SET007 Axioms: SET007+681.ax


%------------------------------------------------------------------------------
% File     : SET007+681 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Lower Tolerance. Preliminaries to Wroclaw Taxonomy
% 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    : taxonom1 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   61 (   0 unit)
%            Number of atoms       :  440 (  37 equality)
%            Maximal formula depth :   19 (   9 average)
%            Number of connectives :  443 (  64 ~  ;   1  |; 193  &)
%                                         (  12 <=>; 173 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   42 (   0 propositional; 1-4 arity)
%            Number of functors    :   32 (   4 constant; 0-5 arity)
%            Number of variables   :  162 (   0 singleton; 155 !;   7 ?)
%            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(rc1_taxonom1,axiom,(
    ? [A] :
      ( v1_xreal_0(A)
      & ~ v3_xreal_0(A)
      & v1_xcmplx_0(A) ) )).

fof(fc1_taxonom1,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(A)
        & v6_metric_1(A)
        & v8_metric_1(A)
        & v9_metric_1(A)
        & l1_metric_1(A)
        & ~ v1_xboole_0(B)
        & v6_tbsp_1(B,A)
        & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
     => ( v1_xreal_0(k2_tbsp_1(A,B))
        & ~ v3_xreal_0(k2_tbsp_1(A,B))
        & v1_xcmplx_0(k2_tbsp_1(A,B)) ) ) )).

fof(t1_taxonom1,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_finseq_1(A) )
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ( r2_hidden(k1_nat_1(B,np__1),k4_finseq_1(A))
           => ( r2_hidden(B,k4_finseq_1(A))
              | B = np__0 ) ) ) ) )).

fof(t2_taxonom1,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_finseq_1(A) )
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => ( ( r2_hidden(B,k4_finseq_1(A))
                  & r2_hidden(C,k4_finseq_1(A))
                  & ! [D] :
                      ( m2_subset_1(D,k1_numbers,k5_numbers)
                     => ( ( r2_hidden(D,k4_finseq_1(A))
                          & r2_hidden(k1_nat_1(D,np__1),k4_finseq_1(A)) )
                       => k1_funct_1(A,D) = k1_funct_1(A,k1_nat_1(D,np__1)) ) ) )
               => k1_funct_1(A,B) = k1_funct_1(A,C) ) ) ) ) )).

fof(t3_taxonom1,axiom,(
    ! [A,B] :
      ( m2_relset_1(B,A,A)
     => ( r1_relat_2(B,A)
       => k4_relset_1(A,A,B) = A ) ) )).

fof(t4_taxonom1,axiom,(
    ! [A,B] :
      ( m2_relset_1(B,A,A)
     => ( r1_relat_2(B,A)
       => k5_relset_1(A,A,B) = A ) ) )).

fof(t5_taxonom1,axiom,(
    ! [A,B] :
      ( m2_relset_1(B,A,A)
     => ( r1_relat_2(B,A)
       => r1_relat_2(k17_finseq_1(B),A) ) ) )).

fof(t6_taxonom1,axiom,(
    ! [A,B,C,D] :
      ( m2_relset_1(D,A,A)
     => ( ( r1_relat_2(D,A)
          & r1_rewrite1(D,B,C)
          & r2_hidden(B,A) )
       => r2_hidden(k4_tarski(B,C),k17_finseq_1(D)) ) ) )).

fof(t7_taxonom1,axiom,(
    ! [A,B] :
      ( m2_relset_1(B,A,A)
     => ( ( r1_relat_2(B,A)
          & r3_relat_2(B,A) )
       => r3_relat_2(k17_finseq_1(B),A) ) ) )).

fof(t8_taxonom1,axiom,(
    ! [A,B] :
      ( m2_relset_1(B,A,A)
     => ( r1_relat_2(B,A)
       => r8_relat_2(k17_finseq_1(B),A) ) ) )).

fof(t9_taxonom1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m2_relset_1(B,A,A)
         => ( ( r1_relat_2(B,A)
              & r3_relat_2(B,A) )
           => ( v3_relat_2(k17_lang1(A,B))
              & v8_relat_2(k17_lang1(A,B))
              & v1_partfun1(k17_lang1(A,B),A,A)
              & m2_relset_1(k17_lang1(A,B),A,A) ) ) ) ) )).

