SET007 Axioms: SET007+325.ax


%------------------------------------------------------------------------------
% File     : SET007+325 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Sequences in Metric 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    : metric_6 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   65 (  24 unit)
%            Number of atoms       :  543 (  13 equality)
%            Maximal formula depth :   17 (   7 average)
%            Number of connectives :  531 (  53 ~  ;   0  |; 312  &)
%                                         (  15 <=>; 151 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   34 (   1 propositional; 0-3 arity)
%            Number of functors    :   26 (   4 constant; 0-5 arity)
%            Number of variables   :  140 (   0 singleton; 130 !;  10 ?)
%            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(cc1_metric_6,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_relset_1(B,k5_numbers,u1_struct_0(A))
         => ( ( v1_funct_1(B)
              & v1_funct_2(B,k5_numbers,u1_struct_0(A))
              & v5_seqm_3(B) )
           => ( v1_funct_1(B)
              & v1_funct_2(B,k5_numbers,u1_struct_0(A))
              & v2_tbsp_1(B,A) ) ) ) ) )).

fof(cc2_metric_6,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_relset_1(B,k5_numbers,u1_struct_0(A))
         => ( ( v1_funct_1(B)
              & v1_funct_2(B,k5_numbers,u1_struct_0(A))
              & v3_tbsp_1(B,A) )
           => ( v1_funct_1(B)
              & v1_funct_2(B,k5_numbers,u1_struct_0(A))
              & v1_metric_6(B,A) ) ) ) ) )).

fof(rc1_metric_6,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_relset_1(B,k5_numbers,u1_struct_0(A))
          & v1_relat_1(B)
          & v1_funct_1(B)
          & v1_funct_2(B,k5_numbers,u1_struct_0(A))
          & v2_tbsp_1(B,A)
          & v3_tbsp_1(B,A)
          & v5_seqm_3(B)
          & v1_metric_6(B,A) ) ) )).

fof(t1_metric_6,axiom,(
    ! [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(A))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ! [D] :
                  ( m1_subset_1(D,u1_struct_0(A))
                 => r1_xreal_0(k18_complex1(k5_real_1(k4_metric_1(A,B,C),k4_metric_1(A,D,C))),k4_metric_1(A,B,D)) ) ) ) ) )).

fof(t2_metric_6,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k2_zfmisc_1(A,A),k1_numbers)
            & m2_relset_1(B,k2_zfmisc_1(A,A),k1_numbers) )
         => ( r1_pcomps_1(A,B)
           => ! [C] :
                ( m1_subset_1(C,A)
               => ! [D] :
                    ( m1_subset_1(D,A)
                   => r1_xreal_0(np__0,k1_metric_1(A,A,B,C,D)) ) ) ) ) ) )).

fof(t3_metric_6,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k2_zfmisc_1(A,A),k1_numbers)
            & m2_relset_1(B,k2_zfmisc_1(A,A),k1_numbers) )
         => ( r1_pcomps_1(A,B)
          <=> ( v2_metric_1(B,A)
              & v3_metric_1(B,A)
              & v4_metric_1(B,A)
              & v5_metric_1(B,A) ) ) ) ) )).

fof(t4_metric_6,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v1_metric_1(A)
        & v6_metric_1(A)
        & v7_metric_1(A)
        & v8_metric_1(A)
        & v9_metric_1(A)
        & l1_metric_1(A) )
     => ( v2_metric_1(u1_metric_1(A),u1_struct_0(A))
        & v3_metric_1(u1_metric_1(A),u1_struct_0(A))
        & v4_metric_1(u1_metric_1(A),u1_struct_0(A))
        & v5_metric_1(u1_metric_1(A),u1_struct_0(A)) ) ) )).

fof(t5_metric_6,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k2_zfmisc_1(A,A),k1_numbers)
            & m2_relset_1(B,k2_zfmisc_1(A,A),k1_numbers) )
         => ( r1_pcomps_1(A,B)
          <=> ( v2_metric_1(B,A)
              & v3_metric_1(B,A)
              & ! [C] :
                  ( m1_subset_1(C,A)
                 => ! [D] :
                      ( m1_subset_1(D,A)
                     => ! [E] :
                          ( m1_subset_1(E,A)
                         => r1_xreal_0(k1_metric_1(A,A,B,D,E),k3_real_1(k1_metric_1(A,A,B,C,D),k1_metric_1(A,A,B,C,E))) ) ) ) ) ) ) ) )).

