SET007 Axioms: SET007+845.ax


%------------------------------------------------------------------------------
% File     : SET007+845 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : The Nagata-Smirnov Theorem. Part I
% 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    : nagata_1 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   54 (   0 unit)
%            Number of atoms       :  558 (  23 equality)
%            Maximal formula depth :   20 (  10 average)
%            Number of connectives :  571 (  67 ~  ;   6  |; 299  &)
%                                         (  12 <=>; 187 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   57 (   0 propositional; 1-4 arity)
%            Number of functors    :   43 (   4 constant; 0-5 arity)
%            Number of variables   :  180 (   0 singleton; 170 !;  10 ?)
%            Maximal term depth    :    6 (   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(rc1_nagata_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_nagata_1(B,A) ) ) )).

fof(rc2_nagata_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_xboole_0(B)
          & v1_finset_1(B)
          & v1_membered(B)
          & v2_membered(B)
          & v3_membered(B)
          & v4_membered(B)
          & v5_membered(B)
          & v1_card_4(B)
          & v1_nagata_1(B,A) ) ) )).

fof(fc1_nagata_1,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A)
        & v1_nagata_1(B,A)
        & m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A)))) )
     => v1_nagata_1(k3_pcomps_1(A,B),A) ) )).

fof(rc3_nagata_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ? [B] :
          ( m1_relset_1(B,k5_numbers,k1_pcomps_1(k1_pcomps_1(u1_struct_0(A))))
          & v1_relat_1(B)
          & v1_funct_1(B)
          & v1_funct_2(B,k5_numbers,k1_pcomps_1(k1_pcomps_1(u1_struct_0(A))))
          & v2_nagata_1(B,A) ) ) )).

fof(cc1_nagata_1,axiom,(
    ! [A] :
      ( ~ v1_card_4(A)
     => ~ v1_xboole_0(A) ) )).

fof(rc4_nagata_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ? [B] :
          ( m1_relset_1(B,k5_numbers,k1_pcomps_1(k1_pcomps_1(u1_struct_0(A))))
          & v1_relat_1(B)
          & v1_funct_1(B)
          & v1_funct_2(B,k5_numbers,k1_pcomps_1(k1_pcomps_1(u1_struct_0(A))))
          & v3_nagata_1(B,A) ) ) )).

fof(d1_nagata_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))))
         => ( v1_nagata_1(B,A)
          <=> ! [C] :
                ( m1_subset_1(C,u1_struct_0(A))
               => ? [D] :
                    ( v3_pre_topc(D,A)
                    & m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
                    & r2_hidden(C,D)
                    & ! [E] :
                        ( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(A)))
                       => ! [F] :
                            ( m1_subset_1(F,k1_zfmisc_1(u1_struct_0(A)))
                           => ( ( r2_hidden(E,B)
                                & r2_hidden(F,B) )
                             => ( r1_xboole_0(D,E)
                                | r1_xboole_0(D,F)
                                | E = F ) ) ) ) ) ) ) ) ) )).

fof(t1_nagata_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)))
                & B = k6_domain_1(k1_zfmisc_1(u1_struct_0(A)),C) )
           => v1_nagata_1(B,A) ) ) ) )).

fof(t2_nagata_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_nagata_1(C,A) )
               => v1_nagata_1(B,A) ) ) ) ) )).

fof(t3_nagata_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))))
             => ( v1_nagata_1(B,A)
               => v1_nagata_1(k5_subset_1(k1_zfmisc_1(u1_struct_0(A)),B,C),A) ) ) ) ) )).

fof(t4_nagata_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))))
             => ( v1_nagata_1(B,A)
               => v1_nagata_1(k6_subset_1(k1_zfmisc_1(u1_struct_0(A)),B,C),A) ) ) ) ) )).

fof(t5_nagata_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))))
             => ! [D] :
                  ( m1_subset_1(D,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
                 => ( ( v1_nagata_1(B,A)
                      & v1_nagata_1(C,A)
                      & k3_setfam_1(B,C) = D )
                   => v1_nagata_1(D,A) ) ) ) ) ) )).

