SET007 Axioms: SET007+757.ax


%------------------------------------------------------------------------------
% File     : SET007+757 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : On the Hausdorff Distance Between Compact Subsets
% 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    : hausdorf [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   58 (   0 unit)
%            Number of atoms       :  626 (  39 equality)
%            Maximal formula depth :   17 (  10 average)
%            Number of connectives :  684 ( 116 ~  ;   1  |; 362  &)
%                                         (   7 <=>; 198 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   28 (   0 propositional; 1-3 arity)
%            Number of functors    :   35 (   4 constant; 0-4 arity)
%            Number of variables   :  189 (   0 singleton; 182 !;   7 ?)
%            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_hausdorf,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_struct_0(k5_pcomps_1(A))
        & v1_pre_topc(k5_pcomps_1(A))
        & v2_pre_topc(k5_pcomps_1(A))
        & v3_compts_1(k5_pcomps_1(A)) ) ) )).

fof(fc2_hausdorf,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(A)
        & v6_metric_1(A)
        & v7_metric_1(A)
        & v8_metric_1(A)
        & v9_metric_1(A)
        & l1_metric_1(A)
        & m1_subset_1(B,u1_struct_0(A)) )
     => ( v1_relat_1(k6_weierstr(A,B))
        & v1_funct_1(k6_weierstr(A,B))
        & v1_funct_2(k6_weierstr(A,B),u1_struct_0(k5_pcomps_1(A)),u1_struct_0(k3_topmetr))
        & v5_pre_topc(k6_weierstr(A,B),k5_pcomps_1(A),k3_topmetr) ) ) )).

fof(fc3_hausdorf,axiom,(
    ! [A,B] :
      ( ( ~ 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_xboole_0(B)
        & v6_compts_1(B,k5_pcomps_1(A))
        & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k5_pcomps_1(A)))) )
     => ( v1_relat_1(k7_weierstr(A,B))
        & v1_funct_1(k7_weierstr(A,B))
        & v1_funct_2(k7_weierstr(A,B),u1_struct_0(k5_pcomps_1(A)),u1_struct_0(k3_topmetr))
        & v5_pre_topc(k7_weierstr(A,B),k5_pcomps_1(A),k3_topmetr) ) ) )).

fof(fc4_hausdorf,axiom,(
    ! [A,B] :
      ( ( ~ 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_xboole_0(B)
        & v6_compts_1(B,k5_pcomps_1(A))
        & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k5_pcomps_1(A)))) )
     => ( v1_relat_1(k8_weierstr(A,B))
        & v1_funct_1(k8_weierstr(A,B))
        & v1_funct_2(k8_weierstr(A,B),u1_struct_0(k5_pcomps_1(A)),u1_struct_0(k3_topmetr))
        & v5_pre_topc(k8_weierstr(A,B),k5_pcomps_1(A),k3_topmetr) ) ) )).

fof(t1_hausdorf,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ( ( r1_xreal_0(np__0,A)
              & r1_xreal_0(np__0,B)
              & k2_square_1(A,B) = np__0 )
           => A = np__0 ) ) ) )).

fof(t2_hausdorf,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_funct_1(k6_weierstr(A,B),B) = np__0 ) ) )).

fof(t3_hausdorf,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(k5_pcomps_1(A))))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ( r2_hidden(C,B)
               => r2_hidden(np__0,k4_pre_topc(k5_pcomps_1(A),k3_topmetr,k6_weierstr(A,C),B)) ) ) ) ) )).

fof(t4_hausdorf,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(k5_pcomps_1(A))))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ! [D] :
                  ( v1_xreal_0(D)
                 => ( r2_hidden(D,k4_pre_topc(k5_pcomps_1(A),k3_topmetr,k6_weierstr(A,C),B))
                   => r1_xreal_0(np__0,D) ) ) ) ) ) )).

fof(t5_hausdorf,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(k5_pcomps_1(A))))
         => ! [C] :
              ( r2_hidden(C,B)
             => k1_funct_1(k8_weierstr(A,B),C) = np__0 ) ) ) )).

