SET007 Axioms: SET007+853.ax


%------------------------------------------------------------------------------
% File     : SET007+853 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : The Nagata-Smirnov Theorem. Part II
% 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_2 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   29 (   1 unit)
%            Number of atoms       :  353 (  30 equality)
%            Maximal formula depth :   24 (  11 average)
%            Number of connectives :  365 (  41 ~  ;   2  |; 178  &)
%                                         (   6 <=>; 138 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   36 (   0 propositional; 1-3 arity)
%            Number of functors    :   45 (   6 constant; 0-6 arity)
%            Number of variables   :  126 (   0 singleton; 117 !;   9 ?)
%            Maximal term depth    :    6 (   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(t1_nagata_2,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ~ ( ~ r1_xreal_0(A,np__0)
          & ! [B] :
              ( m2_subset_1(B,k1_numbers,k5_numbers)
             => ! [C] :
                  ( m2_subset_1(C,k1_numbers,k5_numbers)
                 => A != k4_real_1(k3_prepower(np__2,B),k1_nat_1(k2_nat_1(np__2,C),np__1)) ) ) ) ) )).

fof(d1_nagata_2,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k2_zfmisc_1(k5_numbers,k5_numbers),k5_numbers)
        & m2_relset_1(A,k2_zfmisc_1(k5_numbers,k5_numbers),k5_numbers) )
     => ( A = k1_nagata_2
      <=> ! [B] :
            ( m2_subset_1(B,k1_numbers,k5_numbers)
           => ! [C] :
                ( m2_subset_1(C,k1_numbers,k5_numbers)
               => k8_funct_2(k2_zfmisc_1(k5_numbers,k5_numbers),k5_numbers,A,k1_domain_1(k5_numbers,k5_numbers,B,C)) = k5_real_1(k4_real_1(k3_prepower(np__2,B),k1_nat_1(k2_nat_1(np__2,C),np__1)),np__1) ) ) ) ) )).

fof(t2_nagata_2,axiom,(
    v3_funct_2(k1_nagata_2,k2_zfmisc_1(k5_numbers,k5_numbers),k5_numbers) )).

fof(d2_nagata_2,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) )
     => ! [C] :
          ( m1_subset_1(C,A)
         => ! [D] :
              ( ( v1_funct_1(D)
                & v1_funct_2(D,A,k1_numbers)
                & m2_relset_1(D,A,k1_numbers) )
             => ( D = k2_nagata_2(A,B,C)
              <=> ! [E] :
                    ( m1_subset_1(E,A)
                   => k2_seq_1(A,k1_numbers,D,E) = k1_metric_1(A,A,B,C,E) ) ) ) ) ) )).

fof(t3_nagata_2,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] :
              ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k6_borsuk_1(A,B))))
             => ( v3_pre_topc(C,k6_borsuk_1(A,B))
               => ! [D] :
                    ( m1_subset_1(D,u1_struct_0(A))
                   => ! [E] :
                        ( m1_subset_1(E,u1_struct_0(B))
                       => ! [F] :
                            ( m1_subset_1(F,k1_zfmisc_1(u1_struct_0(A)))
                           => ! [G] :
                                ( m1_subset_1(G,k1_zfmisc_1(u1_struct_0(B)))
                               => ( ( F = k10_relset_1(k2_zfmisc_1(u1_struct_0(A),u1_struct_0(B)),u1_struct_0(A),k9_funct_3(u1_struct_0(A),u1_struct_0(B)),k3_xboole_0(C,k2_zfmisc_1(u1_struct_0(A),k1_struct_0(B,E))))
                                   => v3_pre_topc(F,A) )
                                  & ( G = k10_relset_1(k2_zfmisc_1(u1_struct_0(A),u1_struct_0(B)),u1_struct_0(B),k10_funct_3(u1_struct_0(A),u1_struct_0(B)),k3_xboole_0(C,k2_zfmisc_1(k1_struct_0(A,D),u1_struct_0(B))))
                                   => v3_pre_topc(G,B) ) ) ) ) ) ) ) ) ) ) )).

fof(t4_nagata_2,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(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) )
         => ( ! [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) )
               => ( B = C
                 => v9_pscomp_1(C,k6_borsuk_1(A,A)) ) )
           => ! [C] :
                ( m1_subset_1(C,u1_struct_0(A))
               => v9_pscomp_1(k2_nagata_2(u1_struct_0(A),B,C),A) ) ) ) ) )).