fof(d1_metric_6,axiom,(
    $true )).

fof(d2_metric_6,axiom,(
    $true )).

fof(d3_metric_6,axiom,(
    $true )).

fof(d4_metric_6,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k2_zfmisc_1(A,A),k1_numbers)
            & m2_relset_1(B,k2_zfmisc_1(A,A),k1_numbers) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k2_zfmisc_1(A,A),k1_numbers)
                & m2_relset_1(C,k2_zfmisc_1(A,A),k1_numbers) )
             => ( C = k1_metric_6(A,B)
              <=> ! [D] :
                    ( m1_subset_1(D,A)
                   => ! [E] :
                        ( m1_subset_1(E,A)
                       => k1_metric_1(A,A,C,D,E) = k6_real_1(k1_metric_1(A,A,B,D,E),k3_real_1(np__1,k1_metric_1(A,A,B,D,E))) ) ) ) ) ) ) )).

fof(t6_metric_6,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k2_zfmisc_1(A,A),k1_numbers)
            & m2_relset_1(B,k2_zfmisc_1(A,A),k1_numbers) )
         => ( r1_pcomps_1(A,B)
           => r1_pcomps_1(A,k1_metric_6(A,B)) ) ) ) )).

fof(t7_metric_6,axiom,(
    $true )).

fof(t8_metric_6,axiom,(
    $true )).

fof(t9_metric_6,axiom,(
    $true )).

fof(t10_metric_6,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(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))
              & k3_tbsp_1(A,C) = k1_struct_0(A,B) ) ) ) )).

fof(d5_metric_6,axiom,(
    $true )).

fof(d6_metric_6,axiom,(
    $true )).

fof(d7_metric_6,axiom,(
    $true )).

fof(d8_metric_6,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(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))
             => ( r1_metric_6(A,B,C)
              <=> ! [D] :
                    ( m1_subset_1(D,k1_numbers)
                   => ~ ( ~ r1_xreal_0(D,np__0)
                        & ! [E] :
                            ( m2_subset_1(E,k1_numbers,k5_numbers)
                           => ? [F] :
                                ( m2_subset_1(F,k1_numbers,k5_numbers)
                                & r1_xreal_0(E,F)
                                & r1_xreal_0(D,k2_metric_1(A,k2_normsp_1(A,B,F),C)) ) ) ) ) ) ) ) ) )).

fof(d9_metric_6,axiom,(
    $true )).

fof(d10_metric_6,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v8_metric_1(A)
        & v9_metric_1(A)
        & l1_metric_1(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => ( v6_tbsp_1(B,A)
          <=> ? [C] :
                ( m1_subset_1(C,k1_numbers)
                & ? [D] :
                    ( m1_subset_1(D,u1_struct_0(A))
                    & ~ r1_xreal_0(C,np__0)
                    & r1_tarski(B,k9_metric_1(A,D,C)) ) ) ) ) ) )).

fof(d11_metric_6,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(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_metric_6(B,A)
          <=> ? [C] :
                ( m1_subset_1(C,k1_numbers)
                & ? [D] :
                    ( m1_subset_1(D,u1_struct_0(A))
                    & ~ r1_xreal_0(C,np__0)
                    & r1_tarski(k3_tbsp_1(A,B),k9_metric_1(A,D,C)) ) ) ) ) ) )).

fof(d12_metric_6,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(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)) )
             => ( r2_metric_6(A,B,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,C,E),B) ) ) ) ) ) ) ) )).

fof(t11_metric_6,axiom,(
    $true )).

fof(t12_metric_6,axiom,(
    $true )).

fof(t13_metric_6,axiom,(
    $true )).

fof(t14_metric_6,axiom,(
    $true )).

fof(t15_metric_6,axiom,(
    $true )).

fof(t16_metric_6,axiom,(
    $true )).

fof(t17_metric_6,axiom,(
    $true )).

fof(t18_metric_6,axiom,(
    $true )).

fof(t19_metric_6,axiom,(
    $true )).