fof(t10_taxonom1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m2_relset_1(B,A,A)
         => ! [C] :
              ( m2_relset_1(C,A,A)
             => ( r1_tarski(B,C)
               => r1_tarski(k17_lang1(A,B),k17_lang1(A,C)) ) ) ) ) )).

fof(t11_taxonom1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => r1_setfam_1(k3_pua2mss1(A),k1_tarski(A)) ) )).

fof(d1_taxonom1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(k1_partit1(A)))
         => ( m1_taxonom1(B,A)
          <=> ! [C] :
                ( m1_eqrel_1(C,A)
               => ! [D] :
                    ( m1_eqrel_1(D,A)
                   => ~ ( r2_hidden(C,B)
                        & r2_hidden(D,B)
                        & ~ r1_setfam_1(C,D)
                        & ~ r1_setfam_1(D,C) ) ) ) ) ) ) )).

fof(t12_taxonom1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => m1_taxonom1(k1_tarski(k1_tarski(A)),A) ) )).

fof(t13_taxonom1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => m1_taxonom1(k1_tarski(k3_pua2mss1(A)),A) ) )).

fof(t14_taxonom1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(k1_partit1(A)))
         => ( B = k2_tarski(k1_tarski(A),k3_pua2mss1(A))
           => m1_taxonom1(B,A) ) ) ) )).

fof(d2_taxonom1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(k1_partit1(A)))
         => ( m2_taxonom1(B,A)
          <=> ( m1_taxonom1(B,A)
              & r2_hidden(k1_tarski(A),B)
              & r2_hidden(k3_pua2mss1(A),B) ) ) ) ) )).

fof(t15_taxonom1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(k1_partit1(A)))
         => ( B = k2_tarski(k1_tarski(A),k3_pua2mss1(A))
           => m2_taxonom1(B,A) ) ) ) )).

fof(d3_taxonom1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ( v1_funct_1(B)
            & m2_relset_1(B,k2_zfmisc_1(A,A),k1_numbers) )
         => ! [C] :
              ( v1_xreal_0(C)
             => ! [D] :
                  ( m2_relset_1(D,A,A)
                 => ( D = k1_taxonom1(A,B,C)
                  <=> ! [E] :
                        ( m1_subset_1(E,A)
                       => ! [F] :
                            ( m1_subset_1(F,A)
                           => ( r2_hidden(k4_tarski(E,F),D)
                            <=> r1_xreal_0(k1_metric_1(A,A,B,E,F),C) ) ) ) ) ) ) ) ) )).

fof(t16_taxonom1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ( v1_funct_1(B)
            & m2_relset_1(B,k2_zfmisc_1(A,A),k1_numbers) )
         => ! [C] :
              ( v1_xreal_0(C)
             => ( ( v2_metric_1(B,A)
                  & r1_xreal_0(np__0,C) )
               => r1_relat_2(k1_taxonom1(A,B,C),A) ) ) ) ) )).

fof(t17_taxonom1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ( v1_funct_1(B)
            & m2_relset_1(B,k2_zfmisc_1(A,A),k1_numbers) )
         => ! [C] :
              ( v1_xreal_0(C)
             => ( v4_metric_1(B,A)
               => r3_relat_2(k1_taxonom1(A,B,C),A) ) ) ) ) )).

fof(t18_taxonom1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ( v1_funct_1(B)
            & m2_relset_1(B,k2_zfmisc_1(A,A),k1_numbers) )
         => ! [C] :
              ( v1_xreal_0(C)
             => ( ( r1_xreal_0(np__0,C)
                  & v2_metric_1(B,A)
                  & v4_metric_1(B,A) )
               => ( v1_relat_2(k1_taxonom1(A,B,C))
                  & v3_relat_2(k1_taxonom1(A,B,C))
                  & v1_partfun1(k1_taxonom1(A,B,C),A,A)
                  & m2_relset_1(k1_taxonom1(A,B,C),A,A) ) ) ) ) ) )).