fof(t6_nagata_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)))
             => ! [D] :
                  ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
                 => ~ ( v1_nagata_1(B,A)
                      & r2_hidden(C,B)
                      & r2_hidden(D,B)
                      & C != D
                      & ~ r1_xboole_0(C,D) ) ) ) ) ) )).

fof(t7_nagata_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_nagata_1(B,A)
           => ! [C] :
                ( m1_subset_1(C,u1_struct_0(A))
               => ? [D] :
                    ( v3_pre_topc(D,A)
                    & m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
                    & r2_hidden(C,D)
                    & v1_realset1(k4_xboole_0(k3_setfam_1(k6_domain_1(k1_zfmisc_1(u1_struct_0(A)),D),B),k1_tarski(k1_xboole_0))) ) ) ) ) ) )).

fof(t8_nagata_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_nagata_1(B,A)
          <=> ( ! [C] :
                  ( m1_subset_1(C,u1_struct_0(A))
                 => ? [D] :
                      ( v3_pre_topc(D,A)
                      & m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
                      & r2_hidden(C,D)
                      & v1_realset1(k4_xboole_0(k3_setfam_1(k6_domain_1(k1_zfmisc_1(u1_struct_0(A)),D),B),k1_tarski(k1_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(A)))
                     => ~ ( r2_hidden(C,B)
                          & r2_hidden(D,B)
                          & C != D
                          & ~ r1_xboole_0(C,D) ) ) ) ) ) ) ) )).

fof(t9_nagata_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_nagata_1(B,A)
           => ! [C] :
                ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
               => ! [D] :
                    ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
                   => ( ( r2_hidden(C,B)
                        & r2_hidden(D,B) )
                     => k6_pre_topc(A,k5_subset_1(u1_struct_0(A),C,D)) = k5_subset_1(u1_struct_0(A),k6_pre_topc(A,C),k6_pre_topc(A,D)) ) ) ) ) ) ) )).

fof(t10_nagata_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_nagata_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(t11_nagata_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_nagata_1(B,A)
           => v1_pcomps_1(B,A) ) ) ) )).

fof(d2_nagata_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,k1_pcomps_1(k1_pcomps_1(u1_struct_0(A))))
            & m2_relset_1(B,k5_numbers,k1_pcomps_1(k1_pcomps_1(u1_struct_0(A)))) )
         => ( v2_nagata_1(B,A)
          <=> ! [C] :
                ( m2_subset_1(C,k1_numbers,k5_numbers)
               => v1_nagata_1(k1_nagata_1(A,B,C),A) ) ) ) ) )).

fof(d3_nagata_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,k1_pcomps_1(k1_pcomps_1(u1_struct_0(A))))
            & m2_relset_1(B,k5_numbers,k1_pcomps_1(k1_pcomps_1(u1_struct_0(A)))) )
         => ( v3_nagata_1(B,A)
          <=> ! [C] :
                ( m2_subset_1(C,k1_numbers,k5_numbers)
               => v1_pcomps_1(k1_nagata_1(A,B,C),A) ) ) ) ) )).

fof(d4_nagata_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))))
         => ( v4_nagata_1(B,A)
          <=> ? [C] :
                ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,k1_pcomps_1(k1_pcomps_1(u1_struct_0(A))))
                & v2_nagata_1(C,A)
                & m2_relset_1(C,k5_numbers,k1_pcomps_1(k1_pcomps_1(u1_struct_0(A))))
                & B = k2_nagata_1(A,C) ) ) ) ) )).

fof(t12_nagata_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,k1_pcomps_1(k1_pcomps_1(u1_struct_0(A))))
            & m2_relset_1(B,k5_numbers,k1_pcomps_1(k1_pcomps_1(u1_struct_0(A)))) )
         => ( v2_nagata_1(B,A)
           => v3_nagata_1(B,A) ) ) ) )).

