SET007 Axioms: SET007+438.ax


%------------------------------------------------------------------------------
% File     : SET007+438 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Dyadic Numbers and T_4 Topological Spaces
% Version  : [Urb08] axioms.
% English  :

% Refs     : [Mat90] Matuszewski (1990), Formalized Mathematics
%          : [Urb07] Urban (2007), MPTP 0.2: Design, Implementation, and In
%          : [Urb08] Urban (2006), Email to G. Sutcliffe
% Source   : [Urb08]
% Names    : urysohn1 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   58 (  14 unit)
%            Number of atoms       :  336 (  32 equality)
%            Maximal formula depth :   34 (   8 average)
%            Number of connectives :  335 (  57 ~  ;   4  |; 138  &)
%                                         (  16 <=>; 120 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   27 (   1 propositional; 0-4 arity)
%            Number of functors    :   35 (  11 constant; 0-4 arity)
%            Number of variables   :  122 (   0 singleton; 117 !;   5 ?)
%            Maximal term depth    :    5 (   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(fc1_urysohn1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k5_numbers)
     => ( ~ v1_xboole_0(k3_urysohn1(A))
        & v1_membered(k3_urysohn1(A))
        & v2_membered(k3_urysohn1(A)) ) ) )).

fof(fc2_urysohn1,axiom,
    ( ~ v1_xboole_0(k4_urysohn1)
    & v1_membered(k4_urysohn1)
    & v2_membered(k4_urysohn1) )).

fof(fc3_urysohn1,axiom,
    ( ~ v1_xboole_0(k5_urysohn1)
    & v1_membered(k5_urysohn1)
    & v2_membered(k5_urysohn1) )).

fof(d1_urysohn1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_zfmisc_1(k1_numbers))
     => ( A = k1_urysohn1
      <=> ! [B] :
            ( m1_subset_1(B,k1_numbers)
           => ( r2_hidden(B,A)
            <=> ~ r1_xreal_0(np__0,B) ) ) ) ) )).

fof(d2_urysohn1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_zfmisc_1(k1_numbers))
     => ( A = k2_urysohn1
      <=> ! [B] :
            ( m1_subset_1(B,k1_numbers)
           => ( r2_hidden(B,A)
            <=> ~ r1_xreal_0(B,np__1) ) ) ) ) )).

fof(d3_urysohn1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(k1_numbers))
         => ( B = k3_urysohn1(A)
          <=> ! [C] :
                ( m1_subset_1(C,k1_numbers)
               => ( r2_hidden(C,B)
                <=> ? [D] :
                      ( m2_subset_1(D,k1_numbers,k5_numbers)
                      & r1_xreal_0(np__0,D)
                      & r1_xreal_0(D,k1_card_4(np__2,A))
                      & C = k6_real_1(D,k1_card_4(np__2,A)) ) ) ) ) ) ) )).

fof(d4_urysohn1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_zfmisc_1(k1_numbers))
     => ( A = k4_urysohn1
      <=> ! [B] :
            ( m1_subset_1(B,k1_numbers)
           => ( r2_hidden(B,A)
            <=> ? [C] :
                  ( m2_subset_1(C,k1_numbers,k5_numbers)
                  & r2_hidden(B,k3_urysohn1(C)) ) ) ) ) ) )).

fof(d5_urysohn1,axiom,(
    k5_urysohn1 = k4_subset_1(k1_numbers,k4_subset_1(k1_numbers,k1_urysohn1,k4_urysohn1),k2_urysohn1) )).

fof(t1_urysohn1,axiom,(
    $true )).

fof(t2_urysohn1,axiom,(
    $true )).

fof(t3_urysohn1,axiom,(
    $true )).

fof(t4_urysohn1,axiom,(
    $true )).

fof(t5_urysohn1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ( r2_hidden(B,k3_urysohn1(A))
           => ( r1_xreal_0(np__0,B)
              & r1_xreal_0(B,np__1) ) ) ) ) )).

fof(t6_urysohn1,axiom,(
    k3_urysohn1(np__0) = k7_domain_1(k1_numbers,np__0,np__1) )).