fof(t19_taxonom1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ( v1_funct_1(B)
            & m2_relset_1(B,k2_zfmisc_1(A,A),k1_numbers) )
         => ! [C] :
              ( v1_xreal_0(C)
             => ! [D] :
                  ( v1_xreal_0(D)
                 => ( r1_xreal_0(C,D)
                   => r1_tarski(k1_taxonom1(A,B,C),k1_taxonom1(A,B,D)) ) ) ) ) ) )).

fof(d4_taxonom1,axiom,(
    ! [A,B] :
      ( ( v1_funct_1(B)
        & m2_relset_1(B,k2_zfmisc_1(A,A),k1_numbers) )
     => ( v1_taxonom1(B,A)
      <=> ! [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(t20_taxonom1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ( v1_funct_1(B)
            & m2_relset_1(B,k2_zfmisc_1(A,A),k1_numbers) )
         => ! [C,D] :
              ( ( v1_taxonom1(B,A)
                & v2_metric_1(B,A)
                & v3_metric_1(B,A)
                & r2_hidden(k4_tarski(C,D),k1_taxonom1(A,B,np__0)) )
             => C = D ) ) ) )).

fof(t21_taxonom1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ( v1_funct_1(B)
            & m2_relset_1(B,k2_zfmisc_1(A,A),k1_numbers) )
         => ! [C] :
              ( m1_subset_1(C,A)
             => ( ( v2_metric_1(B,A)
                  & v3_metric_1(B,A) )
               => r2_hidden(k4_tarski(C,C),k1_taxonom1(A,B,np__0)) ) ) ) ) )).

fof(t22_taxonom1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ( v1_funct_1(B)
            & m2_relset_1(B,k2_zfmisc_1(A,A),k1_numbers) )
         => ! [C] :
              ( v1_xreal_0(C)
             => ( ( r1_relat_2(k1_taxonom1(A,B,C),A)
                  & v4_metric_1(B,A) )
               => ( v3_relat_2(k17_lang1(A,k1_taxonom1(A,B,C)))
                  & v8_relat_2(k17_lang1(A,k1_taxonom1(A,B,C)))
                  & v1_partfun1(k17_lang1(A,k1_taxonom1(A,B,C)),A,A)
                  & m2_relset_1(k17_lang1(A,k1_taxonom1(A,B,C)),A,A) ) ) ) ) ) )).

fof(t23_taxonom1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ( v1_funct_1(B)
            & m2_relset_1(B,k2_zfmisc_1(A,A),k1_numbers) )
         => ( ( v1_taxonom1(B,A)
              & v2_metric_1(B,A)
              & v3_metric_1(B,A) )
           => k17_lang1(A,k1_taxonom1(A,B,np__0)) = k1_taxonom1(A,B,np__0) ) ) ) )).

fof(t24_taxonom1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ( v1_funct_1(B)
            & m2_relset_1(B,k2_zfmisc_1(A,A),k1_numbers) )
         => ! [C] :
              ( ( v3_relat_2(C)
                & v8_relat_2(C)
                & v1_partfun1(C,A,A)
                & m2_relset_1(C,A,A) )
             => ( ( C = k17_lang1(A,k1_taxonom1(A,B,np__0))
                  & v1_taxonom1(B,A)
                  & v2_metric_1(B,A)
                  & v3_metric_1(B,A) )
               => C = k6_relat_1(A) ) ) ) ) )).

fof(t25_taxonom1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ( v1_funct_1(B)
            & m2_relset_1(B,k2_zfmisc_1(A,A),k1_numbers) )
         => ! [C] :
              ( ( v3_relat_2(C)
                & v8_relat_2(C)
                & v1_partfun1(C,A,A)
                & m2_relset_1(C,A,A) )
             => ( ( C = k17_lang1(A,k1_taxonom1(A,B,np__0))
                  & v1_taxonom1(B,A)
                  & v2_metric_1(B,A)
                  & v3_metric_1(B,A) )
               => k8_eqrel_1(A,C) = k3_pua2mss1(A) ) ) ) ) )).