fof(t20_metric_6,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)) )
         => ( v1_metric_6(B,A)
          <=> ? [C] :
                ( m1_subset_1(C,k1_numbers)
                & ? [D] :
                    ( m1_subset_1(D,u1_struct_0(A))
                    & ~ r1_xreal_0(C,np__0)
                    & ! [E] :
                        ( m2_subset_1(E,k1_numbers,k5_numbers)
                       => r2_hidden(k2_normsp_1(A,B,E),k9_metric_1(A,D,C)) ) ) ) ) ) ) )).

fof(t21_metric_6,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(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_metric_6(A,C,B)
               => v2_tbsp_1(C,A) ) ) ) ) )).

fof(t22_metric_6,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)) )
         => ~ ( v2_tbsp_1(B,A)
              & ! [C] :
                  ( m1_subset_1(C,u1_struct_0(A))
                 => ~ r1_metric_6(A,B,C) ) ) ) ) )).

fof(d13_metric_6,axiom,(
    $true )).

fof(d14_metric_6,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,k1_numbers)
                    & m2_relset_1(D,k5_numbers,k1_numbers) )
                 => ( D = k2_metric_6(A,B,C)
                  <=> ! [E] :
                        ( m2_subset_1(E,k1_numbers,k5_numbers)
                       => k2_seq_1(k5_numbers,k1_numbers,D,E) = k4_metric_1(A,k2_normsp_1(A,B,E),C) ) ) ) ) ) ) )).

fof(d15_metric_6,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(A))
                & m2_relset_1(C,k5_numbers,u1_struct_0(A)) )
             => ! [D] :
                  ( ( v1_funct_1(D)
                    & v1_funct_2(D,k5_numbers,k1_numbers)
                    & m2_relset_1(D,k5_numbers,k1_numbers) )
                 => ( D = k3_metric_6(A,B,C)
                  <=> ! [E] :
                        ( m2_subset_1(E,k1_numbers,k5_numbers)
                       => k2_seq_1(k5_numbers,k1_numbers,D,E) = k4_metric_1(A,k2_normsp_1(A,B,E),k2_normsp_1(A,C,E)) ) ) ) ) ) ) )).

fof(t23_metric_6,axiom,(
    $true )).

fof(t24_metric_6,axiom,(
    $true )).

fof(t25_metric_6,axiom,(
    $true )).

fof(t26_metric_6,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(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_metric_6(A,C,B)
               => k1_tbsp_1(A,C) = B ) ) ) ) )).

fof(t27_metric_6,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(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_metric_6(A,C,B)
              <=> ( v2_tbsp_1(C,A)
                  & k1_tbsp_1(A,C) = B ) ) ) ) ) )).

fof(t28_metric_6,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)) )
         => ~ ( v2_tbsp_1(B,A)
              & ! [C] :
                  ( m1_subset_1(C,u1_struct_0(A))
                 => ~ ( r1_metric_6(A,B,C)
                      & k1_tbsp_1(A,B) = C ) ) ) ) ) )).

fof(t29_metric_6,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(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_metric_6(A,C,B)
              <=> ( v4_seq_2(k2_metric_6(A,C,B))
                  & k2_seq_2(k2_metric_6(A,C,B)) = np__0 ) ) ) ) ) )).

fof(t30_metric_6,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(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_metric_6(A,C,B)
               => ! [D] :
                    ( m1_subset_1(D,k1_numbers)
                   => ( ~ r1_xreal_0(D,np__0)
                     => r2_metric_6(A,k9_metric_1(A,B,D),C) ) ) ) ) ) ) )).

fof(t31_metric_6,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(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,k1_numbers)
                   => ( ~ r1_xreal_0(D,np__0)
                     => r2_metric_6(A,k9_metric_1(A,B,D),C) ) )
               => ! [D] :
                    ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
                   => ( ( r2_hidden(B,D)
                        & r2_hidden(D,k4_pcomps_1(A)) )
                     => r2_metric_6(A,D,C) ) ) ) ) ) ) )).

fof(t32_metric_6,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(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,k1_zfmisc_1(u1_struct_0(A)))
                   => ( ( r2_hidden(B,D)
                        & r2_hidden(D,k4_pcomps_1(A)) )
                     => r2_metric_6(A,D,C) ) )
               => r1_metric_6(A,C,B) ) ) ) ) )).