fof(d3_nagata_2,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) )
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(A))
             => ! [D] :
                  ( ( v1_funct_1(D)
                    & v1_funct_2(D,A,k1_numbers)
                    & m2_relset_1(D,A,k1_numbers) )
                 => ( D = k3_nagata_2(A,B,C)
                  <=> ! [E] :
                        ( m1_subset_1(E,A)
                       => k8_funct_2(A,k1_numbers,D,E) = k4_pscomp_1(k10_relset_1(A,k1_numbers,k2_nagata_2(A,B,E),C)) ) ) ) ) ) ) )).

fof(t5_nagata_2,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_nagata_1(A,B)
           => ! [C] :
                ( ( ~ v1_xboole_0(C)
                  & m1_subset_1(C,k1_zfmisc_1(A)) )
               => ! [D] :
                    ( m1_subset_1(D,A)
                   => r1_xreal_0(np__0,k8_funct_2(A,k1_numbers,k3_nagata_2(A,B,C),D)) ) ) ) ) ) )).

fof(t6_nagata_2,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_nagata_1(A,B)
           => ! [C] :
                ( m1_subset_1(C,k1_zfmisc_1(A))
               => ! [D] :
                    ( m1_subset_1(D,A)
                   => ( r2_hidden(D,C)
                     => k8_funct_2(A,k1_numbers,k3_nagata_2(A,B,C),D) = np__0 ) ) ) ) ) ) )).

fof(t7_nagata_2,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(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_nagata_1(u1_struct_0(A),B)
           => ! [C] :
                ( m1_subset_1(C,u1_struct_0(A))
               => ! [D] :
                    ( m1_subset_1(D,u1_struct_0(A))
                   => ! [E] :
                        ( ( ~ v1_xboole_0(E)
                          & m1_subset_1(E,k1_zfmisc_1(u1_struct_0(A))) )
                       => r1_xreal_0(k18_complex1(k5_real_1(k8_funct_2(u1_struct_0(A),k1_numbers,k3_nagata_2(u1_struct_0(A),B,E),C),k8_funct_2(u1_struct_0(A),k1_numbers,k3_nagata_2(u1_struct_0(A),B,E),D))),k1_metric_1(u1_struct_0(A),u1_struct_0(A),B,C,D)) ) ) ) ) ) ) )).

fof(t8_nagata_2,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(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_nagata_1(u1_struct_0(A),B)
              & ! [C] :
                  ( m1_subset_1(C,u1_struct_0(A))
                 => v9_pscomp_1(k2_nagata_2(u1_struct_0(A),B,C),A) ) )
           => ! [C] :
                ( ( ~ v1_xboole_0(C)
                  & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
               => v9_pscomp_1(k3_nagata_2(u1_struct_0(A),B,C),A) ) ) ) ) )).

fof(t9_nagata_2,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)
       => r1_nagata_1(A,B) ) ) )).

fof(t11_nagata_2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ! [B] :
          ( m1_seqfunc(B,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),k1_numbers)
         => ( ( ! [C] :
                  ( m2_subset_1(C,k1_numbers,k5_numbers)
                 => ? [D] :
                      ( v1_funct_1(D)
                      & v1_funct_2(D,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),k1_numbers)
                      & m2_relset_1(D,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),k1_numbers)
                      & k1_seqfunc(k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),k1_numbers,B,C) = D
                      & r1_nagata_1(u1_struct_0(A),D) ) )
              & ! [C] :
                  ( m1_subset_1(C,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)))
                 => v1_series_1(k10_seqfunc(k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),B,C)) ) )
           => ! [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) )
               => ( ! [D] :
                      ( m1_subset_1(D,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)))
                     => k8_funct_2(k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),k1_numbers,C,D) = k2_series_1(k10_seqfunc(k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),B,D)) )
                 => r1_nagata_1(u1_struct_0(A),C) ) ) ) ) ) )).