fof(t26_taxonom1,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_finset_1(A)
        & m1_subset_1(A,k1_zfmisc_1(k1_numbers)) )
     => ! [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_xboole_0(C)
                & v1_finset_1(C)
                & m1_subset_1(C,k1_zfmisc_1(k1_numbers)) )
             => ! [D] :
                  ( v1_xreal_0(D)
                 => ( ( C = k5_relset_1(k2_zfmisc_1(A,A),k1_numbers,B)
                      & r1_xreal_0(k1_pre_circ(C),D) )
                   => ! [E] :
                        ( m2_subset_1(E,k1_numbers,A)
                       => ! [F] :
                            ( m2_subset_1(F,k1_numbers,A)
                           => r1_xreal_0(k1_metric_1(A,A,B,E,F),D) ) ) ) ) ) ) ) )).

fof(t27_taxonom1,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_finset_1(A)
        & m1_subset_1(A,k1_zfmisc_1(k1_numbers)) )
     => ! [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_xboole_0(C)
                & v1_finset_1(C)
                & m1_subset_1(C,k1_zfmisc_1(k1_numbers)) )
             => ! [D] :
                  ( v1_xreal_0(D)
                 => ( ( C = k5_relset_1(k2_zfmisc_1(A,A),k1_numbers,B)
                      & r1_xreal_0(k1_pre_circ(C),D) )
                   => ! [E] :
                        ( ( v3_relat_2(E)
                          & v8_relat_2(E)
                          & v1_partfun1(E,A,A)
                          & m2_relset_1(E,A,A) )
                       => ( E = k17_lang1(A,k1_taxonom1(A,B,D))
                         => k8_eqrel_1(A,E) = k1_tarski(A) ) ) ) ) ) ) ) )).

fof(t28_taxonom1,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_finset_1(A)
        & m1_subset_1(A,k1_zfmisc_1(k1_numbers)) )
     => ! [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_xboole_0(C)
                & v1_finset_1(C)
                & m1_subset_1(C,k1_zfmisc_1(k1_numbers)) )
             => ! [D] :
                  ( v1_xreal_0(D)
                 => ( ( C = k5_relset_1(k2_zfmisc_1(A,A),k1_numbers,B)
                      & r1_xreal_0(k1_pre_circ(C),D) )
                   => k17_lang1(A,k1_taxonom1(A,B,D)) = k1_taxonom1(A,B,D) ) ) ) ) ) )).

fof(d5_taxonom1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ( v1_funct_1(B)
            & m2_relset_1(B,k2_zfmisc_1(A,A),k1_numbers) )
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(k1_partit1(A)))
             => ( C = k2_taxonom1(A,B)
              <=> ! [D] :
                    ( r2_hidden(D,C)
                  <=> ? [E] :
                        ( v1_xreal_0(E)
                        & ~ v3_xreal_0(E)
                        & ? [F] :
                            ( v3_relat_2(F)
                            & v8_relat_2(F)
                            & v1_partfun1(F,A,A)
                            & m2_relset_1(F,A,A)
                            & F = k17_lang1(A,k1_taxonom1(A,B,E))
                            & k8_eqrel_1(A,F) = D ) ) ) ) ) ) ) )).

fof(t29_taxonom1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ( v1_funct_1(B)
            & m2_relset_1(B,k2_zfmisc_1(A,A),k1_numbers) )
         => ! [C] :
              ( ( v1_xreal_0(C)
                & ~ v3_xreal_0(C) )
             => ( ( r1_relat_2(k1_taxonom1(A,B,C),A)
                  & v4_metric_1(B,A) )
               => ~ v1_xboole_0(k2_taxonom1(A,B)) ) ) ) ) )).

fof(t30_taxonom1,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_finset_1(A)
        & m1_subset_1(A,k1_zfmisc_1(k1_numbers)) )
     => ! [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) )
         => ( ( v4_metric_1(B,A)
              & v1_taxonom1(B,A) )
           => r2_hidden(k1_tarski(A),k2_taxonom1(A,B)) ) ) ) )).

fof(t31_taxonom1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ( v1_funct_1(B)
            & m2_relset_1(B,k2_zfmisc_1(A,A),k1_numbers) )
         => m1_taxonom1(k2_taxonom1(A,B),A) ) ) )).

fof(t32_taxonom1,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_finset_1(A)
        & m1_subset_1(A,k1_zfmisc_1(k1_numbers)) )
     => ! [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) )
         => ( ( r2_hidden(k3_pua2mss1(A),k2_taxonom1(A,B))
              & v4_metric_1(B,A)
              & v1_taxonom1(B,A) )
           => m2_taxonom1(k2_taxonom1(A,B),A) ) ) ) )).

