SET007 Axioms: SET007+301.ax


%------------------------------------------------------------------------------
% File     : SET007+301 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Paracompact and Metrizable 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    : pcomps_1 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   65 (  11 unit)
%            Number of atoms       :  365 (  29 equality)
%            Maximal formula depth :   26 (   7 average)
%            Number of connectives :  352 (  52 ~  ;   0  |; 150  &)
%                                         (  11 <=>; 139 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   38 (   1 propositional; 0-3 arity)
%            Number of functors    :   31 (   6 constant; 0-5 arity)
%            Number of variables   :  144 (   0 singleton; 133 !;  11 ?)
%            Maximal term depth    :    5 (   2 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(fc1_pcomps_1,axiom,(
    ! [A] :
      ( v1_pre_topc(k2_pcomps_1(A))
      & v2_pre_topc(k2_pcomps_1(A)) ) )).

fof(fc2_pcomps_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ( ~ v3_struct_0(k2_pcomps_1(A))
        & v1_pre_topc(k2_pcomps_1(A))
        & v2_pre_topc(k2_pcomps_1(A)) ) ) )).

fof(rc1_pcomps_1,axiom,(
    ? [A] :
      ( l1_pre_topc(A)
      & ~ v3_struct_0(A)
      & v2_pre_topc(A)
      & v2_pcomps_1(A) ) )).

fof(fc3_pcomps_1,axiom,(
    ! [A] :
      ( l1_metric_1(A)
     => ( v1_pre_topc(k5_pcomps_1(A))
        & v2_pre_topc(k5_pcomps_1(A)) ) ) )).

fof(fc4_pcomps_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_metric_1(A) )
     => ( ~ v3_struct_0(k5_pcomps_1(A))
        & v1_pre_topc(k5_pcomps_1(A))
        & v2_pre_topc(k5_pcomps_1(A)) ) ) )).

fof(rc2_pcomps_1,axiom,(
    ? [A] :
      ( l1_pre_topc(A)
      & ~ v3_struct_0(A)
      & v1_pre_topc(A)
      & v2_pre_topc(A)
      & v3_compts_1(A) ) )).

fof(rc3_pcomps_1,axiom,(
    ? [A] :
      ( l1_pre_topc(A)
      & ~ v3_struct_0(A)
      & v1_pre_topc(A)
      & v2_pre_topc(A)
      & v3_pcomps_1(A) ) )).

fof(t1_pcomps_1,axiom,(
    ! [A] :
      ( l1_metric_1(A)
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( v1_xreal_0(C)
             => ! [D] :
                  ( v1_xreal_0(D)
                 => ( r1_xreal_0(C,D)
                   => r1_tarski(k9_metric_1(A,B,C),k9_metric_1(A,B,D)) ) ) ) ) ) )).

fof(t2_pcomps_1,axiom,(
    ! [A] :
      ( ( v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => ( ~ ( k6_pre_topc(A,B) != k1_xboole_0
                & B = k1_xboole_0 )
            & ~ ( B != k1_xboole_0
                & k6_pre_topc(A,B) = k1_xboole_0 ) ) ) ) )).

fof(t3_pcomps_1,axiom,(
    $true )).

fof(t4_pcomps_1,axiom,(
    $true )).

fof(t5_pcomps_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
         => ( r1_pre_topc(A,B)
           => ! [C] :
                ( m1_subset_1(C,u1_struct_0(A))
               => ? [D] :
                    ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
                    & r2_hidden(C,D)
                    & r2_hidden(D,B) ) ) ) ) ) )).

fof(d1_pcomps_1,axiom,(
    ! [A] : k2_pcomps_1(A) = g1_pre_topc(A,k1_pcomps_1(A)) )).

fof(t6_pcomps_1,axiom,(
    $true )).

fof(t7_pcomps_1,axiom,(
    ! [A] : u1_pre_topc(k2_pcomps_1(A)) = k1_pcomps_1(A) )).