fof(t12_nagata_2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,k1_numbers)
                & m2_relset_1(C,k5_numbers,k1_numbers) )
             => ( ! [D] :
                    ( m2_subset_1(D,k1_numbers,k5_numbers)
                   => ( r1_xreal_0(D,B)
                     => r1_xreal_0(k8_funct_2(k5_numbers,k1_numbers,C,D),A) ) )
               => ! [D] :
                    ( m2_subset_1(D,k1_numbers,k5_numbers)
                   => ( r1_xreal_0(D,B)
                     => r1_xreal_0(k8_funct_2(k5_numbers,k1_numbers,k1_series_1(C),D),k4_real_1(A,k1_nat_1(D,np__1))) ) ) ) ) ) ) )).

fof(t13_nagata_2,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & m2_relset_1(A,k5_numbers,k1_numbers) )
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => r1_xreal_0(k18_complex1(k8_funct_2(k5_numbers,k1_numbers,k1_series_1(A),B)),k8_funct_2(k5_numbers,k1_numbers,k1_series_1(k22_seq_1(A)),B)) ) ) )).

fof(t14_nagata_2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ! [B] :
          ( m1_seqfunc(B,u1_struct_0(A),k1_numbers)
         => ( ! [C] :
                ( m2_subset_1(C,k1_numbers,k5_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_seqfunc(u1_struct_0(A),k1_numbers,B,C) = D
                    & v9_pscomp_1(D,A)
                    & ! [E] :
                        ( m1_subset_1(E,u1_struct_0(A))
                       => r1_xreal_0(np__0,k8_funct_2(u1_struct_0(A),k1_numbers,D,E)) ) ) )
           => ( ! [C] :
                  ( ( v1_funct_1(C)
                    & v1_funct_2(C,k5_numbers,k1_numbers)
                    & m2_relset_1(C,k5_numbers,k1_numbers) )
                 => ~ ( v1_series_1(C)
                      & ! [D] :
                          ( m2_subset_1(D,k1_numbers,k5_numbers)
                         => ! [E] :
                              ( m1_subset_1(E,u1_struct_0(A))
                             => r1_xreal_0(k8_funct_2(k5_numbers,k1_numbers,k10_seqfunc(u1_struct_0(A),B,E),D),k8_funct_2(k5_numbers,k1_numbers,C,D)) ) ) ) )
              | ! [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] :
                        ( m1_subset_1(D,u1_struct_0(A))
                       => k8_funct_2(u1_struct_0(A),k1_numbers,C,D) = k2_series_1(k10_seqfunc(u1_struct_0(A),B,D)) )
                   => v9_pscomp_1(C,A) ) ) ) ) ) ) )).

fof(t15_nagata_2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ! [C] :
              ( m1_seqfunc(C,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),k1_numbers)
             => ( ! [D] :
                    ( m2_subset_1(D,k1_numbers,k5_numbers)
                   => ? [E] :
                        ( v1_funct_1(E)
                        & v1_funct_2(E,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),k1_numbers)
                        & m2_relset_1(E,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),k1_numbers)
                        & k1_seqfunc(k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),k1_numbers,C,D) = E
                        & r1_nagata_1(u1_struct_0(A),E)
                        & ! [F] :
                            ( m1_subset_1(F,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)))
                           => r1_xreal_0(k8_funct_2(k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),k1_numbers,E,F),B) )
                        & ! [F] :
                            ( ( v1_funct_1(F)
                              & v1_funct_2(F,u1_struct_0(k6_borsuk_1(A,A)),k1_numbers)
                              & m2_relset_1(F,u1_struct_0(k6_borsuk_1(A,A)),k1_numbers) )
                           => ( E = F
                             => v9_pscomp_1(F,k6_borsuk_1(A,A)) ) ) ) )
               => ! [D] :
                    ( ( v1_funct_1(D)
                      & v1_funct_2(D,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),k1_numbers)
                      & m2_relset_1(D,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),k1_numbers) )
                   => ( ! [E] :
                          ( m1_subset_1(E,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)))
                         => k8_funct_2(k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),k1_numbers,D,E) = k2_series_1(k11_seq_1(k2_prepower(k6_real_1(np__1,np__2)),k10_seqfunc(k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),C,E))) )
                     => ( r1_nagata_1(u1_struct_0(A),D)
                        & ! [E] :
                            ( ( v1_funct_1(E)
                              & v1_funct_2(E,u1_struct_0(k6_borsuk_1(A,A)),k1_numbers)
                              & m2_relset_1(E,u1_struct_0(k6_borsuk_1(A,A)),k1_numbers) )
                           => ( D = E
                             => v9_pscomp_1(E,k6_borsuk_1(A,A)) ) ) ) ) ) ) ) ) ) )).