fof(t13_nagata_1,axiom,(
    ! [A] :
      ( ~ v1_card_4(A)
     => ? [B] :
          ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(k2_pcomps_1(k2_zfmisc_1(A,A))))))
          & v1_pcomps_1(B,k2_pcomps_1(k2_zfmisc_1(A,A)))
          & ~ v4_nagata_1(B,k2_pcomps_1(k2_zfmisc_1(A,A))) ) ) )).

fof(d5_nagata_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,k1_pcomps_1(k1_pcomps_1(u1_struct_0(A))))
            & m2_relset_1(B,k5_numbers,k1_pcomps_1(k1_pcomps_1(u1_struct_0(A)))) )
         => ( v5_nagata_1(B,A)
          <=> ( v2_nagata_1(B,A)
              & m1_cantor_1(k2_nagata_1(A,B),A) ) ) ) ) )).

fof(d6_nagata_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,k1_pcomps_1(k1_pcomps_1(u1_struct_0(A))))
            & m2_relset_1(B,k5_numbers,k1_pcomps_1(k1_pcomps_1(u1_struct_0(A)))) )
         => ( v6_nagata_1(B,A)
          <=> ( v3_nagata_1(B,A)
              & m1_cantor_1(k2_nagata_1(A,B),A) ) ) ) ) )).

fof(t14_nagata_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_metric_1(A) )
     => ! [B] :
          ( v1_xreal_0(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) )
           => ! [C] :
                ( m1_subset_1(C,u1_struct_0(A))
               => r2_hidden(k6_subset_1(u1_struct_0(A),k2_pre_topc(A),k10_metric_1(A,C,B)),k4_pcomps_1(A)) ) ) ) ) )).

fof(t15_nagata_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ( v3_pcomps_1(A)
       => ( v4_compts_1(A)
          & v1_urysohn1(A) ) ) ) )).

fof(t16_nagata_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ~ ( v3_pcomps_1(A)
          & ! [B] :
              ( ( v1_funct_1(B)
                & v1_funct_2(B,k5_numbers,k1_pcomps_1(k1_pcomps_1(u1_struct_0(A))))
                & m2_relset_1(B,k5_numbers,k1_pcomps_1(k1_pcomps_1(u1_struct_0(A)))) )
             => ~ v6_nagata_1(B,A) ) ) ) )).

fof(t17_nagata_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,k1_pcomps_1(u1_struct_0(A)))
            & m2_relset_1(B,k5_numbers,k1_pcomps_1(u1_struct_0(A))) )
         => ( ! [C] :
                ( m2_subset_1(C,k1_numbers,k5_numbers)
               => v3_pre_topc(k5_urysohn3(k5_numbers,A,B,C),A) )
           => v3_pre_topc(k2_prob_1(u1_struct_0(A),B),A) ) ) ) )).

fof(t18_nagata_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] :
                ( ( v3_pre_topc(C,A)
                  & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
               => ~ ( v4_pre_topc(B,A)
                    & v3_pre_topc(C,A)
                    & r1_tarski(B,C)
                    & ! [D] :
                        ( ( v1_funct_1(D)
                          & v1_funct_2(D,k5_numbers,k1_pcomps_1(u1_struct_0(A)))
                          & m2_relset_1(D,k5_numbers,k1_pcomps_1(u1_struct_0(A))) )
                       => ~ ( r1_tarski(B,k2_prob_1(u1_struct_0(A),D))
                            & r1_tarski(k2_prob_1(u1_struct_0(A),D),C)
                            & ! [E] :
                                ( m2_subset_1(E,k1_numbers,k5_numbers)
                               => ( r1_tarski(k6_pre_topc(A,k5_urysohn3(k5_numbers,A,D,E)),C)
                                  & v3_pre_topc(k5_urysohn3(k5_numbers,A,D,E),A) ) ) ) ) ) ) )
       => v5_compts_1(A) ) ) )).

