SET007 Axioms: SET007+266.ax


%------------------------------------------------------------------------------
% File     : SET007+266 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : On Pseudometric 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_2 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   99 (  20 unit)
%            Number of atoms       :  617 (  47 equality)
%            Maximal formula depth :   19 (   7 average)
%            Number of connectives :  602 (  84 ~  ;   0  |; 308  &)
%                                         (  35 <=>; 175 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   23 (   1 propositional; 0-4 arity)
%            Number of functors    :   35 (   2 constant; 0-6 arity)
%            Number of variables   :  239 (   0 singleton; 193 !;  46 ?)
%            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_2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v6_metric_1(A)
        & v8_metric_1(A)
        & v9_metric_1(A)
        & l1_metric_1(A) )
     => ! [B] :
          ( m1_metric_2(B,A)
         => ~ v1_xboole_0(B) ) ) )).

fof(fc1_metric_2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_metric_1(A) )
     => ~ v1_xboole_0(k2_metric_2(A)) ) )).

fof(fc2_metric_2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v6_metric_1(A)
        & v8_metric_1(A)
        & v9_metric_1(A)
        & l1_metric_1(A) )
     => ( ~ v3_struct_0(k11_metric_2(A))
        & v1_metric_1(k11_metric_2(A))
        & v6_metric_1(k11_metric_2(A))
        & v7_metric_1(k11_metric_2(A))
        & v8_metric_1(k11_metric_2(A))
        & v9_metric_1(k11_metric_2(A)) ) ) )).

fof(d1_metric_2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_metric_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ( r1_metric_2(A,B,C)
              <=> k2_metric_1(A,B,C) = np__0 ) ) ) ) )).

fof(d3_metric_2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_metric_1(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => ( m1_metric_2(B,A)
          <=> ? [C] :
                ( m1_subset_1(C,u1_struct_0(A))
                & B = k1_metric_2(A,C) ) ) ) ) )).

fof(t1_metric_2,axiom,(
    $true )).

fof(t2_metric_2,axiom,(
    $true )).

fof(t3_metric_2,axiom,(
    $true )).

fof(t4_metric_2,axiom,(
    $true )).

fof(t5_metric_2,axiom,(
    $true )).

fof(t6_metric_2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v6_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))
                 => ( ( r3_metric_2(A,B,C)
                      & r3_metric_2(A,C,D) )
                   => r3_metric_2(A,B,D) ) ) ) ) ) )).

fof(t7_metric_2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v6_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))
             => ( r2_hidden(C,k1_metric_2(A,B))
              <=> r3_metric_2(A,C,B) ) ) ) ) )).

fof(t8_metric_2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v6_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))
                 => ( ( r2_hidden(C,k1_metric_2(A,B))
                      & r2_hidden(D,k1_metric_2(A,B)) )
                   => r3_metric_2(A,C,D) ) ) ) ) ) )).

fof(t9_metric_2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v6_metric_1(A)
        & v8_metric_1(A)
        & v9_metric_1(A)
        & l1_metric_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => r2_hidden(B,k1_metric_2(A,B)) ) ) )).

fof(t10_metric_2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v6_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))
             => ( r2_hidden(B,k1_metric_2(A,C))
              <=> r2_hidden(C,k1_metric_2(A,B)) ) ) ) ) )).

fof(t11_metric_2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v6_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))
                 => ( ( r2_hidden(B,k1_metric_2(A,C))
                      & r3_metric_2(A,C,D) )
                   => r2_hidden(B,k1_metric_2(A,D)) ) ) ) ) ) )).

fof(t12_metric_2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v6_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))
             => ( r2_hidden(C,k1_metric_2(A,B))
               => k1_metric_2(A,B) = k1_metric_2(A,C) ) ) ) ) )).

fof(t13_metric_2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v6_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))
             => ( k1_metric_2(A,B) = k1_metric_2(A,C)
              <=> r3_metric_2(A,B,C) ) ) ) ) )).

fof(t14_metric_2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v6_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))
             => ( ~ r1_xboole_0(k1_metric_2(A,B),k1_metric_2(A,C))
              <=> r3_metric_2(A,B,C) ) ) ) ) )).

fof(t15_metric_2,axiom,(
    $true )).

fof(t16_metric_2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v6_metric_1(A)
        & v8_metric_1(A)
        & v9_metric_1(A)
        & l1_metric_1(A) )
     => ! [B] :
          ( m1_metric_2(B,A)
         => ~ v1_xboole_0(B) ) ) )).