fof(t8_pcomps_1,axiom,(
    ! [A] : u1_struct_0(k2_pcomps_1(A)) = A )).

fof(t9_pcomps_1,axiom,(
    ! [A] : v2_compts_1(k2_pcomps_1(k1_tarski(A))) )).

fof(t10_pcomps_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ( v3_compts_1(A)
           => v4_pre_topc(k1_struct_0(A,B),A) ) ) ) )).

fof(t12_pcomps_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
             => ( ( r1_tarski(B,C)
                  & v1_pcomps_1(C,A) )
               => v1_pcomps_1(B,A) ) ) ) ) )).

fof(t13_pcomps_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
         => ( v1_finset_1(B)
           => v1_pcomps_1(B,A) ) ) ) )).

fof(d3_pcomps_1,axiom,(
    ! [A] :
      ( l1_pre_topc(A)
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
             => ( C = k3_pcomps_1(A,B)
              <=> ! [D] :
                    ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
                   => ( r2_hidden(D,C)
                    <=> ? [E] :
                          ( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(A)))
                          & D = k6_pre_topc(A,E)
                          & r2_hidden(E,B) ) ) ) ) ) ) ) )).

fof(t14_pcomps_1,axiom,(
    ! [A] :
      ( ( v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
         => v2_tops_2(k3_pcomps_1(A,B),A) ) ) )).

fof(t15_pcomps_1,axiom,(
    ! [A] :
      ( ( v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
         => ( B = k1_xboole_0
           => k3_pcomps_1(A,B) = k1_xboole_0 ) ) ) )).

fof(t16_pcomps_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
             => ( C = k1_tarski(B)
               => k3_pcomps_1(A,C) = k1_tarski(k6_pre_topc(A,B)) ) ) ) ) )).

fof(t17_pcomps_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
             => ( r1_tarski(B,C)
               => r1_tarski(k3_pcomps_1(A,B),k3_pcomps_1(A,C)) ) ) ) ) )).

fof(t18_pcomps_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
             => k3_pcomps_1(A,k4_subset_1(k1_zfmisc_1(u1_struct_0(A)),B,C)) = k4_subset_1(k1_zfmisc_1(u1_struct_0(A)),k3_pcomps_1(A,B),k3_pcomps_1(A,C)) ) ) ) )).

fof(t19_pcomps_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
         => ( v1_finset_1(B)
           => k6_pre_topc(A,k5_setfam_1(u1_struct_0(A),B)) = k5_setfam_1(u1_struct_0(A),k3_pcomps_1(A,B)) ) ) ) )).

fof(t20_pcomps_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
         => r1_setfam_1(B,k3_pcomps_1(A,B)) ) ) )).

fof(t21_pcomps_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
         => ( v1_pcomps_1(B,A)
           => v1_pcomps_1(k3_pcomps_1(A,B),A) ) ) ) )).

fof(t22_pcomps_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
         => r1_tarski(k5_setfam_1(u1_struct_0(A),B),k5_setfam_1(u1_struct_0(A),k3_pcomps_1(A,B))) ) ) )).

fof(t23_pcomps_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
         => ( v1_pcomps_1(B,A)
           => k6_pre_topc(A,k5_setfam_1(u1_struct_0(A),B)) = k5_setfam_1(u1_struct_0(A),k3_pcomps_1(A,B)) ) ) ) )).

fof(t24_pcomps_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
         => ( ( v1_pcomps_1(B,A)
              & v2_tops_2(B,A) )
           => v4_pre_topc(k5_setfam_1(u1_struct_0(A),B),A) ) ) ) )).

fof(d4_pcomps_1,axiom,(
    ! [A] :
      ( l1_pre_topc(A)
     => ( v2_pcomps_1(A)
      <=> ! [B] :
            ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
           => ~ ( r1_pre_topc(A,B)
                & v1_tops_2(B,A)
                & ! [C] :
                    ( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
                   => ~ ( v1_tops_2(C,A)
                        & r1_pre_topc(A,C)
                        & r1_setfam_1(C,B)
                        & v1_pcomps_1(C,A) ) ) ) ) ) ) )).