fof(t6_hausdorf,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(k5_pcomps_1(A)))
             => ! [D] :
                  ( v1_xreal_0(D)
                 => ( B = C
                   => ( r1_xreal_0(D,np__0)
                      | m1_connsp_2(k9_metric_1(A,B,D),k5_pcomps_1(A),C) ) ) ) ) ) ) )).

fof(t7_hausdorf,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(k5_pcomps_1(A))))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ( r2_hidden(C,k6_pre_topc(k5_pcomps_1(A),B))
              <=> ! [D] :
                    ( v1_xreal_0(D)
                   => ~ ( ~ r1_xreal_0(D,np__0)
                        & r1_xboole_0(k9_metric_1(A,C,D),B) ) ) ) ) ) ) )).

fof(t8_hausdorf,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,k1_zfmisc_1(u1_struct_0(k5_pcomps_1(A))))
             => ( r2_hidden(B,k6_pre_topc(k5_pcomps_1(A),C))
              <=> ! [D] :
                    ( v1_xreal_0(D)
                   => ~ ( ~ r1_xreal_0(D,np__0)
                        & ! [E] :
                            ( m1_subset_1(E,u1_struct_0(A))
                           => ~ ( r2_hidden(E,C)
                                & ~ r1_xreal_0(D,k4_metric_1(A,B,E)) ) ) ) ) ) ) ) ) )).

fof(t9_hausdorf,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_xboole_0(B)
            & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k5_pcomps_1(A)))) )
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ( k1_funct_1(k8_weierstr(A,B),C) = np__0
              <=> ! [D] :
                    ( v1_xreal_0(D)
                   => ~ ( ~ r1_xreal_0(D,np__0)
                        & ! [E] :
                            ( m1_subset_1(E,u1_struct_0(A))
                           => ~ ( r2_hidden(E,B)
                                & ~ r1_xreal_0(D,k4_metric_1(A,C,E)) ) ) ) ) ) ) ) ) )).

fof(t10_hausdorf,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_xboole_0(B)
            & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k5_pcomps_1(A)))) )
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ( r2_hidden(C,k6_pre_topc(k5_pcomps_1(A),B))
              <=> k1_funct_1(k8_weierstr(A,B),C) = np__0 ) ) ) ) )).

fof(t11_hausdorf,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_xboole_0(B)
            & v4_pre_topc(B,k5_pcomps_1(A))
            & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k5_pcomps_1(A)))) )
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ( r2_hidden(C,B)
              <=> k1_funct_1(k8_weierstr(A,B),C) = np__0 ) ) ) ) )).

fof(t12_hausdorf,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k3_topmetr))) )
     => ? [B] :
          ( ~ v1_xboole_0(B)
          & m1_subset_1(B,k1_zfmisc_1(k1_numbers))
          & A = B
          & k5_weierstr(A) = k4_pscomp_1(B) ) ) )).

fof(t13_hausdorf,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k3_topmetr))) )
     => ? [B] :
          ( ~ v1_xboole_0(B)
          & m1_subset_1(B,k1_zfmisc_1(k1_numbers))
          & A = B
          & k4_weierstr(A) = k3_pscomp_1(B) ) ) )).

fof(t14_hausdorf,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_xboole_0(B)
            & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k5_pcomps_1(A)))) )
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ! [D] :
                  ( m1_subset_1(D,k1_zfmisc_1(k1_numbers))
                 => ( D = k4_pre_topc(k5_pcomps_1(A),k3_topmetr,k6_weierstr(A,C),B)
                   => v2_seq_4(D) ) ) ) ) ) )).

fof(t15_hausdorf,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_xboole_0(B)
            & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k5_pcomps_1(A)))) )
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ! [D] :
                  ( m1_subset_1(D,u1_struct_0(A))
                 => ( r2_hidden(D,B)
                   => r1_xreal_0(k1_funct_1(k8_weierstr(A,B),C),k4_metric_1(A,C,D)) ) ) ) ) ) )).