fof(t17_metric_2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v6_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))
                 => ( ( r2_hidden(C,k1_metric_2(A,B))
                      & r2_hidden(D,k1_metric_2(A,B)) )
                   => k4_metric_1(A,C,D) = np__0 ) ) ) ) ) )).

fof(t18_metric_2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v6_metric_1(A)
        & v7_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))
             => ( r2_metric_2(A,B,C)
              <=> B = C ) ) ) ) )).

fof(t19_metric_2,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))
             => ( r2_hidden(C,k1_metric_2(A,B))
              <=> C = B ) ) ) ) )).

fof(t20_metric_2,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))
         => k1_metric_2(A,B) = k1_struct_0(A,B) ) ) )).

fof(t21_metric_2,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)))
         => ( m1_metric_2(B,A)
          <=> ? [C] :
                ( m1_subset_1(C,u1_struct_0(A))
                & B = k1_struct_0(A,C) ) ) ) ) )).

fof(t22_metric_2,axiom,(
    $true )).

fof(t23_metric_2,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(B)
        & l1_metric_1(B) )
     => ( r2_hidden(A,k2_metric_2(B))
      <=> ? [C] :
            ( m1_subset_1(C,u1_struct_0(B))
            & A = k1_metric_2(B,C) ) ) ) )).

fof(t24_metric_2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_metric_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => r2_hidden(k1_metric_2(A,B),k2_metric_2(A)) ) ) )).

fof(t25_metric_2,axiom,(
    $true )).

fof(t26_metric_2,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(B)
        & l1_metric_1(B) )
     => ( r2_hidden(A,k2_metric_2(B))
      <=> m1_metric_2(A,B) ) ) )).

fof(t27_metric_2,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))
         => r2_hidden(k1_struct_0(A,B),k2_metric_2(A)) ) ) )).

fof(t28_metric_2,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(B)
        & v6_metric_1(B)
        & v7_metric_1(B)
        & v8_metric_1(B)
        & v9_metric_1(B)
        & l1_metric_1(B) )
     => ( r2_hidden(A,k2_metric_2(B))
      <=> ? [C] :
            ( m1_subset_1(C,u1_struct_0(B))
            & A = k1_struct_0(B,C) ) ) ) )).

fof(t29_metric_2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v6_metric_1(A)
        & v8_metric_1(A)
        & v9_metric_1(A)
        & l1_metric_1(A) )
     => ! [B] :
          ( m1_subset_1(B,k2_metric_2(A))
         => ! [C] :
              ( m1_subset_1(C,k2_metric_2(A))
             => ! [D] :
                  ( m1_subset_1(D,u1_struct_0(A))
                 => ! [E] :
                      ( m1_subset_1(E,u1_struct_0(A))
                     => ! [F] :
                          ( m1_subset_1(F,u1_struct_0(A))
                         => ! [G] :
                              ( m1_subset_1(G,u1_struct_0(A))
                             => ( ( r2_hidden(D,B)
                                  & r2_hidden(F,C)
                                  & r2_hidden(E,B)
                                  & r2_hidden(G,C) )
                               => k4_metric_1(A,D,F) = k4_metric_1(A,E,G) ) ) ) ) ) ) ) ) )).

fof(d5_metric_2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_metric_1(A) )
     => ! [B] :
          ( m1_subset_1(B,k2_metric_2(A))
         => ! [C] :
              ( m1_subset_1(C,k2_metric_2(A))
             => ! [D] :
                  ( m1_subset_1(D,k1_numbers)
                 => ( r4_metric_2(A,B,C,D)
                  <=> ! [E] :
                        ( m1_subset_1(E,u1_struct_0(A))
                       => ! [F] :
                            ( m1_subset_1(F,u1_struct_0(A))
                           => ( ( r2_hidden(E,B)
                                & r2_hidden(F,C) )
                             => k2_metric_1(A,E,F) = D ) ) ) ) ) ) ) ) )).

fof(t30_metric_2,axiom,(
    $true )).