fof(t25_pcomps_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ( v2_compts_1(A)
       => v2_pcomps_1(A) ) ) )).

fof(t26_pcomps_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
             => ~ ( v2_pcomps_1(A)
                  & v4_pre_topc(B,A)
                  & v4_pre_topc(C,A)
                  & r1_xboole_0(B,C)
                  & ! [D] :
                      ( m1_subset_1(D,u1_struct_0(A))
                     => ~ ( r2_hidden(D,C)
                          & ! [E] :
                              ( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(A)))
                             => ! [F] :
                                  ( m1_subset_1(F,k1_zfmisc_1(u1_struct_0(A)))
                                 => ~ ( v3_pre_topc(E,A)
                                      & v3_pre_topc(F,A)
                                      & r1_tarski(B,E)
                                      & r2_hidden(D,F)
                                      & r1_xboole_0(E,F) ) ) ) ) )
                  & ! [D] :
                      ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
                     => ! [E] :
                          ( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(A)))
                         => ~ ( v3_pre_topc(D,A)
                              & v3_pre_topc(E,A)
                              & r1_tarski(B,D)
                              & r1_tarski(C,E)
                              & r1_xboole_0(D,E) ) ) ) ) ) ) ) )).

fof(t27_pcomps_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ( ( v3_compts_1(A)
          & v2_pcomps_1(A) )
       => v4_compts_1(A) ) ) )).

fof(t28_pcomps_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ( ( v3_compts_1(A)
          & v2_pcomps_1(A) )
       => v5_compts_1(A) ) ) )).

fof(d5_pcomps_1,axiom,(
    ! [A] :
      ( l1_metric_1(A)
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
         => ( B = k4_pcomps_1(A)
          <=> ! [C] :
                ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
               => ( r2_hidden(C,B)
                <=> ! [D] :
                      ( m1_subset_1(D,u1_struct_0(A))
                     => ~ ( r2_hidden(D,C)
                          & ! [E] :
                              ( m1_subset_1(E,k1_numbers)
                             => ~ ( ~ r1_xreal_0(E,np__0)
                                  & r1_tarski(k9_metric_1(A,D,E),C) ) ) ) ) ) ) ) ) ) )).

fof(t29_pcomps_1,axiom,(
    ! [A] :
      ( l1_metric_1(A)
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ? [C] :
              ( m1_subset_1(C,k1_numbers)
              & ~ r1_xreal_0(C,np__0)
              & r1_tarski(k9_metric_1(A,B,C),u1_struct_0(A)) ) ) ) )).

fof(t30_pcomps_1,axiom,(
    ! [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_xreal_0(D)
                 => ~ ( v9_metric_1(A)
                      & r2_hidden(B,k9_metric_1(A,C,D))
                      & ! [E] :
                          ( m1_subset_1(E,k1_numbers)
                         => ~ ( ~ r1_xreal_0(E,np__0)
                              & r1_tarski(k9_metric_1(A,B,E),k9_metric_1(A,C,D)) ) ) ) ) ) ) ) )).

fof(t31_pcomps_1,axiom,(
    ! [A] :
      ( l1_metric_1(A)
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ! [C] :
              ( m1_subset_1(C,k1_numbers)
             => ! [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))
                         => ~ ( v9_metric_1(A)
                              & r2_hidden(D,k5_subset_1(u1_struct_0(A),k9_metric_1(A,E,B),k9_metric_1(A,F,C)))
                              & ! [G] :
                                  ( m1_subset_1(G,k1_numbers)
                                 => ~ ( r1_tarski(k9_metric_1(A,D,G),k9_metric_1(A,E,B))
                                      & r1_tarski(k9_metric_1(A,D,G),k9_metric_1(A,F,C)) ) ) ) ) ) ) ) ) ) )).