fof(t19_nagata_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ( v4_compts_1(A)
       => ! [B] :
            ( ( v1_funct_1(B)
              & v1_funct_2(B,k5_numbers,k1_pcomps_1(k1_pcomps_1(u1_struct_0(A))))
              & m2_relset_1(B,k5_numbers,k1_pcomps_1(k1_pcomps_1(u1_struct_0(A)))) )
           => ( m1_cantor_1(k2_nagata_1(A,B),A)
             => ! [C] :
                  ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
                 => ! [D] :
                      ( m1_subset_1(D,u1_struct_0(A))
                     => ~ ( v3_pre_topc(C,A)
                          & r2_hidden(D,C)
                          & ! [E] :
                              ( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(A)))
                             => ~ ( r2_hidden(D,E)
                                  & r1_tarski(k6_pre_topc(A,E),C)
                                  & r2_hidden(E,k2_nagata_1(A,B)) ) ) ) ) ) ) ) ) ) )).

fof(t20_nagata_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ( ( v4_compts_1(A)
          & v1_urysohn1(A) )
       => ( ! [B] :
              ( ( v1_funct_1(B)
                & v1_funct_2(B,k5_numbers,k1_pcomps_1(k1_pcomps_1(u1_struct_0(A))))
                & m2_relset_1(B,k5_numbers,k1_pcomps_1(k1_pcomps_1(u1_struct_0(A)))) )
             => ~ v6_nagata_1(B,A) )
          | v5_compts_1(A) ) ) ) )).

fof(d7_nagata_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,u1_struct_0(A),k1_numbers)
            & m2_relset_1(B,u1_struct_0(A),k1_numbers) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,u1_struct_0(A),k1_numbers)
                & m2_relset_1(C,u1_struct_0(A),k1_numbers) )
             => ! [D] :
                  ( ( v1_funct_1(D)
                    & v1_funct_2(D,u1_struct_0(A),k1_numbers)
                    & m2_relset_1(D,u1_struct_0(A),k1_numbers) )
                 => ( D = k3_nagata_1(A,B,C)
                  <=> ! [E] :
                        ( m1_subset_1(E,u1_struct_0(A))
                       => k8_funct_2(u1_struct_0(A),k1_numbers,D,E) = k3_real_1(k8_funct_2(u1_struct_0(A),k1_numbers,B,E),k8_funct_2(u1_struct_0(A),k1_numbers,C,E)) ) ) ) ) ) ) )).

fof(t21_nagata_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,u1_struct_0(A),k1_numbers)
            & m2_relset_1(B,u1_struct_0(A),k1_numbers) )
         => ( v9_pscomp_1(B,A)
           => ! [C] :
                ( ( v1_funct_1(C)
                  & v1_funct_2(C,u1_struct_0(k6_borsuk_1(A,A)),k1_numbers)
                  & m2_relset_1(C,u1_struct_0(k6_borsuk_1(A,A)),k1_numbers) )
               => ( ! [D] :
                      ( m1_subset_1(D,u1_struct_0(A))
                     => ! [E] :
                          ( m1_subset_1(E,u1_struct_0(A))
                         => k8_funct_2(u1_struct_0(k6_borsuk_1(A,A)),k1_numbers,C,k8_borsuk_1(A,A,D,E)) = k18_complex1(k5_real_1(k8_funct_2(u1_struct_0(A),k1_numbers,B,D),k8_funct_2(u1_struct_0(A),k1_numbers,B,E))) ) )
                 => v9_pscomp_1(C,k6_borsuk_1(A,A)) ) ) ) ) ) )).

fof(t22_nagata_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,u1_struct_0(A),k1_numbers)
            & m2_relset_1(B,u1_struct_0(A),k1_numbers) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,u1_struct_0(A),k1_numbers)
                & m2_relset_1(C,u1_struct_0(A),k1_numbers) )
             => ( ( v9_pscomp_1(B,A)
                  & v9_pscomp_1(C,A) )
               => v9_pscomp_1(k3_nagata_1(A,B,C),A) ) ) ) ) )).