fof(t16_nagata_2,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(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_nagata_1(u1_struct_0(A),B)
              & ! [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) )
                 => ( B = C
                   => v9_pscomp_1(C,k6_borsuk_1(A,A)) ) ) )
           => ! [C] :
                ( ( ~ v1_xboole_0(C)
                  & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
               => ! [D] :
                    ( m1_subset_1(D,u1_struct_0(A))
                   => ( r2_hidden(D,k6_pre_topc(A,C))
                     => k8_funct_2(u1_struct_0(A),k1_numbers,k3_nagata_2(u1_struct_0(A),B,C),D) = np__0 ) ) ) ) ) ) )).

fof(t17_nagata_2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ( v1_urysohn1(A)
       => ! [B] :
            ( m1_subset_1(B,k1_numbers)
           => ! [C] :
                ( m1_seqfunc(C,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),k1_numbers)
               => ( ( ! [D] :
                        ( m2_subset_1(D,k1_numbers,k5_numbers)
                       => ? [E] :
                            ( v1_funct_1(E)
                            & v1_funct_2(E,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),k1_numbers)
                            & m2_relset_1(E,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),k1_numbers)
                            & k1_seqfunc(k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),k1_numbers,C,D) = E
                            & r1_nagata_1(u1_struct_0(A),E)
                            & ! [F] :
                                ( m1_subset_1(F,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)))
                               => r1_xreal_0(k8_funct_2(k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),k1_numbers,E,F),B) )
                            & ! [F] :
                                ( ( v1_funct_1(F)
                                  & v1_funct_2(F,u1_struct_0(k6_borsuk_1(A,A)),k1_numbers)
                                  & m2_relset_1(F,u1_struct_0(k6_borsuk_1(A,A)),k1_numbers) )
                               => ( E = F
                                 => v9_pscomp_1(F,k6_borsuk_1(A,A)) ) ) ) )
                    & ! [D] :
                        ( m1_subset_1(D,u1_struct_0(A))
                       => ! [E] :
                            ( ( ~ v1_xboole_0(E)
                              & m1_subset_1(E,k1_zfmisc_1(u1_struct_0(A))) )
                           => ~ ( ~ r2_hidden(D,E)
                                & v4_pre_topc(E,A)
                                & ! [F] :
                                    ( m2_subset_1(F,k1_numbers,k5_numbers)
                                   => ? [G] :
                                        ( v1_funct_1(G)
                                        & v1_funct_2(G,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),k1_numbers)
                                        & m2_relset_1(G,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),k1_numbers)
                                        & k1_seqfunc(k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),k1_numbers,C,F) = G
                                        & r1_xreal_0(k8_funct_2(u1_struct_0(A),k1_numbers,k3_nagata_2(u1_struct_0(A),G,E),D),np__0) ) ) ) ) ) )
                 => ( ? [D] :
                        ( v1_funct_1(D)
                        & v1_funct_2(D,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),k1_numbers)
                        & m2_relset_1(D,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),k1_numbers)
                        & r1_pcomps_1(u1_struct_0(A),D)
                        & ! [E] :
                            ( m1_subset_1(E,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)))
                           => k8_funct_2(k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),k1_numbers,D,E) = k2_series_1(k11_seq_1(k2_prepower(k6_real_1(np__1,np__2)),k10_seqfunc(k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),C,E))) ) )
                    & v3_pcomps_1(A) ) ) ) ) ) ) )).