fof(t31_metric_2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v6_metric_1(A)
        & v8_metric_1(A)
        & v9_metric_1(A)
        & l1_metric_1(A) )
     => ! [B] :
          ( m1_subset_1(B,k2_metric_2(A))
         => ! [C] :
              ( m1_subset_1(C,k2_metric_2(A))
             => ! [D] :
                  ( m1_subset_1(D,k1_numbers)
                 => ( r4_metric_2(A,B,C,D)
                  <=> ? [E] :
                        ( m1_subset_1(E,u1_struct_0(A))
                        & ? [F] :
                            ( m1_subset_1(F,u1_struct_0(A))
                            & r2_hidden(E,B)
                            & r2_hidden(F,C)
                            & k4_metric_1(A,E,F) = D ) ) ) ) ) ) ) )).

fof(t32_metric_2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v6_metric_1(A)
        & v8_metric_1(A)
        & v9_metric_1(A)
        & l1_metric_1(A) )
     => ! [B] :
          ( m1_subset_1(B,k2_metric_2(A))
         => ! [C] :
              ( m1_subset_1(C,k2_metric_2(A))
             => ! [D] :
                  ( m1_subset_1(D,k1_numbers)
                 => ( r4_metric_2(A,B,C,D)
                  <=> r4_metric_2(A,C,B,D) ) ) ) ) ) )).

fof(t33_metric_2,axiom,(
    $true )).

fof(t34_metric_2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v6_metric_1(A)
        & v8_metric_1(A)
        & v9_metric_1(A)
        & l1_metric_1(A) )
     => ! [B] :
          ( m1_subset_1(B,k2_metric_2(A))
         => ! [C] :
              ( m1_subset_1(C,k2_metric_2(A))
             => ! [D] :
                  ( m1_subset_1(D,k1_numbers)
                 => ( r2_hidden(D,k3_metric_2(A,B,C))
                  <=> r4_metric_2(A,B,C,D) ) ) ) ) ) )).

fof(t35_metric_2,axiom,(
    $true )).

fof(t36_metric_2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v6_metric_1(A)
        & v8_metric_1(A)
        & v9_metric_1(A)
        & l1_metric_1(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ! [C] :
              ( m1_subset_1(C,k2_zfmisc_1(k2_metric_2(A),k2_metric_2(A)))
             => ( r2_hidden(C,k4_metric_2(A,B))
              <=> ? [D] :
                    ( m1_subset_1(D,k2_metric_2(A))
                    & ? [E] :
                        ( m1_subset_1(E,k2_metric_2(A))
                        & C = k1_domain_1(k2_metric_2(A),k2_metric_2(A),D,E)
                        & r4_metric_2(A,D,E,B) ) ) ) ) ) ) )).

fof(t37_metric_2,axiom,(
    $true )).

fof(t38_metric_2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v6_metric_1(A)
        & v8_metric_1(A)
        & v9_metric_1(A)
        & l1_metric_1(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ( r2_hidden(B,k5_metric_2(A))
          <=> ? [C] :
                ( m1_subset_1(C,k2_metric_2(A))
                & ? [D] :
                    ( m1_subset_1(D,k2_metric_2(A))
                    & r4_metric_2(A,C,D,B) ) ) ) ) ) )).

fof(t39_metric_2,axiom,(
    $true )).

fof(t40_metric_2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v6_metric_1(A)
        & v8_metric_1(A)
        & v9_metric_1(A)
        & l1_metric_1(A) )
     => ! [B] :
          ( m1_subset_1(B,k2_metric_2(A))
         => ( r2_hidden(B,k6_metric_2(A))
          <=> ? [C] :
                ( m1_subset_1(C,k2_metric_2(A))
                & ? [D] :
                    ( m1_subset_1(D,k1_numbers)
                    & r4_metric_2(A,B,C,D) ) ) ) ) ) )).

fof(t41_metric_2,axiom,(
    $true )).

fof(t42_metric_2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v6_metric_1(A)
        & v8_metric_1(A)
        & v9_metric_1(A)
        & l1_metric_1(A) )
     => ! [B] :
          ( m1_subset_1(B,k2_metric_2(A))
         => ( r2_hidden(B,k7_metric_2(A))
          <=> ? [C] :
                ( m1_subset_1(C,k2_metric_2(A))
                & ? [D] :
                    ( m1_subset_1(D,k1_numbers)
                    & r4_metric_2(A,C,B,D) ) ) ) ) ) )).

fof(t43_metric_2,axiom,(
    $true )).