fof(t23_nagata_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k2_zfmisc_1(k1_fraenkel(u1_struct_0(A),k1_numbers),k1_fraenkel(u1_struct_0(A),k1_numbers)),k1_fraenkel(u1_struct_0(A),k1_numbers))
            & m2_relset_1(B,k2_zfmisc_1(k1_fraenkel(u1_struct_0(A),k1_numbers),k1_fraenkel(u1_struct_0(A),k1_numbers)),k1_fraenkel(u1_struct_0(A),k1_numbers)) )
         => ( ! [C] :
                ( ( v1_funct_1(C)
                  & v1_funct_2(C,u1_struct_0(A),k1_numbers)
                  & m2_relset_1(C,u1_struct_0(A),k1_numbers) )
               => ! [D] :
                    ( ( v1_funct_1(D)
                      & v1_funct_2(D,u1_struct_0(A),k1_numbers)
                      & m2_relset_1(D,u1_struct_0(A),k1_numbers) )
                   => k1_binop_1(B,C,D) = k3_nagata_1(A,C,D) ) )
           => ( v1_setwiseo(B,k1_fraenkel(u1_struct_0(A),k1_numbers))
              & v1_binop_1(B,k1_fraenkel(u1_struct_0(A),k1_numbers))
              & v2_binop_1(B,k1_fraenkel(u1_struct_0(A),k1_numbers)) ) ) ) ) )).

fof(t24_nagata_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k2_zfmisc_1(k1_fraenkel(u1_struct_0(A),k1_numbers),k1_fraenkel(u1_struct_0(A),k1_numbers)),k1_fraenkel(u1_struct_0(A),k1_numbers))
            & m2_relset_1(B,k2_zfmisc_1(k1_fraenkel(u1_struct_0(A),k1_numbers),k1_fraenkel(u1_struct_0(A),k1_numbers)),k1_fraenkel(u1_struct_0(A),k1_numbers)) )
         => ( ! [C] :
                ( ( v1_funct_1(C)
                  & v1_funct_2(C,u1_struct_0(A),k1_numbers)
                  & m2_relset_1(C,u1_struct_0(A),k1_numbers) )
               => ! [D] :
                    ( ( v1_funct_1(D)
                      & v1_funct_2(D,u1_struct_0(A),k1_numbers)
                      & m2_relset_1(D,u1_struct_0(A),k1_numbers) )
                   => k1_binop_1(B,C,D) = k3_nagata_1(A,C,D) ) )
           => ! [C] :
                ( m2_fraenkel(C,u1_struct_0(A),k1_numbers,k1_fraenkel(u1_struct_0(A),k1_numbers))
               => ( r3_binop_1(k1_fraenkel(u1_struct_0(A),k1_numbers),C,B)
                 => v9_pscomp_1(C,A) ) ) ) ) ) )).

fof(d8_nagata_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & v2_pre_topc(B)
            & l1_pre_topc(B) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,u1_struct_0(A),k1_pcomps_1(u1_struct_0(A)))
                & m2_relset_1(C,u1_struct_0(A),k1_pcomps_1(u1_struct_0(A))) )
             => ! [D] :
                  ( ( v1_funct_1(D)
                    & v1_funct_2(D,u1_struct_0(A),k1_fraenkel(u1_struct_0(A),u1_struct_0(B)))
                    & m2_relset_1(D,u1_struct_0(A),k1_fraenkel(u1_struct_0(A),u1_struct_0(B))) )
                 => ! [E] :
                      ( ( v1_funct_1(E)
                        & v1_funct_2(E,u1_struct_0(A),u1_struct_0(B))
                        & m2_relset_1(E,u1_struct_0(A),u1_struct_0(B)) )
                     => ( E = k4_nagata_1(A,B,C,D)
                      <=> ! [F] :
                            ( m1_subset_1(F,u1_struct_0(A))
                           => k7_urysohn3(A,B,E,F) = k1_funct_1(k8_funct_2(u1_struct_0(A),k1_fraenkel(u1_struct_0(A),u1_struct_0(B)),D,F),F) ) ) ) ) ) ) ) )).

