A,C))
                  & k3_frechet2(A,k3_frechet2(A,B)) = k3_frechet2(A,B) ) ) ) ) ) )).

fof(t25_frechet2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ( ( v4_frechet(A)
          & ! [B] :
              ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
             => k3_frechet2(A,k3_frechet2(A,B)) = k3_frechet2(A,B) ) )
       => v3_frechet(A) ) ) )).

fof(t26_frechet2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ( v4_frechet(A)
       => ( v3_frechet(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)))
                 => ( k3_frechet2(A,k1_pre_topc(A)) = k1_xboole_0
                    & r1_tarski(B,k3_frechet2(A,B))
                    & k3_frechet2(A,k4_subset_1(u1_struct_0(A),B,C)) = k4_subset_1(u1_struct_0(A),k3_frechet2(A,B),k3_frechet2(A,C))
                    & k3_frechet2(A,k3_frechet2(A,B)) = k3_frechet2(A,B) ) ) ) ) ) ) )).

fof(d3_frechet2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(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))
                & k2_frechet(A,B) = k1_struct_0(A,C) )
           => ! [C] :
                ( m1_subset_1(C,u1_struct_0(A))
               => ( C = k4_frechet2(A,B)
                <=> r1_frechet(A,B,C) ) ) ) ) ) )).

fof(t27_frechet2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ( v3_compts_1(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))
                   => k2_frechet(A,B) != k1_struct_0(A,C) ) ) ) ) ) )).

fof(t28_frechet2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ( v3_compts_1(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)
                <=> ( v2_frechet(B,A)
                    & C = k4_frechet2(A,B) ) ) ) ) ) ) )).

fof(t29_frechet2,axiom,(
    ! [A] :
      ( l1_metric_1(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)) )
         => ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,u1_struct_0(k5_pcomps_1(A)))
            & m2_relset_1(B,k5_numbers,u1_struct_0(k5_pcomps_1(A))) ) ) ) )).

fof(t30_frechet2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_metric_1(A) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,u1_struct_0(k5_pcomps_1(A)))
            & m2_relset_1(B,k5_numbers,u1_struct_0(k5_pcomps_1(A))) )
         => ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,u1_struct_0(A))
            & m2_relset_1(B,k5_numbers,u1_struct_0(A)) ) ) ) )).

fof(t31_frechet2,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_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))
             => ! [D] :
                  ( ( v1_funct_1(D)
                    & v1_funct_2(D,k5_numbers,u1_struct_0(k5_pcomps_1(A)))
                    & m2_relset_1(D,k5_numbers,u1_struct_0(k5_pcomps_1(A))) )
                 => ! [E] :
                      ( m1_subset_1(E,u1_struct_0(k5_pcomps_1(A)))
                     => ( ( B = D
                          & C = E )
                       => ( r1_metric_6(A,B,C)
                        <=> r1_frechet(k5_pcomps_1(A),D,E) ) ) ) ) ) ) ) )).

fof(t32_frechet2,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_funct_1(B)
            & v1_funct_2(B,k5_numbers,u1_struct_0(A))
            & m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,u1_struct_0(k5_pcomps_1(A)))
                & m2_relset_1(C,k5_numbers,u1_struct_0(k5_pcomps_1(A))) )
             => ( B = C
               => ( v2_tbsp_1(B,A)
                <=> v2_frechet(C,k5_pcomps_1(A)) ) ) ) ) ) )).

fof(t33_frechet2,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_funct_1(B)
            & v1_funct_2(B,k5_numbers,u1_struct_0(A))
            & m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,u1_struct_0(k5_pcomps_1(A)))
                & m2_relset_1(C,k5_numbers,u1_struct_0(k5_pcomps_1(A))) )
             => ( ( B = C
                  & v2_tbsp_1(B,A) )
               => k1_tbsp_1(A,B) = k4_frechet2(k5_pcomps_1(A),C) ) ) ) ) )).

fof(d4_frechet2,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_frechet2(A,B,C)
              <=> ! [D] :
                    ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
                   => ! [E] :
                        ( m2_subset_1(E,k1_numbers,k5_numbers)
                       => ~ ( v3_pre_topc(D,A)
                            & r2_hidden(C,D)
                            & ! [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(t34_frechet2,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,u1_struct_0(A))
             => ( ? [D] :
                    ( m1_bhsp_3(D,A,B)
                    & r1_frechet(A,D,C) )
               => r1_frechet2(A,B,C) ) ) ) ) )).