fof(t44_metric_2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v6_metric_1(A)
        & v8_metric_1(A)
        & v9_metric_1(A)
        & l1_metric_1(A) )
     => ! [B] :
          ( m1_subset_1(B,k2_zfmisc_1(k2_metric_2(A),k2_metric_2(A)))
         => ( r2_hidden(B,k8_metric_2(A))
          <=> ? [C] :
                ( m1_subset_1(C,k2_metric_2(A))
                & ? [D] :
                    ( m1_subset_1(D,k2_metric_2(A))
                    & ? [E] :
                        ( m1_subset_1(E,k1_numbers)
                        & B = k1_domain_1(k2_metric_2(A),k2_metric_2(A),C,D)
                        & r4_metric_2(A,C,D,E) ) ) ) ) ) ) )).

fof(t45_metric_2,axiom,(
    $true )).

fof(t46_metric_2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v6_metric_1(A)
        & v8_metric_1(A)
        & v9_metric_1(A)
        & l1_metric_1(A) )
     => ! [B] :
          ( m1_subset_1(B,k3_zfmisc_1(k2_metric_2(A),k2_metric_2(A),k1_numbers))
         => ( r2_hidden(B,k9_metric_2(A))
          <=> ? [C] :
                ( m1_subset_1(C,k2_metric_2(A))
                & ? [D] :
                    ( m1_subset_1(D,k2_metric_2(A))
                    & ? [E] :
                        ( m1_subset_1(E,k1_numbers)
                        & B = k3_mcart_1(C,D,E)
                        & r4_metric_2(A,C,D,E) ) ) ) ) ) ) )).

fof(t47_metric_2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v6_metric_1(A)
        & v8_metric_1(A)
        & v9_metric_1(A)
        & l1_metric_1(A) )
     => k6_metric_2(A) = k7_metric_2(A) ) )).

fof(t48_metric_2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v6_metric_1(A)
        & v8_metric_1(A)
        & v9_metric_1(A)
        & l1_metric_1(A) )
     => r1_tarski(k9_metric_2(A),k13_mcart_1(k2_metric_2(A),k2_metric_2(A),k1_numbers,k6_metric_2(A),k7_metric_2(A),k5_metric_2(A))) ) )).

fof(t49_metric_2,axiom,(
    $true )).

fof(t50_metric_2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v6_metric_1(A)
        & v8_metric_1(A)
        & v9_metric_1(A)
        & l1_metric_1(A) )
     => ! [B] :
          ( m1_subset_1(B,k2_metric_2(A))
         => ! [C] :
              ( m1_subset_1(C,k2_metric_2(A))
             => ! [D] :
                  ( m1_subset_1(D,k1_numbers)
                 => ! [E] :
                      ( m1_subset_1(E,k1_numbers)
                     => ( ( r4_metric_2(A,B,C,D)
                          & r4_metric_2(A,B,C,E) )
                       => D = E ) ) ) ) ) ) )).

fof(t51_metric_2,axiom,(
    $true )).

fof(t52_metric_2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v6_metric_1(A)
        & v8_metric_1(A)
        & v9_metric_1(A)
        & l1_metric_1(A) )
     => ! [B] :
          ( m1_subset_1(B,k2_metric_2(A))
         => ! [C] :
              ( m1_subset_1(C,k2_metric_2(A))
             => ? [D] :
                  ( m1_subset_1(D,k1_numbers)
                  & r4_metric_2(A,B,C,D) ) ) ) ) )).

fof(d13_metric_2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v6_metric_1(A)
        & v8_metric_1(A)
        & v9_metric_1(A)
        & l1_metric_1(A) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k2_zfmisc_1(k2_metric_2(A),k2_metric_2(A)),k1_numbers)
            & m2_relset_1(B,k2_zfmisc_1(k2_metric_2(A),k2_metric_2(A)),k1_numbers) )
         => ( B = k10_metric_2(A)
          <=> ! [C] :
                ( m1_subset_1(C,k2_metric_2(A))
               => ! [D] :
                    ( m1_subset_1(D,k2_metric_2(A))
                   => ! [E] :
                        ( m1_subset_1(E,u1_struct_0(A))
                       => ! [F] :
                            ( m1_subset_1(F,u1_struct_0(A))
                           => ( ( r2_hidden(E,C)
                                & r2_hidden(F,D) )
                             => k1_metric_1(k2_metric_2(A),k2_metric_2(A),B,C,D) = k4_metric_1(A,E,F) ) ) ) ) ) ) ) ) )).

fof(t53_metric_2,axiom,(
    $true )).