fof(t25_nagata_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k2_zfmisc_1(k1_fraenkel(u1_struct_0(A),k1_numbers),k1_fraenkel(u1_struct_0(A),k1_numbers)),k1_fraenkel(u1_struct_0(A),k1_numbers))
            & m2_relset_1(B,k2_zfmisc_1(k1_fraenkel(u1_struct_0(A),k1_numbers),k1_fraenkel(u1_struct_0(A),k1_numbers)),k1_fraenkel(u1_struct_0(A),k1_numbers)) )
         => ( ! [C] :
                ( ( v1_funct_1(C)
                  & v1_funct_2(C,u1_struct_0(A),k1_numbers)
                  & m2_relset_1(C,u1_struct_0(A),k1_numbers) )
               => ! [D] :
                    ( ( v1_funct_1(D)
                      & v1_funct_2(D,u1_struct_0(A),k1_numbers)
                      & m2_relset_1(D,u1_struct_0(A),k1_numbers) )
                   => k1_binop_1(B,C,D) = k3_nagata_1(A,C,D) ) )
           => ! [C] :
                ( m2_finseq_1(C,k1_fraenkel(u1_struct_0(A),k1_numbers))
               => ( ! [D] :
                      ( m2_subset_1(D,k1_numbers,k5_numbers)
                     => ( r1_xreal_0(D,k3_finseq_1(C))
                       => ( np__0 = D
                          | ( v1_funct_1(k1_funct_1(C,D))
                            & v1_funct_2(k1_funct_1(C,D),u1_struct_0(A),k1_numbers)
                            & v9_pscomp_1(k1_funct_1(C,D),A)
                            & m2_relset_1(k1_funct_1(C,D),u1_struct_0(A),k1_numbers) ) ) ) )
                 => ( v1_funct_1(k2_finsop_1(k1_fraenkel(u1_struct_0(A),k1_numbers),C,B))
                    & v1_funct_2(k2_finsop_1(k1_fraenkel(u1_struct_0(A),k1_numbers),C,B),u1_struct_0(A),k1_numbers)
                    & v9_pscomp_1(k2_finsop_1(k1_fraenkel(u1_struct_0(A),k1_numbers),C,B),A)
                    & m2_relset_1(k2_finsop_1(k1_fraenkel(u1_struct_0(A),k1_numbers),C,B),u1_struct_0(A),k1_numbers) ) ) ) ) ) ) )).

fof(t26_nagata_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & v2_pre_topc(B)
            & l1_pre_topc(B) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,u1_struct_0(A),k1_fraenkel(u1_struct_0(A),u1_struct_0(B)))
                & m2_relset_1(C,u1_struct_0(A),k1_fraenkel(u1_struct_0(A),u1_struct_0(B))) )
             => ( ! [D] :
                    ( m1_subset_1(D,u1_struct_0(A))
                   => ( v1_funct_1(k8_funct_2(u1_struct_0(A),k1_fraenkel(u1_struct_0(A),u1_struct_0(B)),C,D))
                      & v1_funct_2(k8_funct_2(u1_struct_0(A),k1_fraenkel(u1_struct_0(A),u1_struct_0(B)),C,D),u1_struct_0(A),u1_struct_0(B))
                      & v5_pre_topc(k8_funct_2(u1_struct_0(A),k1_fraenkel(u1_struct_0(A),u1_struct_0(B)),C,D),A,B)
                      & m2_relset_1(k8_funct_2(u1_struct_0(A),k1_fraenkel(u1_struct_0(A),u1_struct_0(B)),C,D),u1_struct_0(A),u1_struct_0(B)) ) )
               => ! [D] :
                    ( ( v1_funct_1(D)
                      & v1_funct_2(D,u1_struct_0(A),k1_pcomps_1(u1_struct_0(A)))
                      & m2_relset_1(D,u1_struct_0(A),k1_pcomps_1(u1_struct_0(A))) )
                   => ( ! [E] :
                          ( m1_subset_1(E,u1_struct_0(A))
                         => ( r2_hidden(E,k5_urysohn3(u1_struct_0(A),A,D,E))
                            & v3_pre_topc(k5_urysohn3(u1_struct_0(A),A,D,E),A)
                            & ! [F] :
                                ( m1_subset_1(F,u1_struct_0(A))
                               => ! [G] :
                                    ( m1_subset_1(G,u1_struct_0(A))
                                   => ( r2_hidden(F,k5_urysohn3(u1_struct_0(A),A,D,G))
                                     => k1_funct_1(k8_funct_2(u1_struct_0(A),k1_fraenkel(u1_struct_0(A),u1_struct_0(B)),C,F),F) = k1_funct_1(k8_funct_2(u1_struct_0(A),k1_fraenkel(u1_struct_0(A),u1_struct_0(B)),C,G),F) ) ) ) ) )
                     => v5_pre_topc(k4_nagata_1(A,B,D,C),A,B) ) ) ) ) ) ) )).