fof(t33_metric_6,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(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_metric_6(A,C,B)
              <=> ! [D] :
                    ( m1_subset_1(D,k1_numbers)
                   => ( ~ r1_xreal_0(D,np__0)
                     => r2_metric_6(A,k9_metric_1(A,B,D),C) ) ) ) ) ) ) )).

fof(t34_metric_6,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(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_metric_6(A,C,B)
              <=> ! [D] :
                    ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
                   => ( ( r2_hidden(B,D)
                        & r2_hidden(D,k4_pcomps_1(A)) )
                     => r2_metric_6(A,D,C) ) ) ) ) ) ) )).

fof(t35_metric_6,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(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,k1_numbers)
                   => ( ~ r1_xreal_0(D,np__0)
                     => r2_metric_6(A,k9_metric_1(A,B,D),C) ) )
              <=> ! [D] :
                    ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
                   => ( ( r2_hidden(B,D)
                        & r2_hidden(D,k4_pcomps_1(A)) )
                     => r2_metric_6(A,D,C) ) ) ) ) ) ) )).

fof(t36_metric_6,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(A))
                & m2_relset_1(C,k5_numbers,u1_struct_0(A)) )
             => ( ( v2_tbsp_1(B,A)
                  & v2_tbsp_1(C,A) )
               => k4_metric_1(A,k1_tbsp_1(A,B),k1_tbsp_1(A,C)) = k2_seq_2(k3_metric_6(A,B,C)) ) ) ) ) )).

fof(t37_metric_6,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(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)) )
                 => ( ( r1_metric_6(A,D,B)
                      & r1_metric_6(A,D,C) )
                   => B = C ) ) ) ) ) )).

fof(t38_metric_6,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)) )
         => ( v5_seqm_3(B)
           => v2_tbsp_1(B,A) ) ) ) )).

fof(t39_metric_6,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(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] :
                  ( ( 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_metric_6(A,C,B)
                      & m1_bhsp_3(D,A,C) )
                   => r1_metric_6(A,D,B) ) ) ) ) ) )).

fof(t40_metric_6,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(A))
                & m2_relset_1(C,k5_numbers,u1_struct_0(A)) )
             => ( ( v3_tbsp_1(B,A)
                  & m1_bhsp_3(C,A,B) )
               => v3_tbsp_1(C,A) ) ) ) ) )).

fof(t41_metric_6,axiom,(
    $true )).

fof(t42_metric_6,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)) )
         => ( v5_seqm_3(B)
           => v3_tbsp_1(B,A) ) ) ) )).

fof(t43_metric_6,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)) )
         => ( v2_tbsp_1(B,A)
           => v1_metric_6(B,A) ) ) ) )).

fof(t44_metric_6,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)) )
         => ( v3_tbsp_1(B,A)
           => v1_metric_6(B,A) ) ) ) )).

fof(dt_k1_metric_6,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(A)
        & v1_funct_1(B)
        & v1_funct_2(B,k2_zfmisc_1(A,A),k1_numbers)
        & m1_relset_1(B,k2_zfmisc_1(A,A),k1_numbers) )
     => ( v1_funct_1(k1_metric_6(A,B))
        & v1_funct_2(k1_metric_6(A,B),k2_zfmisc_1(A,A),k1_numbers)
        & m2_relset_1(k1_metric_6(A,B),k2_zfmisc_1(A,A),k1_numbers) ) ) )).

fof(dt_k2_metric_6,axiom,(
    ! [A,B,C] :
      ( ( ~ 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_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(C,u1_struct_0(A)) )
     => ( v1_funct_1(k2_metric_6(A,B,C))
        & v1_funct_2(k2_metric_6(A,B,C),k5_numbers,k1_numbers)
        & m2_relset_1(k2_metric_6(A,B,C),k5_numbers,k1_numbers) ) ) )).

fof(dt_k3_metric_6,axiom,(
    ! [A,B,C] :
      ( ( ~ 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_funct_1(B)
        & v1_funct_2(B,k5_numbers,u1_struct_0(A))
        & m1_relset_1(B,k5_numbers,u1_struct_0(A))
        & 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(k3_metric_6(A,B,C))
        & v1_funct_2(k3_metric_6(A,B,C),k5_numbers,k1_numbers)
        & m2_relset_1(k3_metric_6(A,B,C),k5_numbers,k1_numbers) ) ) )).
%------------------------------------------------------------------------------