fof(t54_metric_2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v6_metric_1(A)
        & v8_metric_1(A)
        & v9_metric_1(A)
        & l1_metric_1(A) )
     => ! [B] :
          ( m1_subset_1(B,k2_metric_2(A))
         => ! [C] :
              ( m1_subset_1(C,k2_metric_2(A))
             => ( k1_metric_1(k2_metric_2(A),k2_metric_2(A),k10_metric_2(A),B,C) = np__0
              <=> B = C ) ) ) ) )).

fof(t55_metric_2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v6_metric_1(A)
        & v8_metric_1(A)
        & v9_metric_1(A)
        & l1_metric_1(A) )
     => ! [B] :
          ( m1_subset_1(B,k2_metric_2(A))
         => ! [C] :
              ( m1_subset_1(C,k2_metric_2(A))
             => k1_metric_1(k2_metric_2(A),k2_metric_2(A),k10_metric_2(A),B,C) = k1_metric_1(k2_metric_2(A),k2_metric_2(A),k10_metric_2(A),C,B) ) ) ) )).

fof(t56_metric_2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v6_metric_1(A)
        & v8_metric_1(A)
        & v9_metric_1(A)
        & l1_metric_1(A) )
     => ! [B] :
          ( m1_subset_1(B,k2_metric_2(A))
         => ! [C] :
              ( m1_subset_1(C,k2_metric_2(A))
             => ! [D] :
                  ( m1_subset_1(D,k2_metric_2(A))
                 => r1_xreal_0(k1_metric_1(k2_metric_2(A),k2_metric_2(A),k10_metric_2(A),B,D),k2_xcmplx_0(k1_metric_1(k2_metric_2(A),k2_metric_2(A),k10_metric_2(A),B,C),k1_metric_1(k2_metric_2(A),k2_metric_2(A),k10_metric_2(A),C,D))) ) ) ) ) )).

fof(d14_metric_2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v6_metric_1(A)
        & v8_metric_1(A)
        & v9_metric_1(A)
        & l1_metric_1(A) )
     => k11_metric_2(A) = g1_metric_1(k2_metric_2(A),k10_metric_2(A)) ) )).

fof(dt_m1_metric_2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_metric_1(A) )
     => ! [B] :
          ( m1_metric_2(B,A)
         => m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) ) ) )).

fof(existence_m1_metric_2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_metric_1(A) )
     => ? [B] : m1_metric_2(B,A) ) )).

fof(reflexivity_r2_metric_2,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(A)
        & v6_metric_1(A)
        & l1_metric_1(A)
        & m1_subset_1(B,u1_struct_0(A))
        & m1_subset_1(C,u1_struct_0(A)) )
     => r2_metric_2(A,B,B) ) )).

fof(redefinition_r2_metric_2,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(A)
        & v6_metric_1(A)
        & l1_metric_1(A)
        & m1_subset_1(B,u1_struct_0(A))
        & m1_subset_1(C,u1_struct_0(A)) )
     => ( r2_metric_2(A,B,C)
      <=> r1_metric_2(A,B,C) ) ) )).

fof(symmetry_r3_metric_2,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(A)
        & v8_metric_1(A)
        & l1_metric_1(A)
        & m1_subset_1(B,u1_struct_0(A))
        & m1_subset_1(C,u1_struct_0(A)) )
     => ( r3_metric_2(A,B,C)
       => r3_metric_2(A,C,B) ) ) )).

fof(redefinition_r3_metric_2,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(A)
        & v8_metric_1(A)
        & l1_metric_1(A)
        & m1_subset_1(B,u1_struct_0(A))
        & m1_subset_1(C,u1_struct_0(A)) )
     => ( r3_metric_2(A,B,C)
      <=> r1_metric_2(A,B,C) ) ) )).

fof(dt_k1_metric_2,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(A)
        & l1_metric_1(A)
        & m1_subset_1(B,u1_struct_0(A)) )
     => m1_subset_1(k1_metric_2(A,B),k1_zfmisc_1(u1_struct_0(A))) ) )).

fof(dt_k2_metric_2,axiom,(
    $true )).

fof(dt_k3_metric_2,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(A)
        & l1_metric_1(A)
        & m1_subset_1(B,k2_metric_2(A))
        & m1_subset_1(C,k2_metric_2(A)) )
     => m1_subset_1(k3_metric_2(A,B,C),k1_zfmisc_1(k1_numbers)) ) )).