fof(d9_nagata_1,axiom,(
    ! [A,B] :
      ( m1_subset_1(B,k1_numbers)
     => ! [C] :
          ( ( v1_funct_1(C)
            & v1_funct_2(C,A,k1_numbers)
            & m2_relset_1(C,A,k1_numbers) )
         => ! [D] :
              ( ( v1_funct_1(D)
                & v1_funct_2(D,A,k1_numbers)
                & m2_relset_1(D,A,k1_numbers) )
             => ( D = k5_nagata_1(A,B,C)
              <=> ! [E] :
                    ( r2_hidden(E,A)
                   => k1_funct_1(D,E) = k1_square_1(B,k1_funct_1(C,E)) ) ) ) ) ) )).

fof(t27_nagata_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,u1_struct_0(A),k1_numbers)
                & m2_relset_1(C,u1_struct_0(A),k1_numbers) )
             => ( v9_pscomp_1(C,A)
               => v9_pscomp_1(k5_nagata_1(u1_struct_0(A),B,C),A) ) ) ) ) )).

fof(d10_nagata_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_nagata_1(A,B)
      <=> ( v2_metric_1(B,A)
          & v4_metric_1(B,A)
          & v5_metric_1(B,A) ) ) ) )).

fof(t28_nagata_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_nagata_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,C) = np__0
                      & 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,E,D))) ) ) ) ) ) ) )).

fof(t29_nagata_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_nagata_1(A,B)
       => ! [C] :
            ( m1_subset_1(C,A)
           => ! [D] :
                ( m1_subset_1(D,A)
               => r1_xreal_0(np__0,k1_metric_1(A,A,B,C,D)) ) ) ) ) )).

fof(t30_nagata_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),k1_numbers)
                & m2_relset_1(C,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),k1_numbers) )
             => ( r1_nagata_1(u1_struct_0(A),C)
               => ( r1_xreal_0(B,np__0)
                  | r1_nagata_1(u1_struct_0(A),k5_nagata_1(k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),B,C)) ) ) ) ) ) )).

fof(t31_nagata_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),k1_numbers)
                & m2_relset_1(C,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),k1_numbers) )
             => ( r1_pcomps_1(u1_struct_0(A),C)
               => ( r1_xreal_0(B,np__0)
                  | r1_pcomps_1(u1_struct_0(A),k5_nagata_1(k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),B,C)) ) ) ) ) ) )).

fof(dt_k1_nagata_1,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A)
        & v1_funct_1(B)
        & v1_funct_2(B,k5_numbers,k1_pcomps_1(k1_pcomps_1(u1_struct_0(A))))
        & m1_relset_1(B,k5_numbers,k1_pcomps_1(k1_pcomps_1(u1_struct_0(A))))
        & m1_subset_1(C,k5_numbers) )
     => m1_subset_1(k1_nagata_1(A,B,C),k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A)))) ) )).