fof(t16_hausdorf,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_xboole_0(B)
            & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k5_pcomps_1(A)))) )
         => ! [C] :
              ( v1_xreal_0(C)
             => ! [D] :
                  ( m1_subset_1(D,u1_struct_0(A))
                 => ( ! [E] :
                        ( m1_subset_1(E,u1_struct_0(A))
                       => ( r2_hidden(E,B)
                         => r1_xreal_0(C,k4_metric_1(A,D,E)) ) )
                   => r1_xreal_0(C,k1_funct_1(k8_weierstr(A,B),D)) ) ) ) ) ) )).

fof(t17_hausdorf,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_xboole_0(B)
            & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k5_pcomps_1(A)))) )
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ! [D] :
                  ( m1_subset_1(D,u1_struct_0(A))
                 => r1_xreal_0(k1_funct_1(k8_weierstr(A,B),C),k2_xcmplx_0(k4_metric_1(A,C,D),k1_funct_1(k8_weierstr(A,B),D))) ) ) ) ) )).

fof(t18_hausdorf,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(k5_pcomps_1(A))))
         => ! [C] :
              ( ( ~ v1_xboole_0(C)
                & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
             => ( B = C
               => k3_pre_topc(k5_pcomps_1(A),B) = k5_pcomps_1(k1_topmetr(A,C)) ) ) ) ) )).

fof(t19_hausdorf,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_xboole_0(C)
                & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
             => ! [D] :
                  ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k1_topmetr(A,C))))
                 => ( ( r1_tarski(B,C)
                      & B = D
                      & v6_tbsp_1(D,k1_topmetr(A,C)) )
                   => v6_tbsp_1(B,A) ) ) ) ) ) )).

fof(t20_hausdorf,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] :
              ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k5_pcomps_1(A))))
             => ( ( C = B
                  & v6_compts_1(C,k5_pcomps_1(A)) )
               => v6_tbsp_1(B,A) ) ) ) ) )).

fof(t21_hausdorf,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_xboole_0(B)
            & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k5_pcomps_1(A)))) )
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ? [D] :
                  ( m1_subset_1(D,u1_struct_0(A))
                  & r2_hidden(D,B)
                  & r1_xreal_0(k1_funct_1(k8_weierstr(A,B),C),k4_metric_1(A,D,C)) ) ) ) ) )).

fof(t22_hausdorf,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_xboole_0(B)
            & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k5_pcomps_1(A)))) )
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ! [D] :
                  ( m1_subset_1(D,u1_struct_0(A))
                 => ( ( r2_hidden(D,B)
                      & v6_compts_1(B,k5_pcomps_1(A)) )
                   => r1_xreal_0(k4_metric_1(A,C,D),k1_funct_1(k7_weierstr(A,B),C)) ) ) ) ) ) )).

fof(t23_hausdorf,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_xboole_0(B)
            & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k5_pcomps_1(A)))) )
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ~ ( v6_compts_1(B,k5_pcomps_1(A))
                  & ! [D] :
                      ( m1_subset_1(D,u1_struct_0(A))
                     => ~ ( r2_hidden(D,B)
                          & r1_xreal_0(k4_metric_1(A,D,C),k1_funct_1(k7_weierstr(A,B),C)) ) ) ) ) ) ) )).

fof(t24_hausdorf,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_xboole_0(B)
            & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k5_pcomps_1(A)))) )
         => ! [C] :
              ( ( ~ v1_xboole_0(C)
                & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k5_pcomps_1(A)))) )
             => ! [D] :
                  ( m1_subset_1(D,u1_struct_0(A))
                 => ( ( v6_compts_1(B,k5_pcomps_1(A))
                      & v6_compts_1(C,k5_pcomps_1(A))
                      & r2_hidden(D,C) )
                   => r1_xreal_0(k1_funct_1(k8_weierstr(A,B),D),k12_weierstr(A,B,C)) ) ) ) ) ) )).

fof(t25_hausdorf,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_xboole_0(B)
            & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k5_pcomps_1(A)))) )
         => ! [C] :
              ( ( ~ v1_xboole_0(C)
                & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k5_pcomps_1(A)))) )
             => ! [D] :
                  ( m1_subset_1(D,u1_struct_0(A))
                 => ( ( v6_compts_1(B,k5_pcomps_1(A))
                      & v6_compts_1(C,k5_pcomps_1(A))
                      & r2_hidden(D,C) )
                   => r1_xreal_0(k1_funct_1(k7_weierstr(A,B),D),k12_weierstr(A,B,C)) ) ) ) ) ) )).