fof(t7_urysohn1,axiom,(
    k3_urysohn1(np__1) = k8_domain_1(k1_numbers,np__0,k6_real_1(np__1,np__2),np__1) )).

fof(t8_urysohn1,axiom,(
    $true )).

fof(t9_urysohn1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ? [B] :
          ( v1_relat_1(B)
          & v1_funct_1(B)
          & v1_finseq_1(B)
          & k4_finseq_1(B) = k2_finseq_1(k1_nat_1(k1_card_4(np__2,A),np__1))
          & ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => ( r2_hidden(C,k4_finseq_1(B))
               => k1_funct_1(B,C) = k6_real_1(k5_real_1(C,np__1),k1_card_4(np__2,A)) ) ) ) ) )).

fof(d6_urysohn1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ( v1_relat_1(B)
            & v1_funct_1(B)
            & v1_finseq_1(B) )
         => ( B = k7_urysohn1(A)
          <=> ( k4_finseq_1(B) = k2_finseq_1(k1_nat_1(k1_card_4(np__2,A),np__1))
              & ! [C] :
                  ( m2_subset_1(C,k1_numbers,k5_numbers)
                 => ( r2_hidden(C,k4_finseq_1(B))
                   => k1_funct_1(B,C) = k6_real_1(k5_real_1(C,np__1),k1_card_4(np__2,A)) ) ) ) ) ) ) )).

fof(t10_urysohn1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ( k4_finseq_1(k7_urysohn1(A)) = k2_finseq_1(k1_nat_1(k1_card_4(np__2,A),np__1))
        & k2_relat_1(k7_urysohn1(A)) = k3_urysohn1(A) ) ) )).

fof(t11_urysohn1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => r1_tarski(k3_urysohn1(A),k3_urysohn1(k1_nat_1(A,np__1))) ) )).

fof(t12_urysohn1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ( r2_hidden(np__0,k3_urysohn1(A))
        & r2_hidden(np__1,k3_urysohn1(A)) ) ) )).

fof(t13_urysohn1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ( r1_xreal_0(B,k1_card_4(np__2,A))
           => ( r1_xreal_0(B,np__0)
              | r2_hidden(k6_real_1(k5_real_1(k2_nat_1(B,np__2),np__1),k1_card_4(np__2,k1_nat_1(A,np__1))),k6_subset_1(k1_numbers,k3_urysohn1(k1_nat_1(A,np__1)),k3_urysohn1(A))) ) ) ) ) )).

fof(t14_urysohn1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ( r1_xreal_0(np__0,B)
           => ( r1_xreal_0(k1_card_4(np__2,A),B)
              | r2_hidden(k6_real_1(k1_nat_1(k2_nat_1(B,np__2),np__1),k1_card_4(np__2,k1_nat_1(A,np__1))),k6_subset_1(k1_numbers,k3_urysohn1(k1_nat_1(A,np__1)),k3_urysohn1(A))) ) ) ) ) )).

fof(t15_urysohn1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => r2_hidden(k6_real_1(np__1,k1_card_4(np__2,k1_nat_1(A,np__1))),k6_subset_1(k1_numbers,k3_urysohn1(k1_nat_1(A,np__1)),k3_urysohn1(A))) ) )).

fof(d7_urysohn1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k3_urysohn1(A))
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => ( C = k8_urysohn1(A,B)
              <=> B = k6_real_1(C,k1_card_4(np__2,A)) ) ) ) ) )).

fof(t16_urysohn1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k3_urysohn1(A))
         => ( B = k6_real_1(k8_urysohn1(A,B),k1_card_4(np__2,A))
            & r1_xreal_0(np__0,k8_urysohn1(A,B))
            & r1_xreal_0(k8_urysohn1(A,B),k1_card_4(np__2,A)) ) ) ) )).

fof(t17_urysohn1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k3_urysohn1(A))
         => ( ~ r1_xreal_0(B,k6_real_1(k5_real_1(k8_urysohn1(A,B),np__1),k1_card_4(np__2,A)))
            & ~ r1_xreal_0(k6_real_1(k1_nat_1(k8_urysohn1(A,B),np__1),k1_card_4(np__2,A)),B) ) ) ) )).