fof(redefinition_k1_nagata_1,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A)
        & v1_funct_1(B)
        & v1_funct_2(B,k5_numbers,k1_pcomps_1(k1_pcomps_1(u1_struct_0(A))))
        & m1_relset_1(B,k5_numbers,k1_pcomps_1(k1_pcomps_1(u1_struct_0(A))))
        & m1_subset_1(C,k5_numbers) )
     => k1_nagata_1(A,B,C) = k1_funct_1(B,C) ) )).

fof(dt_k2_nagata_1,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A)
        & v1_funct_1(B)
        & v1_funct_2(B,k5_numbers,k1_pcomps_1(k1_pcomps_1(u1_struct_0(A))))
        & m1_relset_1(B,k5_numbers,k1_pcomps_1(k1_pcomps_1(u1_struct_0(A)))) )
     => m1_subset_1(k2_nagata_1(A,B),k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A)))) ) )).

fof(redefinition_k2_nagata_1,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A)
        & v1_funct_1(B)
        & v1_funct_2(B,k5_numbers,k1_pcomps_1(k1_pcomps_1(u1_struct_0(A))))
        & m1_relset_1(B,k5_numbers,k1_pcomps_1(k1_pcomps_1(u1_struct_0(A)))) )
     => k2_nagata_1(A,B) = k3_card_3(B) ) )).

fof(dt_k3_nagata_1,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A)
        & v1_funct_1(B)
        & v1_funct_2(B,u1_struct_0(A),k1_numbers)
        & m1_relset_1(B,u1_struct_0(A),k1_numbers)
        & v1_funct_1(C)
        & v1_funct_2(C,u1_struct_0(A),k1_numbers)
        & m1_relset_1(C,u1_struct_0(A),k1_numbers) )
     => ( v1_funct_1(k3_nagata_1(A,B,C))
        & v1_funct_2(k3_nagata_1(A,B,C),u1_struct_0(A),k1_numbers)
        & m2_relset_1(k3_nagata_1(A,B,C),u1_struct_0(A),k1_numbers) ) ) )).

fof(dt_k4_nagata_1,axiom,(
    ! [A,B,C,D] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A)
        & ~ v3_struct_0(B)
        & v2_pre_topc(B)
        & l1_pre_topc(B)
        & v1_funct_1(C)
        & v1_funct_2(C,u1_struct_0(A),k1_pcomps_1(u1_struct_0(A)))
        & m1_relset_1(C,u1_struct_0(A),k1_pcomps_1(u1_struct_0(A)))
        & v1_funct_1(D)
        & v1_funct_2(D,u1_struct_0(A),k1_fraenkel(u1_struct_0(A),u1_struct_0(B)))
        & m1_relset_1(D,u1_struct_0(A),k1_fraenkel(u1_struct_0(A),u1_struct_0(B))) )
     => ( v1_funct_1(k4_nagata_1(A,B,C,D))
        & v1_funct_2(k4_nagata_1(A,B,C,D),u1_struct_0(A),u1_struct_0(B))
        & m2_relset_1(k4_nagata_1(A,B,C,D),u1_struct_0(A),u1_struct_0(B)) ) ) )).

fof(dt_k5_nagata_1,axiom,(
    ! [A,B,C] :
      ( ( m1_subset_1(B,k1_numbers)
        & v1_funct_1(C)
        & v1_funct_2(C,A,k1_numbers)
        & m1_relset_1(C,A,k1_numbers) )
     => ( v1_funct_1(k5_nagata_1(A,B,C))
        & v1_funct_2(k5_nagata_1(A,B,C),A,k1_numbers)
        & m2_relset_1(k5_nagata_1(A,B,C),A,k1_numbers) ) ) )).
%------------------------------------------------------------------------------