fof(t26_hausdorf,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_xboole_0(B)
            & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k5_pcomps_1(A)))) )
         => ! [C] :
              ( ( ~ v1_xboole_0(C)
                & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k5_pcomps_1(A)))) )
             => ! [D] :
                  ( m1_subset_1(D,k1_zfmisc_1(k1_numbers))
                 => ( ( D = k4_pre_topc(k5_pcomps_1(A),k3_topmetr,k7_weierstr(A,B),C)
                      & v6_compts_1(B,k5_pcomps_1(A))
                      & v6_compts_1(C,k5_pcomps_1(A)) )
                   => v1_seq_4(D) ) ) ) ) ) )).

fof(t27_hausdorf,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_xboole_0(B)
            & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k5_pcomps_1(A)))) )
         => ! [C] :
              ( ( ~ v1_xboole_0(C)
                & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k5_pcomps_1(A)))) )
             => ! [D] :
                  ( m1_subset_1(D,k1_zfmisc_1(k1_numbers))
                 => ( ( D = k4_pre_topc(k5_pcomps_1(A),k3_topmetr,k8_weierstr(A,B),C)
                      & v6_compts_1(B,k5_pcomps_1(A))
                      & v6_compts_1(C,k5_pcomps_1(A)) )
                   => v1_seq_4(D) ) ) ) ) ) )).

fof(t28_hausdorf,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_xboole_0(B)
            & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k5_pcomps_1(A)))) )
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ( v6_compts_1(B,k5_pcomps_1(A))
               => r1_xreal_0(k1_funct_1(k8_weierstr(A,B),C),k1_funct_1(k7_weierstr(A,B),C)) ) ) ) ) )).

fof(t29_hausdorf,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_xboole_0(B)
            & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k5_pcomps_1(A)))) )
         => k4_pre_topc(k5_pcomps_1(A),k3_topmetr,k8_weierstr(A,B),B) = k1_hausdorf(np__0) ) ) )).

fof(t30_hausdorf,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_xboole_0(B)
            & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k5_pcomps_1(A)))) )
         => ! [C] :
              ( ( ~ v1_xboole_0(C)
                & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k5_pcomps_1(A)))) )
             => ( ( v6_compts_1(B,k5_pcomps_1(A))
                  & v6_compts_1(C,k5_pcomps_1(A)) )
               => r1_xreal_0(np__0,k10_weierstr(A,B,C)) ) ) ) ) )).

fof(t31_hausdorf,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_xboole_0(B)
            & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k5_pcomps_1(A)))) )
         => k10_weierstr(A,B,B) = np__0 ) ) )).

fof(t32_hausdorf,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_xboole_0(B)
            & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k5_pcomps_1(A)))) )
         => ! [C] :
              ( ( ~ v1_xboole_0(C)
                & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k5_pcomps_1(A)))) )
             => ( ( v6_compts_1(B,k5_pcomps_1(A))
                  & v6_compts_1(C,k5_pcomps_1(A)) )
               => r1_xreal_0(np__0,k11_weierstr(A,B,C)) ) ) ) ) )).

fof(t33_hausdorf,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_xboole_0(B)
            & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k5_pcomps_1(A)))) )
         => ! [C] :
              ( ( ~ v1_xboole_0(C)
                & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k5_pcomps_1(A)))) )
             => ! [D] :
                  ( m1_subset_1(D,u1_struct_0(A))
                 => ( ( v6_compts_1(B,k5_pcomps_1(A))
                      & v6_compts_1(C,k5_pcomps_1(A))
                      & r2_hidden(D,B) )
                   => r1_xreal_0(k1_funct_1(k8_weierstr(A,C),D),k10_weierstr(A,C,B)) ) ) ) ) ) )).

fof(d1_hausdorf,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(k5_pcomps_1(A))))
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k5_pcomps_1(A))))
             => k2_hausdorf(A,B,C) = k4_square_1(k10_weierstr(A,B,C),k10_weierstr(A,C,B)) ) ) ) )).