fof(dt_k4_metric_2,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(A)
        & l1_metric_1(A)
        & m1_subset_1(B,k1_numbers) )
     => m1_subset_1(k4_metric_2(A,B),k1_zfmisc_1(k2_zfmisc_1(k2_metric_2(A),k2_metric_2(A)))) ) )).

fof(dt_k5_metric_2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_metric_1(A) )
     => m1_subset_1(k5_metric_2(A),k1_zfmisc_1(k1_numbers)) ) )).

fof(dt_k6_metric_2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_metric_1(A) )
     => m1_subset_1(k6_metric_2(A),k1_zfmisc_1(k2_metric_2(A))) ) )).

fof(dt_k7_metric_2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_metric_1(A) )
     => m1_subset_1(k7_metric_2(A),k1_zfmisc_1(k2_metric_2(A))) ) )).

fof(dt_k8_metric_2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_metric_1(A) )
     => m1_subset_1(k8_metric_2(A),k1_zfmisc_1(k2_zfmisc_1(k2_metric_2(A),k2_metric_2(A)))) ) )).

fof(dt_k9_metric_2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_metric_1(A) )
     => m1_subset_1(k9_metric_2(A),k1_zfmisc_1(k3_zfmisc_1(k2_metric_2(A),k2_metric_2(A),k1_numbers))) ) )).

fof(dt_k10_metric_2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v6_metric_1(A)
        & v8_metric_1(A)
        & v9_metric_1(A)
        & l1_metric_1(A) )
     => ( v1_funct_1(k10_metric_2(A))
        & v1_funct_2(k10_metric_2(A),k2_zfmisc_1(k2_metric_2(A),k2_metric_2(A)),k1_numbers)
        & m2_relset_1(k10_metric_2(A),k2_zfmisc_1(k2_metric_2(A),k2_metric_2(A)),k1_numbers) ) ) )).

fof(dt_k11_metric_2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v6_metric_1(A)
        & v8_metric_1(A)
        & v9_metric_1(A)
        & l1_metric_1(A) )
     => ( v6_metric_1(k11_metric_2(A))
        & v7_metric_1(k11_metric_2(A))
        & v8_metric_1(k11_metric_2(A))
        & v9_metric_1(k11_metric_2(A))
        & l1_metric_1(k11_metric_2(A)) ) ) )).

fof(d2_metric_2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_metric_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => k1_metric_2(A,B) = a_2_0_metric_2(A,B) ) ) )).

fof(d4_metric_2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_metric_1(A) )
     => k2_metric_2(A) = a_1_0_metric_2(A) ) )).

fof(d6_metric_2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_metric_1(A) )
     => ! [B] :
          ( m1_subset_1(B,k2_metric_2(A))
         => ! [C] :
              ( m1_subset_1(C,k2_metric_2(A))
             => k3_metric_2(A,B,C) = a_3_0_metric_2(A,B,C) ) ) ) )).

fof(d7_metric_2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_metric_1(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => k4_metric_2(A,B) = a_2_1_metric_2(A,B) ) ) )).

fof(d8_metric_2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_metric_1(A) )
     => k5_metric_2(A) = a_1_1_metric_2(A) ) )).

fof(d9_metric_2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_metric_1(A) )
     => k6_metric_2(A) = a_1_2_metric_2(A) ) )).

fof(d10_metric_2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_metric_1(A) )
     => k7_metric_2(A) = a_1_3_metric_2(A) ) )).

fof(d11_metric_2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_metric_1(A) )
     => k8_metric_2(A) = a_1_4_metric_2(A) ) )).

fof(d12_metric_2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_metric_1(A) )
     => k9_metric_2(A) = a_1_5_metric_2(A) ) )).

fof(fraenkel_a_2_0_metric_2,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(B)
        & l1_metric_1(B)
        & m1_subset_1(C,u1_struct_0(B)) )
     => ( r2_hidden(A,a_2_0_metric_2(B,C))
      <=> ? [D] :
            ( m1_subset_1(D,u1_struct_0(B))
            & A = D
            & r1_metric_2(B,C,D) ) ) ) )).

fof(fraenkel_a_1_0_metric_2,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(B)
        & l1_metric_1(B) )
     => ( r2_hidden(A,a_1_0_metric_2(B))
      <=> ? [C] :
            ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(B)))
            & A = C
            & ? [D] :
                ( m1_subset_1(D,u1_struct_0(B))
                & k1_metric_2(B,D) = C ) ) ) ) )).