fof(d6_taxonom1,axiom,(
    ! [A] :
      ( l1_metric_1(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ! [D] :
                  ( m1_subset_1(D,u1_struct_0(A))
                 => ( r1_taxonom1(A,B,C,D)
                  <=> r1_xreal_0(k2_metric_1(A,C,D),B) ) ) ) ) ) )).

fof(d7_taxonom1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_metric_1(A) )
     => ! [B] :
          ( v1_xreal_0(B)
         => ! [C] :
              ( m2_relset_1(C,u1_struct_0(A),u1_struct_0(A))
             => ( C = k3_taxonom1(A,B)
              <=> ! [D] :
                    ( m1_subset_1(D,u1_struct_0(A))
                   => ! [E] :
                        ( m1_subset_1(E,u1_struct_0(A))
                       => ( r2_hidden(k4_tarski(D,E),C)
                        <=> r1_taxonom1(A,B,D,E) ) ) ) ) ) ) ) )).

fof(t33_taxonom1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_metric_1(A) )
     => ! [B] :
          ( v1_xreal_0(B)
         => k3_taxonom1(A,B) = k1_taxonom1(u1_struct_0(A),u1_metric_1(A),B) ) ) )).

fof(t34_taxonom1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v6_metric_1(A)
        & v8_metric_1(A)
        & l1_metric_1(A) )
     => ! [B] :
          ( v1_xreal_0(B)
         => ! [C] :
              ( m2_relset_1(C,u1_struct_0(A),u1_struct_0(A))
             => ( ( C = k3_taxonom1(A,B)
                  & r1_xreal_0(np__0,B) )
               => ( v1_relat_2(C)
                  & v3_relat_2(C)
                  & v1_partfun1(C,u1_struct_0(A),u1_struct_0(A))
                  & m2_relset_1(C,u1_struct_0(A),u1_struct_0(A)) ) ) ) ) ) )).

fof(d8_taxonom1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v6_metric_1(A)
        & v8_metric_1(A)
        & l1_metric_1(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(k1_partit1(u1_struct_0(A))))
         => ( B = k4_taxonom1(A)
          <=> ! [C] :
                ( r2_hidden(C,B)
              <=> ? [D] :
                    ( v1_xreal_0(D)
                    & ~ v3_xreal_0(D)
                    & ? [E] :
                        ( v3_relat_2(E)
                        & v8_relat_2(E)
                        & v1_partfun1(E,u1_struct_0(A),u1_struct_0(A))
                        & m2_relset_1(E,u1_struct_0(A),u1_struct_0(A))
                        & E = k17_lang1(u1_struct_0(A),k3_taxonom1(A,D))
                        & k8_eqrel_1(u1_struct_0(A),E) = C ) ) ) ) ) ) )).

fof(t35_taxonom1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v6_metric_1(A)
        & v8_metric_1(A)
        & l1_metric_1(A) )
     => k4_taxonom1(A) = k2_taxonom1(u1_struct_0(A),u1_metric_1(A)) ) )).

fof(t36_taxonom1,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] :
          ( ( v3_relat_2(B)
            & v8_relat_2(B)
            & v1_partfun1(B,u1_struct_0(A),u1_struct_0(A))
            & m2_relset_1(B,u1_struct_0(A),u1_struct_0(A)) )
         => ( B = k17_lang1(u1_struct_0(A),k3_taxonom1(A,np__0))
           => k8_eqrel_1(u1_struct_0(A),B) = k3_pua2mss1(u1_struct_0(A)) ) ) ) )).

fof(t37_taxonom1,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & v6_metric_1(B)
            & v8_metric_1(B)
            & v5_tbsp_1(B)
            & l1_metric_1(B) )
         => ( r1_xreal_0(k2_tbsp_1(B,k2_pre_topc(B)),A)
           => k3_taxonom1(B,A) = k1_eqrel_1(u1_struct_0(B)) ) ) ) )).