fof(t34_hausdorf,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_xboole_0(B)
            & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k5_pcomps_1(A)))) )
         => ! [C] :
              ( ( ~ v1_xboole_0(C)
                & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k5_pcomps_1(A)))) )
             => ! [D] :
                  ( m1_subset_1(D,u1_struct_0(A))
                 => ( ( v6_compts_1(B,k5_pcomps_1(A))
                      & v6_compts_1(C,k5_pcomps_1(A))
                      & r2_hidden(D,B) )
                   => r1_xreal_0(k1_funct_1(k8_weierstr(A,C),D),k2_hausdorf(A,B,C)) ) ) ) ) ) )).

fof(t35_hausdorf,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_xboole_0(B)
            & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k5_pcomps_1(A)))) )
         => ! [C] :
              ( ( ~ v1_xboole_0(C)
                & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k5_pcomps_1(A)))) )
             => ! [D] :
                  ( ( ~ v1_xboole_0(D)
                    & m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k5_pcomps_1(A)))) )
                 => ( ( v6_compts_1(B,k5_pcomps_1(A))
                      & v6_compts_1(C,k5_pcomps_1(A))
                      & v6_compts_1(D,k5_pcomps_1(A)) )
                   => r1_xreal_0(k10_weierstr(A,B,D),k3_real_1(k2_hausdorf(A,B,C),k2_hausdorf(A,C,D))) ) ) ) ) ) )).

fof(t36_hausdorf,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_xboole_0(B)
            & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k5_pcomps_1(A)))) )
         => ! [C] :
              ( ( ~ v1_xboole_0(C)
                & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k5_pcomps_1(A)))) )
             => ! [D] :
                  ( ( ~ v1_xboole_0(D)
                    & m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k5_pcomps_1(A)))) )
                 => ( ( v6_compts_1(B,k5_pcomps_1(A))
                      & v6_compts_1(C,k5_pcomps_1(A))
                      & v6_compts_1(D,k5_pcomps_1(A)) )
                   => r1_xreal_0(k10_weierstr(A,D,B),k3_real_1(k2_hausdorf(A,B,C),k2_hausdorf(A,C,D))) ) ) ) ) ) )).

fof(t37_hausdorf,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_xboole_0(B)
            & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k5_pcomps_1(A)))) )
         => ! [C] :
              ( ( ~ v1_xboole_0(C)
                & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k5_pcomps_1(A)))) )
             => ( ( v6_compts_1(B,k5_pcomps_1(A))
                  & v6_compts_1(C,k5_pcomps_1(A)) )
               => r1_xreal_0(np__0,k2_hausdorf(A,B,C)) ) ) ) ) )).

fof(t38_hausdorf,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_xboole_0(B)
            & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k5_pcomps_1(A)))) )
         => k2_hausdorf(A,B,B) = np__0 ) ) )).

fof(t39_hausdorf,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_xboole_0(B)
            & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k5_pcomps_1(A)))) )
         => ! [C] :
              ( ( ~ v1_xboole_0(C)
                & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k5_pcomps_1(A)))) )
             => ( ( v6_compts_1(B,k5_pcomps_1(A))
                  & v6_compts_1(C,k5_pcomps_1(A))
                  & k2_hausdorf(A,B,C) = np__0 )
               => B = C ) ) ) ) )).

fof(t40_hausdorf,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_xboole_0(B)
            & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k5_pcomps_1(A)))) )
         => ! [C] :
              ( ( ~ v1_xboole_0(C)
                & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k5_pcomps_1(A)))) )
             => ! [D] :
                  ( ( ~ v1_xboole_0(D)
                    & m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k5_pcomps_1(A)))) )
                 => ( ( v6_compts_1(B,k5_pcomps_1(A))
                      & v6_compts_1(C,k5_pcomps_1(A))
                      & v6_compts_1(D,k5_pcomps_1(A)) )
                   => r1_xreal_0(k2_hausdorf(A,B,D),k3_real_1(k2_hausdorf(A,B,C),k2_hausdorf(A,C,D))) ) ) ) ) ) )).