fof(t32_pcomps_1,axiom,(
    $true )).

fof(t33_pcomps_1,axiom,(
    ! [A] :
      ( l1_metric_1(A)
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( v1_xreal_0(C)
             => ( v9_metric_1(A)
               => r2_hidden(k9_metric_1(A,B,C),k4_pcomps_1(A)) ) ) ) ) )).

fof(t34_pcomps_1,axiom,(
    ! [A] :
      ( l1_metric_1(A)
     => r2_hidden(u1_struct_0(A),k4_pcomps_1(A)) ) )).

fof(t35_pcomps_1,axiom,(
    ! [A] :
      ( l1_metric_1(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)))
             => ( ( r2_hidden(B,k4_pcomps_1(A))
                  & r2_hidden(C,k4_pcomps_1(A)) )
               => r2_hidden(k5_subset_1(u1_struct_0(A),B,C),k4_pcomps_1(A)) ) ) ) ) )).

fof(t36_pcomps_1,axiom,(
    ! [A] :
      ( l1_metric_1(A)
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
         => ( r1_tarski(B,k4_pcomps_1(A))
           => r2_hidden(k5_setfam_1(u1_struct_0(A),B),k4_pcomps_1(A)) ) ) ) )).

fof(t37_pcomps_1,axiom,(
    ! [A] :
      ( l1_metric_1(A)
     => ( v2_pre_topc(g1_pre_topc(u1_struct_0(A),k4_pcomps_1(A)))
        & l1_pre_topc(g1_pre_topc(u1_struct_0(A),k4_pcomps_1(A))) ) ) )).

fof(d6_pcomps_1,axiom,(
    ! [A] :
      ( l1_metric_1(A)
     => k5_pcomps_1(A) = g1_pre_topc(u1_struct_0(A),k4_pcomps_1(A)) ) )).

fof(t38_pcomps_1,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) )
     => v3_compts_1(k5_pcomps_1(A)) ) )).

fof(d7_pcomps_1,axiom,(
    ! [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)
               => ! [E] :
                    ( m1_subset_1(E,A)
                   => ( ( k1_metric_1(A,A,B,C,D) = np__0
                       => C = D )
                      & ( C = D
                       => k1_metric_1(A,A,B,C,D) = np__0 )
                      & k1_metric_1(A,A,B,C,D) = k1_metric_1(A,A,B,D,C)
                      & r1_xreal_0(k1_metric_1(A,A,B,C,E),k3_real_1(k1_metric_1(A,A,B,C,D),k1_metric_1(A,A,B,D,E))) ) ) ) ) ) ) )).

fof(t39_pcomps_1,axiom,(
    ! [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)
      <=> ( v6_metric_1(g1_metric_1(A,B))
          & v7_metric_1(g1_metric_1(A,B))
          & v8_metric_1(g1_metric_1(A,B))
          & v9_metric_1(g1_metric_1(A,B))
          & l1_metric_1(g1_metric_1(A,B)) ) ) ) )).

fof(d8_pcomps_1,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)
           => k6_pcomps_1(A,B) = g1_metric_1(A,B) ) ) ) )).

fof(d9_pcomps_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_pre_topc(A) )
     => ( v3_pcomps_1(A)
      <=> ? [B] :
            ( v1_funct_1(B)
            & v1_funct_2(B,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),k1_numbers)
            & m2_relset_1(B,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),k1_numbers)
            & r1_pcomps_1(u1_struct_0(A),B)
            & k4_pcomps_1(k6_pcomps_1(u1_struct_0(A),B)) = u1_pre_topc(A) ) ) ) )).