fof(t18_nagata_2,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m2_finseq_1(B,A)
         => ! [C] :
              ( m2_finseq_1(C,A)
             => ! [D] :
                  ( ( v1_funct_1(D)
                    & v1_funct_2(D,k2_zfmisc_1(A,A),A)
                    & m2_relset_1(D,k2_zfmisc_1(A,A),A) )
                 => ~ ( v2_funct_1(B)
                      & v2_funct_1(C)
                      & r1_tarski(k5_relset_1(k5_numbers,A,C),k5_relset_1(k5_numbers,A,B))
                      & v1_binop_1(D,A)
                      & v2_binop_1(D,A)
                      & ( v1_setwiseo(D,A)
                        | ( r1_xreal_0(np__1,k3_finseq_1(C))
                          & ~ r1_xreal_0(k3_finseq_1(B),k3_finseq_1(C)) ) )
                      & ! [E] :
                          ( m2_finseq_1(E,A)
                         => ~ ( v2_funct_1(E)
                              & k5_relset_1(k5_numbers,A,E) = k6_subset_1(A,k5_relset_1(k5_numbers,A,B),k5_relset_1(k5_numbers,A,C))
                              & k2_finsop_1(A,B,D) = k2_binop_1(A,A,A,D,k2_finsop_1(A,C,D),k2_finsop_1(A,E,D)) ) ) ) ) ) ) ) )).

fof(t19_nagata_2,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) ) )
      <=> v3_pcomps_1(A) ) ) )).

fof(t20_nagata_2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ( v3_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] :
                    ( ( v1_funct_1(C)
                      & v1_funct_2(C,k5_numbers,k1_pcomps_1(k1_pcomps_1(u1_struct_0(A))))
                      & m2_relset_1(C,k5_numbers,k1_pcomps_1(k1_pcomps_1(u1_struct_0(A)))) )
                   => ~ ( v1_tops_2(k2_nagata_1(A,C),A)
                        & r1_pre_topc(A,k2_nagata_1(A,C))
                        & r1_setfam_1(k2_nagata_1(A,C),B)
                        & v2_nagata_1(C,A) ) ) ) ) ) ) )).

fof(t21_nagata_2,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)))) )
             => ~ v5_nagata_1(B,A) ) ) ) )).

fof(t22_nagata_2,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))))
              & v5_nagata_1(B,A) ) )
      <=> v3_pcomps_1(A) ) ) )).

fof(dt_k1_nagata_2,axiom,
    ( v1_funct_1(k1_nagata_2)
    & v1_funct_2(k1_nagata_2,k2_zfmisc_1(k5_numbers,k5_numbers),k5_numbers)
    & m2_relset_1(k1_nagata_2,k2_zfmisc_1(k5_numbers,k5_numbers),k5_numbers) )).

fof(dt_k2_nagata_2,axiom,(
    ! [A,B,C] :
      ( ( 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)
        & m1_subset_1(C,A) )
     => ( v1_funct_1(k2_nagata_2(A,B,C))
        & v1_funct_2(k2_nagata_2(A,B,C),A,k1_numbers)
        & m2_relset_1(k2_nagata_2(A,B,C),A,k1_numbers) ) ) )).

fof(dt_k3_nagata_2,axiom,(
    ! [A,B,C] :
      ( ( ~ 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)
        & m1_subset_1(C,k1_zfmisc_1(A)) )
     => ( v1_funct_1(k3_nagata_2(A,B,C))
        & v1_funct_2(k3_nagata_2(A,B,C),A,k1_numbers)
        & m2_relset_1(k3_nagata_2(A,B,C),A,k1_numbers) ) ) )).

fof(t10_nagata_2,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(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)
              & ! [C] :
                  ( ( ~ v1_xboole_0(C)
                    & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
                 => k6_pre_topc(A,C) = a_3_0_nagata_2(A,B,C) ) )
           => v3_pcomps_1(A) ) ) ) )).

fof(fraenkel_a_3_0_nagata_2,axiom,(
    ! [A,B,C,D] :
      ( ( ~ v3_struct_0(B)
        & v2_pre_topc(B)
        & l1_pre_topc(B)
        & v1_funct_1(C)
        & v1_funct_2(C,k2_zfmisc_1(u1_struct_0(B),u1_struct_0(B)),k1_numbers)
        & m2_relset_1(C,k2_zfmisc_1(u1_struct_0(B),u1_struct_0(B)),k1_numbers)
        & ~ v1_xboole_0(D)
        & m1_subset_1(D,k1_zfmisc_1(u1_struct_0(B))) )
     => ( r2_hidden(A,a_3_0_nagata_2(B,C,D))
      <=> ? [E] :
            ( m1_subset_1(E,u1_struct_0(B))
            & A = E
            & k8_funct_2(u1_struct_0(B),k1_numbers,k3_nagata_2(u1_struct_0(B),C,D),E) = np__0 ) ) ) )).
%------------------------------------------------------------------------------