fof(t35_frechet2,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,u1_struct_0(A))
             => ( r1_frechet(A,B,C)
               => r1_frechet2(A,B,C) ) ) ) ) )).

fof(t37_frechet2,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,u1_struct_0(A))
             => ! [D] :
                  ( m1_subset_1(D,u1_struct_0(A))
                 => ( ! [E] :
                        ( m2_subset_1(E,k1_numbers,k5_numbers)
                       => ( k2_normsp_1(A,B,E) = D
                          & r1_frechet(A,B,C) ) )
                   => r2_hidden(C,k6_pre_topc(A,k1_struct_0(A,D))) ) ) ) ) ) )).

fof(t39_frechet2,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] :
              ( ( 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),k1_frechet(A,B))
                  & v2_funct_1(C)
                  & ! [D] :
                      ( ( v1_funct_1(D)
                        & v1_funct_2(D,k5_numbers,k5_numbers)
                        & v3_funct_2(D,k5_numbers,k5_numbers)
                        & m2_relset_1(D,k5_numbers,k5_numbers) )
                     => ~ m1_bhsp_3(k1_frechet2(A,D,C),A,B) ) ) ) ) ) )).

fof(t40_frechet2,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] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,k5_numbers)
                & v3_funct_2(C,k5_numbers,k5_numbers)
                & m2_relset_1(C,k5_numbers,k5_numbers) )
             => ! [D] :
                  ( m1_subset_1(D,u1_struct_0(A))
                 => ( r1_frechet(A,B,D)
                   => r1_frechet(A,k1_frechet2(A,C,B),D) ) ) ) ) ) )).

fof(t41_frechet2,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ? [B] :
          ( v1_funct_1(B)
          & v1_funct_2(B,k5_numbers,k1_numbers)
          & v1_seqm_3(B)
          & v7_seqm_3(B)
          & m2_relset_1(B,k5_numbers,k1_numbers)
          & ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => k2_seqm_3(B,C) = k1_nat_1(C,A) ) ) ) )).

fof(t42_frechet2,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)) )
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => ? [D] :
                  ( m1_bhsp_3(D,A,B)
                  & ! [E] :
                      ( m2_subset_1(E,k1_numbers,k5_numbers)
                     => k2_normsp_1(A,D,E) = k2_normsp_1(A,B,k1_nat_1(E,C)) ) ) ) ) ) )).

fof(t43_frechet2,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,u1_struct_0(A))
             => ! [D] :
                  ( m1_bhsp_3(D,A,B)
                 => ( r1_frechet2(A,B,C)
                   => ( ! [E] :
                          ( m2_subset_1(E,k1_numbers,k5_numbers)
                         => ~ ! [F] :
                                ( m2_subset_1(F,k1_numbers,k5_numbers)
                               => k2_normsp_1(A,D,F) = k2_normsp_1(A,B,k1_nat_1(F,E)) ) )
                      | r1_frechet2(A,D,C) ) ) ) ) ) ) )).

fof(t44_frechet2,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,u1_struct_0(A))
             => ( r1_frechet2(A,B,C)
               => r2_hidden(C,k6_pre_topc(A,k1_frechet(A,B))) ) ) ) ) )).

fof(t45_frechet2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_pre_topc(A) )
     => ( v3_frechet(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_frechet2(A,B,C)
                    & ! [D] :
                        ( m1_bhsp_3(D,A,B)
                       => ~ r1_frechet(A,D,C) ) ) ) ) ) ) )).

fof(t46_frechet2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ( v1_frechet(A)
       => ! [B] :
            ( m1_subset_1(B,u1_struct_0(A))
           => ? [C] :
                ( m1_yellow_8(C,A,B)
                & ? [D] :
                    ( v1_relat_1(D)
                    & v1_funct_1(D)
                    & k1_relat_1(D) = k5_numbers
                    & k2_relat_1(D) = C
                    & ! [E] :
                        ( m2_subset_1(E,k1_numbers,k5_numbers)
                       => ! [F] :
                            ( m2_subset_1(F,k1_numbers,k5_numbers)
                           => ( r1_xreal_0(E,F)
                             => r1_tarski(k1_funct_1(D,F),k1_funct_1(D,E)) ) ) ) ) ) ) ) ) )).