fof(s1_pcomps_1,axiom,
    ( ! [A] :
        ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(f1_s1_pcomps_1)))
       => ( r2_hidden(A,f2_s1_pcomps_1)
         => r2_hidden(f4_s1_pcomps_1(A),f3_s1_pcomps_1) ) )
   => ? [A] :
        ( v1_funct_1(A)
        & v1_funct_2(A,f2_s1_pcomps_1,f3_s1_pcomps_1)
        & m2_relset_1(A,f2_s1_pcomps_1,f3_s1_pcomps_1)
        & ! [B] :
            ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(f1_s1_pcomps_1)))
           => ( r2_hidden(B,f2_s1_pcomps_1)
             => k1_funct_1(A,B) = f4_s1_pcomps_1(B) ) ) ) )).

fof(dt_k1_pcomps_1,axiom,(
    ! [A] : m1_subset_1(k1_pcomps_1(A),k1_zfmisc_1(k1_zfmisc_1(A))) )).

fof(redefinition_k1_pcomps_1,axiom,(
    ! [A] : k1_pcomps_1(A) = k1_zfmisc_1(A) )).

fof(dt_k2_pcomps_1,axiom,(
    ! [A] : l1_pre_topc(k2_pcomps_1(A)) )).

fof(dt_k3_pcomps_1,axiom,(
    ! [A,B] :
      ( ( l1_pre_topc(A)
        & m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A)))) )
     => m1_subset_1(k3_pcomps_1(A,B),k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A)))) ) )).

fof(dt_k4_pcomps_1,axiom,(
    ! [A] :
      ( l1_metric_1(A)
     => m1_subset_1(k4_pcomps_1(A),k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A)))) ) )).

fof(dt_k5_pcomps_1,axiom,(
    ! [A] :
      ( l1_metric_1(A)
     => l1_pre_topc(k5_pcomps_1(A)) ) )).

fof(dt_k6_pcomps_1,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) )
     => ( ~ v3_struct_0(k6_pcomps_1(A,B))
        & v1_metric_1(k6_pcomps_1(A,B))
        & v6_metric_1(k6_pcomps_1(A,B))
        & v7_metric_1(k6_pcomps_1(A,B))
        & v8_metric_1(k6_pcomps_1(A,B))
        & v9_metric_1(k6_pcomps_1(A,B))
        & l1_metric_1(k6_pcomps_1(A,B)) ) ) )).

fof(d2_pcomps_1,axiom,(
    ! [A] :
      ( l1_pre_topc(A)
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
         => ( v1_pcomps_1(B,A)
          <=> ! [C] :
                ( m1_subset_1(C,u1_struct_0(A))
               => ? [D] :
                    ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
                    & r2_hidden(C,D)
                    & v3_pre_topc(D,A)
                    & v1_finset_1(a_3_0_pcomps_1(A,B,D)) ) ) ) ) ) )).

fof(t11_pcomps_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
             => r1_tarski(a_3_1_pcomps_1(A,B,C),B) ) ) ) )).

fof(fraenkel_a_3_0_pcomps_1,axiom,(
    ! [A,B,C,D] :
      ( ( l1_pre_topc(B)
        & m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(B))))
        & m1_subset_1(D,k1_zfmisc_1(u1_struct_0(B))) )
     => ( r2_hidden(A,a_3_0_pcomps_1(B,C,D))
      <=> ? [E] :
            ( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(B)))
            & A = E
            & r2_hidden(E,C)
            & ~ r1_xboole_0(E,D) ) ) ) )).

fof(fraenkel_a_3_1_pcomps_1,axiom,(
    ! [A,B,C,D] :
      ( ( ~ v3_struct_0(B)
        & v2_pre_topc(B)
        & l1_pre_topc(B)
        & m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(B))))
        & m1_subset_1(D,k1_zfmisc_1(u1_struct_0(B))) )
     => ( r2_hidden(A,a_3_1_pcomps_1(B,C,D))
      <=> ? [E] :
            ( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(B)))
            & A = E
            & r2_hidden(E,C)
            & ~ r1_xboole_0(E,D) ) ) ) )).
%------------------------------------------------------------------------------