fof(t18_urysohn1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k3_urysohn1(A))
         => ~ r1_xreal_0(k6_real_1(k1_nat_1(k8_urysohn1(A,B),np__1),k1_card_4(np__2,A)),k6_real_1(k5_real_1(k8_urysohn1(A,B),np__1),k1_card_4(np__2,A))) ) ) )).

fof(t19_urysohn1,axiom,(
    $true )).

fof(t20_urysohn1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k3_urysohn1(k1_nat_1(A,np__1)))
         => ( ~ r2_hidden(B,k3_urysohn1(A))
           => ( r2_hidden(k6_real_1(k5_real_1(k8_urysohn1(k1_nat_1(A,np__1),B),np__1),k1_card_4(np__2,k1_nat_1(A,np__1))),k3_urysohn1(A))
              & r2_hidden(k6_real_1(k1_nat_1(k8_urysohn1(k1_nat_1(A,np__1),B),np__1),k1_card_4(np__2,k1_nat_1(A,np__1))),k3_urysohn1(A)) ) ) ) ) )).

fof(t21_urysohn1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k3_urysohn1(A))
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k3_urysohn1(A))
             => ~ ( ~ r1_xreal_0(C,B)
                  & r1_xreal_0(k8_urysohn1(A,C),k8_urysohn1(A,B)) ) ) ) ) )).

fof(t22_urysohn1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k3_urysohn1(A))
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k3_urysohn1(A))
             => ( ~ r1_xreal_0(C,B)
               => ( r1_xreal_0(B,k6_real_1(k5_real_1(k8_urysohn1(A,C),np__1),k1_card_4(np__2,A)))
                  & r1_xreal_0(k6_real_1(k1_nat_1(k8_urysohn1(A,B),np__1),k1_card_4(np__2,A)),C) ) ) ) ) ) )).

fof(t23_urysohn1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k3_urysohn1(k1_nat_1(A,np__1)))
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k3_urysohn1(k1_nat_1(A,np__1)))
             => ~ ( ~ r1_xreal_0(C,B)
                  & ~ r2_hidden(B,k3_urysohn1(A))
                  & ~ r2_hidden(C,k3_urysohn1(A))
                  & ~ r1_xreal_0(k6_real_1(k1_nat_1(k8_urysohn1(k1_nat_1(A,np__1),B),np__1),k1_card_4(np__2,k1_nat_1(A,np__1))),k6_real_1(k5_real_1(k8_urysohn1(k1_nat_1(A,np__1),C),np__1),k1_card_4(np__2,k1_nat_1(A,np__1)))) ) ) ) ) )).

fof(d8_urysohn1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
             => ( m1_connsp_2(C,A,B)
              <=> ? [D] :
                    ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
                    & v3_pre_topc(D,A)
                    & r2_hidden(B,D)
                    & r1_tarski(D,C) ) ) ) ) ) )).

fof(t24_urysohn1,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)))
         => ( v3_pre_topc(B,A)
          <=> ! [C] :
                ( m1_subset_1(C,u1_struct_0(A))
               => ~ ( r2_hidden(C,B)
                    & ! [D] :
                        ( m1_connsp_2(D,A,C)
                       => ~ r1_tarski(D,B) ) ) ) ) ) ) )).

fof(t25_urysohn1,axiom,(
    $true )).

fof(t26_urysohn1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => ( ! [C] :
                ( m1_subset_1(C,u1_struct_0(A))
               => ( r2_hidden(C,B)
                 => m1_connsp_2(B,A,C) ) )
           => v3_pre_topc(B,A) ) ) ) )).