fof(t47_frechet2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ( v1_urysohn1(A)
      <=> ! [B] :
            ( m1_subset_1(B,u1_struct_0(A))
           => k6_pre_topc(A,k1_struct_0(A,B)) = k1_struct_0(A,B) ) ) ) )).

fof(t48_frechet2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ( v3_compts_1(A)
       => v1_urysohn1(A) ) ) )).

fof(t49_frechet2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ~ ( ~ v1_urysohn1(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,k5_numbers,u1_struct_0(A))
                        & m2_relset_1(D,k5_numbers,u1_struct_0(A)) )
                     => ~ ( D = k3_finsop_1(u1_struct_0(A),k5_numbers,B)
                          & B != C
                          & r1_frechet(A,D,C) ) ) ) ) ) ) )).

fof(t50_frechet2,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ~ ( ~ v1_finset_1(k1_relat_1(A))
          & v2_funct_1(A)
          & v1_finset_1(k2_relat_1(A)) ) ) )).

fof(s1_frechet2,axiom,
    ( ! [A] :
        ( m2_subset_1(A,k1_numbers,k5_numbers)
       => ? [B] :
            ( m2_subset_1(B,k1_numbers,k5_numbers)
            & ? [C] :
                ( m1_subset_1(C,u1_struct_0(f1_s1_frechet2))
                & r1_xreal_0(A,B)
                & C = k2_normsp_1(f1_s1_frechet2,f2_s1_frechet2,B)
                & p1_s1_frechet2(C) ) ) )
   => ? [A] :
        ( m1_bhsp_3(A,f1_s1_frechet2,f2_s1_frechet2)
        & ! [B] :
            ( m2_subset_1(B,k1_numbers,k5_numbers)
           => p1_s1_frechet2(k2_normsp_1(f1_s1_frechet2,A,B)) ) ) )).

fof(s2_frechet2,axiom,
    ( ! [A] :
        ( m2_subset_1(A,k1_numbers,k5_numbers)
       => ? [B] :
            ( m2_subset_1(B,k1_numbers,k5_numbers)
            & ? [C] :
                ( m1_subset_1(C,u1_struct_0(f1_s2_frechet2))
                & r1_xreal_0(A,B)
                & C = k2_normsp_1(f1_s2_frechet2,f2_s2_frechet2,B)
                & p1_s2_frechet2(C) ) ) )
   => ? [A] :
        ( m1_bhsp_3(A,f1_s2_frechet2,f2_s2_frechet2)
        & ! [B] :
            ( m2_subset_1(B,k1_numbers,k5_numbers)
           => p1_s2_frechet2(k2_normsp_1(f1_s2_frechet2,A,B)) ) ) )).

fof(s3_frechet2,axiom,
    ( ? [A] :
        ( m2_subset_1(A,k1_numbers,k5_numbers)
        & ! [B] :
            ( m2_subset_1(B,k1_numbers,k5_numbers)
           => ! [C] :
                ( m1_subset_1(C,u1_struct_0(f1_s3_frechet2))
               => ( ( r1_xreal_0(A,B)
                    & C = k2_normsp_1(f1_s3_frechet2,f2_s3_frechet2,B) )
                 => p1_s3_frechet2(C) ) ) ) )
   => ? [A] :
        ( m2_subset_1(A,k1_numbers,k5_numbers)
        & ! [B] :
            ( m2_subset_1(B,k1_numbers,k5_numbers)
           => ( r1_xreal_0(A,B)
             => p1_s3_frechet2(k2_normsp_1(f1_s3_frechet2,k1_frechet2(f1_s3_frechet2,f3_s3_frechet2,f2_s3_frechet2),B)) ) ) ) )).

fof(s4_frechet2,axiom,
    ( ? [A] :
        ( m2_subset_1(A,k1_numbers,k5_numbers)
        & ! [B] :
            ( m2_subset_1(B,k1_numbers,k5_numbers)
           => ! [C] :
                ( m1_subset_1(C,u1_struct_0(f1_s4_frechet2))
               => ( ( r1_xreal_0(A,B)
                    & C = k2_normsp_1(f1_s4_frechet2,f2_s4_frechet2,B) )
                 => p1_s4_frechet2(C) ) ) ) )
   => ? [A] :
        ( m2_subset_1(A,k1_numbers,k5_numbers)
        & ! [B] :
            ( m2_subset_1(B,k1_numbers,k5_numbers)
           => ( r1_xreal_0(A,B)
             => p1_s4_frechet2(k2_normsp_1(f1_s4_frechet2,k1_frechet2(f1_s4_frechet2,f3_s4_frechet2,f2_s4_frechet2),B)) ) ) ) )).