fof(fraenkel_a_3_0_metric_2,axiom,(
    ! [A,B,C,D] :
      ( ( ~ v3_struct_0(B)
        & l1_metric_1(B)
        & m1_subset_1(C,k2_metric_2(B))
        & m1_subset_1(D,k2_metric_2(B)) )
     => ( r2_hidden(A,a_3_0_metric_2(B,C,D))
      <=> ? [E] :
            ( m1_subset_1(E,k1_numbers)
            & A = E
            & r4_metric_2(B,C,D,E) ) ) ) )).

fof(fraenkel_a_2_1_metric_2,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(B)
        & l1_metric_1(B)
        & m1_subset_1(C,k1_numbers) )
     => ( r2_hidden(A,a_2_1_metric_2(B,C))
      <=> ? [D] :
            ( m1_subset_1(D,k2_zfmisc_1(k2_metric_2(B),k2_metric_2(B)))
            & A = D
            & ? [E] :
                ( m1_subset_1(E,k2_metric_2(B))
                & ? [F] :
                    ( m1_subset_1(F,k2_metric_2(B))
                    & D = k1_domain_1(k2_metric_2(B),k2_metric_2(B),E,F)
                    & r4_metric_2(B,E,F,C) ) ) ) ) ) )).

fof(fraenkel_a_1_1_metric_2,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(B)
        & l1_metric_1(B) )
     => ( r2_hidden(A,a_1_1_metric_2(B))
      <=> ? [C] :
            ( m1_subset_1(C,k1_numbers)
            & A = C
            & ? [D] :
                ( m1_subset_1(D,k2_metric_2(B))
                & ? [E] :
                    ( m1_subset_1(E,k2_metric_2(B))
                    & r4_metric_2(B,D,E,C) ) ) ) ) ) )).

fof(fraenkel_a_1_2_metric_2,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(B)
        & l1_metric_1(B) )
     => ( r2_hidden(A,a_1_2_metric_2(B))
      <=> ? [C] :
            ( m1_subset_1(C,k2_metric_2(B))
            & A = C
            & ? [D] :
                ( m1_subset_1(D,k2_metric_2(B))
                & ? [E] :
                    ( m1_subset_1(E,k1_numbers)
                    & r4_metric_2(B,C,D,E) ) ) ) ) ) )).

fof(fraenkel_a_1_3_metric_2,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(B)
        & l1_metric_1(B) )
     => ( r2_hidden(A,a_1_3_metric_2(B))
      <=> ? [C] :
            ( m1_subset_1(C,k2_metric_2(B))
            & A = C
            & ? [D] :
                ( m1_subset_1(D,k2_metric_2(B))
                & ? [E] :
                    ( m1_subset_1(E,k1_numbers)
                    & r4_metric_2(B,D,C,E) ) ) ) ) ) )).

fof(fraenkel_a_1_4_metric_2,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(B)
        & l1_metric_1(B) )
     => ( r2_hidden(A,a_1_4_metric_2(B))
      <=> ? [C] :
            ( m1_subset_1(C,k2_zfmisc_1(k2_metric_2(B),k2_metric_2(B)))
            & A = C
            & ? [D] :
                ( m1_subset_1(D,k2_metric_2(B))
                & ? [E] :
                    ( m1_subset_1(E,k2_metric_2(B))
                    & ? [F] :
                        ( m1_subset_1(F,k1_numbers)
                        & C = k1_domain_1(k2_metric_2(B),k2_metric_2(B),D,E)
                        & r4_metric_2(B,D,E,F) ) ) ) ) ) ) )).

fof(fraenkel_a_1_5_metric_2,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(B)
        & l1_metric_1(B) )
     => ( r2_hidden(A,a_1_5_metric_2(B))
      <=> ? [C] :
            ( m1_subset_1(C,k3_zfmisc_1(k2_metric_2(B),k2_metric_2(B),k1_numbers))
            & A = C
            & ? [D] :
                ( m1_subset_1(D,k2_metric_2(B))
                & ? [E] :
                    ( m1_subset_1(E,k2_metric_2(B))
                    & ? [F] :
                        ( m1_subset_1(F,k1_numbers)
                        & C = k3_mcart_1(D,E,F)
                        & r4_metric_2(B,D,E,F) ) ) ) ) ) ) )).
%------------------------------------------------------------------------------