fof(t38_taxonom1,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & v6_metric_1(B)
            & v8_metric_1(B)
            & v5_tbsp_1(B)
            & l1_metric_1(B) )
         => ( r1_xreal_0(k2_tbsp_1(B,k2_pre_topc(B)),A)
           => k3_taxonom1(B,A) = k17_lang1(u1_struct_0(B),k3_taxonom1(B,A)) ) ) ) )).

fof(t39_taxonom1,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & v6_metric_1(B)
            & v8_metric_1(B)
            & v5_tbsp_1(B)
            & l1_metric_1(B) )
         => ( r1_xreal_0(k2_tbsp_1(B,k2_pre_topc(B)),A)
           => k17_lang1(u1_struct_0(B),k3_taxonom1(B,A)) = k1_eqrel_1(u1_struct_0(B)) ) ) ) )).

fof(t40_taxonom1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v6_metric_1(A)
        & v8_metric_1(A)
        & v5_tbsp_1(A)
        & l1_metric_1(A) )
     => ! [B] :
          ( ( v3_relat_2(B)
            & v8_relat_2(B)
            & v1_partfun1(B,u1_struct_0(A),u1_struct_0(A))
            & m2_relset_1(B,u1_struct_0(A),u1_struct_0(A)) )
         => ! [C] :
              ( ( v1_xreal_0(C)
                & ~ v3_xreal_0(C) )
             => ( ( r1_xreal_0(k2_tbsp_1(A,k2_pre_topc(A)),C)
                  & B = k17_lang1(u1_struct_0(A),k3_taxonom1(A,C)) )
               => k8_eqrel_1(u1_struct_0(A),B) = k1_tarski(u1_struct_0(A)) ) ) ) ) )).

fof(t41_taxonom1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v6_metric_1(A)
        & v7_metric_1(A)
        & v8_metric_1(A)
        & v9_metric_1(A)
        & v5_tbsp_1(A)
        & l1_metric_1(A) )
     => r2_hidden(k1_tarski(u1_struct_0(A)),k4_taxonom1(A)) ) )).

fof(t42_taxonom1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v6_metric_1(A)
        & v8_metric_1(A)
        & l1_metric_1(A) )
     => m1_taxonom1(k4_taxonom1(A),u1_struct_0(A)) ) )).

fof(t43_taxonom1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v6_metric_1(A)
        & v7_metric_1(A)
        & v8_metric_1(A)
        & v9_metric_1(A)
        & v5_tbsp_1(A)
        & l1_metric_1(A) )
     => m2_taxonom1(k4_taxonom1(A),u1_struct_0(A)) ) )).

fof(dt_m1_taxonom1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_taxonom1(B,A)
         => m1_subset_1(B,k1_zfmisc_1(k1_partit1(A))) ) ) )).

fof(existence_m1_taxonom1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ? [B] : m1_taxonom1(B,A) ) )).

fof(dt_m2_taxonom1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m2_taxonom1(B,A)
         => m1_subset_1(B,k1_zfmisc_1(k1_partit1(A))) ) ) )).

fof(existence_m2_taxonom1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ? [B] : m2_taxonom1(B,A) ) )).

fof(dt_k1_taxonom1,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(A)
        & v1_funct_1(B)
        & m1_relset_1(B,k2_zfmisc_1(A,A),k1_numbers)
        & v1_xreal_0(C) )
     => m2_relset_1(k1_taxonom1(A,B,C),A,A) ) )).

fof(dt_k2_taxonom1,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(A)
        & v1_funct_1(B)
        & m1_relset_1(B,k2_zfmisc_1(A,A),k1_numbers) )
     => m1_subset_1(k2_taxonom1(A,B),k1_zfmisc_1(k1_partit1(A))) ) )).

fof(dt_k3_taxonom1,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(A)
        & l1_metric_1(A)
        & v1_xreal_0(B) )
     => m2_relset_1(k3_taxonom1(A,B),u1_struct_0(A),u1_struct_0(A)) ) )).

fof(dt_k4_taxonom1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v6_metric_1(A)
        & v8_metric_1(A)
        & l1_metric_1(A) )
     => m1_subset_1(k4_taxonom1(A),k1_zfmisc_1(k1_partit1(u1_struct_0(A)))) ) )).
%------------------------------------------------------------------------------