fof(d2_hausdorf,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
             => ! [D] :
                  ( m1_subset_1(D,k1_numbers)
                 => ( D = k3_hausdorf(A,B,C)
                  <=> ? [E] :
                        ( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(k5_pcomps_1(k14_euclid(A)))))
                        & ? [F] :
                            ( m1_subset_1(F,k1_zfmisc_1(u1_struct_0(k5_pcomps_1(k14_euclid(A)))))
                            & B = E
                            & C = F
                            & D = k10_weierstr(k14_euclid(A),E,F) ) ) ) ) ) ) ) )).

fof(d3_hausdorf,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
             => ! [D] :
                  ( m1_subset_1(D,k1_numbers)
                 => ( D = k4_hausdorf(A,B,C)
                  <=> ? [E] :
                        ( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(k5_pcomps_1(k14_euclid(A)))))
                        & ? [F] :
                            ( m1_subset_1(F,k1_zfmisc_1(u1_struct_0(k5_pcomps_1(k14_euclid(A)))))
                            & B = E
                            & C = F
                            & D = k2_hausdorf(k14_euclid(A),E,F) ) ) ) ) ) ) ) )).

fof(t41_hausdorf,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(A)))) )
         => ! [C] :
              ( ( ~ v1_xboole_0(C)
                & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(A)))) )
             => ( ( v6_compts_1(B,k15_euclid(A))
                  & v6_compts_1(C,k15_euclid(A)) )
               => r1_xreal_0(np__0,k4_hausdorf(A,B,C)) ) ) ) ) )).

fof(t42_hausdorf,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(A)))) )
         => k4_hausdorf(A,B,B) = np__0 ) ) )).

fof(t43_hausdorf,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(A)))) )
         => ! [C] :
              ( ( ~ v1_xboole_0(C)
                & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(A)))) )
             => ( ( v6_compts_1(B,k15_euclid(A))
                  & v6_compts_1(C,k15_euclid(A))
                  & k4_hausdorf(A,B,C) = np__0 )
               => B = C ) ) ) ) )).

fof(t44_hausdorf,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(A)))) )
         => ! [C] :
              ( ( ~ v1_xboole_0(C)
                & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(A)))) )
             => ! [D] :
                  ( ( ~ v1_xboole_0(D)
                    & m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k15_euclid(A)))) )
                 => ( ( v6_compts_1(B,k15_euclid(A))
                      & v6_compts_1(C,k15_euclid(A))
                      & v6_compts_1(D,k15_euclid(A)) )
                   => r1_xreal_0(k4_hausdorf(A,B,D),k3_real_1(k4_hausdorf(A,B,C),k4_hausdorf(A,C,D))) ) ) ) ) ) )).

fof(dt_k1_hausdorf,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => m1_subset_1(k1_hausdorf(A),k1_zfmisc_1(k1_numbers)) ) )).

fof(redefinition_k1_hausdorf,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => k1_hausdorf(A) = k1_tarski(A) ) )).

fof(dt_k2_hausdorf,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)
        & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k5_pcomps_1(A))))
        & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k5_pcomps_1(A)))) )
     => m1_subset_1(k2_hausdorf(A,B,C),k1_numbers) ) )).

fof(commutativity_k2_hausdorf,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)
        & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k5_pcomps_1(A))))
        & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k5_pcomps_1(A)))) )
     => k2_hausdorf(A,B,C) = k2_hausdorf(A,C,B) ) )).

fof(dt_k3_hausdorf,axiom,(
    ! [A,B,C] :
      ( ( m1_subset_1(A,k5_numbers)
        & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
        & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(A)))) )
     => m1_subset_1(k3_hausdorf(A,B,C),k1_numbers) ) )).

fof(dt_k4_hausdorf,axiom,(
    ! [A,B,C] :
      ( ( m1_subset_1(A,k5_numbers)
        & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
        & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(A)))) )
     => m1_subset_1(k4_hausdorf(A,B,C),k1_numbers) ) )).

fof(commutativity_k4_hausdorf,axiom,(
    ! [A,B,C] :
      ( ( m1_subset_1(A,k5_numbers)
        & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
        & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(A)))) )
     => k4_hausdorf(A,B,C) = k4_hausdorf(A,C,B) ) )).
%------------------------------------------------------------------------------