fof(d9_urysohn1,axiom,(
    ! [A] :
      ( l1_pre_topc(A)
     => ( v1_urysohn1(A)
      <=> ! [B] :
            ( m1_subset_1(B,u1_struct_0(A))
           => ! [C] :
                ( m1_subset_1(C,u1_struct_0(A))
               => ~ ( B != C
                    & ! [D] :
                        ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
                       => ! [E] :
                            ( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(A)))
                           => ~ ( v3_pre_topc(D,A)
                                & v3_pre_topc(E,A)
                                & r2_hidden(B,D)
                                & ~ r2_hidden(C,D)
                                & r2_hidden(C,E)
                                & ~ r2_hidden(B,E) ) ) ) ) ) ) ) ) )).

fof(t27_urysohn1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ( v1_urysohn1(A)
      <=> ! [B] :
            ( m1_subset_1(B,u1_struct_0(A))
           => v4_pre_topc(k1_struct_0(A,B),A) ) ) ) )).

fof(t28_urysohn1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ( v5_compts_1(A)
       => ! [B] :
            ( ( v3_pre_topc(B,A)
              & 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))) )
               => ~ ( B != k1_xboole_0
                    & r1_tarski(k6_pre_topc(A,B),C)
                    & ! [D] :
                        ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
                       => ~ ( D != k1_xboole_0
                            & v3_pre_topc(D,A)
                            & r1_tarski(k6_pre_topc(A,B),D)
                            & r1_tarski(k6_pre_topc(A,D),C) ) ) ) ) ) ) ) )).

fof(t29_urysohn1,axiom,(
    ! [A] :
      ( ( v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ( v4_compts_1(A)
      <=> ! [B] :
            ( ( v3_pre_topc(B,A)
              & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
           => ! [C] :
                ( m1_subset_1(C,u1_struct_0(A))
               => ~ ( r2_hidden(C,B)
                    & ! [D] :
                        ( ( v3_pre_topc(D,A)
                          & m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A))) )
                       => ~ ( r2_hidden(C,D)
                            & r1_tarski(k6_pre_topc(A,D),B) ) ) ) ) ) ) ) )).

fof(t30_urysohn1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ( ( v5_compts_1(A)
          & v1_urysohn1(A) )
       => ! [B] :
            ( ( v3_pre_topc(B,A)
              & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
           => ~ ( B != k1_xboole_0
                & ! [C] :
                    ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
                   => ~ ( C != k1_xboole_0
                        & r1_tarski(k6_pre_topc(A,C),B) ) ) ) ) ) ) )).

fof(t31_urysohn1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ( v5_compts_1(A)
       => ! [B] :
            ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
           => ! [C] :
                ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
               => ~ ( v3_pre_topc(B,A)
                    & v4_pre_topc(C,A)
                    & C != k1_xboole_0
                    & r1_tarski(C,B)
                    & ! [D] :
                        ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
                       => ~ ( v3_pre_topc(D,A)
                            & r1_tarski(C,D)
                            & r1_tarski(k6_pre_topc(A,D),B) ) ) ) ) ) ) ) )).

fof(d10_urysohn1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
             => ( ( v5_compts_1(A)
                  & v3_pre_topc(B,A)
                  & v3_pre_topc(C,A)
                  & r1_tarski(k6_pre_topc(A,B),C) )
               => ( B = k1_xboole_0
                  | ! [D] :
                      ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
                     => ( m1_urysohn1(D,A,B,C)
                      <=> ( D != k1_xboole_0
                          & v3_pre_topc(D,A)
                          & r1_tarski(k6_pre_topc(A,B),D)
                          & r1_tarski(k6_pre_topc(A,D),C) ) ) ) ) ) ) ) ) )).