fof(dt_k1_frechet2,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(A)
        & l1_struct_0(A)
        & v1_funct_1(B)
        & v1_funct_2(B,k5_numbers,k5_numbers)
        & m1_relset_1(B,k5_numbers,k5_numbers)
        & v1_funct_1(C)
        & v1_funct_2(C,k5_numbers,u1_struct_0(A))
        & m1_relset_1(C,k5_numbers,u1_struct_0(A)) )
     => ( v1_funct_1(k1_frechet2(A,B,C))
        & v1_funct_2(k1_frechet2(A,B,C),k5_numbers,u1_struct_0(A))
        & m2_relset_1(k1_frechet2(A,B,C),k5_numbers,u1_struct_0(A)) ) ) )).

fof(redefinition_k1_frechet2,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(A)
        & l1_struct_0(A)
        & v1_funct_1(B)
        & v1_funct_2(B,k5_numbers,k5_numbers)
        & m1_relset_1(B,k5_numbers,k5_numbers)
        & v1_funct_1(C)
        & v1_funct_2(C,k5_numbers,u1_struct_0(A))
        & m1_relset_1(C,k5_numbers,u1_struct_0(A)) )
     => k1_frechet2(A,B,C) = k5_relat_1(B,C) ) )).

fof(dt_k2_frechet2,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(A)
        & l1_struct_0(A)
        & v1_funct_1(B)
        & v1_funct_2(B,k5_numbers,k1_numbers)
        & v1_seqm_3(B)
        & v7_seqm_3(B)
        & m1_relset_1(B,k5_numbers,k1_numbers)
        & v1_funct_1(C)
        & v1_funct_2(C,k5_numbers,u1_struct_0(A))
        & m1_relset_1(C,k5_numbers,u1_struct_0(A)) )
     => m1_bhsp_3(k2_frechet2(A,B,C),A,C) ) )).

fof(redefinition_k2_frechet2,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(A)
        & l1_struct_0(A)
        & v1_funct_1(B)
        & v1_funct_2(B,k5_numbers,k1_numbers)
        & v1_seqm_3(B)
        & v7_seqm_3(B)
        & m1_relset_1(B,k5_numbers,k1_numbers)
        & v1_funct_1(C)
        & v1_funct_2(C,k5_numbers,u1_struct_0(A))
        & m1_relset_1(C,k5_numbers,u1_struct_0(A)) )
     => k2_frechet2(A,B,C) = k5_relat_1(B,C) ) )).

fof(dt_k3_frechet2,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(A)
        & l1_pre_topc(A)
        & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
     => m1_subset_1(k3_frechet2(A,B),k1_zfmisc_1(u1_struct_0(A))) ) )).

fof(dt_k4_frechet2,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(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(k4_frechet2(A,B),u1_struct_0(A)) ) )).

fof(t36_frechet2,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,u1_struct_0(A))
             => ! [D] :
                  ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
                 => ( ( D = a_2_0_frechet2(A,C)
                      & r1_tarski(k1_frechet(A,B),D) )
                   => r1_frechet(A,B,C) ) ) ) ) ) )).

fof(t38_frechet2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_pre_topc(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
             => ! [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)) )
                 => ~ ( C = a_2_0_frechet2(A,B)
                      & r1_xboole_0(k1_frechet(A,D),C)
                      & r1_frechet(A,D,B)
                      & ! [E] :
                          ( m1_bhsp_3(E,A,D)
                         => ~ v2_funct_1(E) ) ) ) ) ) ) )).

fof(fraenkel_a_2_0_frechet2,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(B)
        & l1_pre_topc(B)
        & m1_subset_1(C,u1_struct_0(B)) )
     => ( r2_hidden(A,a_2_0_frechet2(B,C))
      <=> ? [D] :
            ( m1_subset_1(D,u1_struct_0(B))
            & A = D
            & r2_hidden(C,k6_pre_topc(B,k1_struct_0(B,D))) ) ) ) )).
%------------------------------------------------------------------------------