fof(t32_urysohn1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ( v5_compts_1(A)
       => ! [B] :
            ( ( v4_pre_topc(B,A)
              & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
           => ! [C] :
                ( ( v4_pre_topc(C,A)
                  & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
               => ( r1_xboole_0(B,C)
                 => ( B = k1_xboole_0
                    | ! [D] :
                        ( m2_subset_1(D,k1_numbers,k5_numbers)
                       => ! [E] :
                            ( ( v1_funct_1(E)
                              & v1_funct_2(E,k3_urysohn1(D),k1_zfmisc_1(u1_struct_0(A)))
                              & m2_relset_1(E,k3_urysohn1(D),k1_zfmisc_1(u1_struct_0(A))) )
                           => ~ ( r1_tarski(B,k1_funct_1(E,np__0))
                                & C = k4_xboole_0(k2_pre_topc(A),k1_funct_1(E,np__1))
                                & ! [F] :
                                    ( m2_subset_1(F,k1_numbers,k3_urysohn1(D))
                                   => ! [G] :
                                        ( m2_subset_1(G,k1_numbers,k3_urysohn1(D))
                                       => ( ~ r1_xreal_0(G,F)
                                         => ( v3_pre_topc(k6_urysohn1(A,k3_urysohn1(D),E,F),A)
                                            & v3_pre_topc(k6_urysohn1(A,k3_urysohn1(D),E,G),A)
                                            & r1_tarski(k6_pre_topc(A,k6_urysohn1(A,k3_urysohn1(D),E,F)),k6_urysohn1(A,k3_urysohn1(D),E,G)) ) ) ) )
                                & ! [F] :
                                    ( ( v1_funct_1(F)
                                      & v1_funct_2(F,k3_urysohn1(k1_nat_1(D,np__1)),k1_zfmisc_1(u1_struct_0(A)))
                                      & m2_relset_1(F,k3_urysohn1(k1_nat_1(D,np__1)),k1_zfmisc_1(u1_struct_0(A))) )
                                   => ~ ( r1_tarski(B,k1_funct_1(F,np__0))
                                        & C = k4_xboole_0(k2_pre_topc(A),k1_funct_1(F,np__1))
                                        & ! [G] :
                                            ( m2_subset_1(G,k1_numbers,k3_urysohn1(k1_nat_1(D,np__1)))
                                           => ! [H] :
                                                ( m2_subset_1(H,k1_numbers,k3_urysohn1(k1_nat_1(D,np__1)))
                                               => ! [I] :
                                                    ( m2_subset_1(I,k1_numbers,k3_urysohn1(k1_nat_1(D,np__1)))
                                                   => ( ~ r1_xreal_0(H,G)
                                                     => ( v3_pre_topc(k6_urysohn1(A,k3_urysohn1(k1_nat_1(D,np__1)),F,G),A)
                                                        & v3_pre_topc(k6_urysohn1(A,k3_urysohn1(k1_nat_1(D,np__1)),F,H),A)
                                                        & r1_tarski(k6_pre_topc(A,k6_urysohn1(A,k3_urysohn1(k1_nat_1(D,np__1)),F,G)),k6_urysohn1(A,k3_urysohn1(k1_nat_1(D,np__1)),F,H))
                                                        & ( r2_hidden(I,k3_urysohn1(D))
                                                         => k6_urysohn1(A,k3_urysohn1(k1_nat_1(D,np__1)),F,I) = k1_funct_1(E,I) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )).

fof(dt_m1_urysohn1,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A)
        & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
        & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
     => ! [D] :
          ( m1_urysohn1(D,A,B,C)
         => m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A))) ) ) )).

fof(existence_m1_urysohn1,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A)
        & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
        & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
     => ? [D] : m1_urysohn1(D,A,B,C) ) )).

fof(dt_k1_urysohn1,axiom,(
    m1_subset_1(k1_urysohn1,k1_zfmisc_1(k1_numbers)) )).

fof(dt_k2_urysohn1,axiom,(
    m1_subset_1(k2_urysohn1,k1_zfmisc_1(k1_numbers)) )).

fof(dt_k3_urysohn1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k5_numbers)
     => m1_subset_1(k3_urysohn1(A),k1_zfmisc_1(k1_numbers)) ) )).

fof(dt_k4_urysohn1,axiom,(
    m1_subset_1(k4_urysohn1,k1_zfmisc_1(k1_numbers)) )).

fof(dt_k5_urysohn1,axiom,(
    m1_subset_1(k5_urysohn1,k1_zfmisc_1(k1_numbers)) )).

fof(dt_k6_urysohn1,axiom,(
    ! [A,B,C,D] :
      ( ( v2_pre_topc(A)
        & l1_pre_topc(A)
        & ~ v1_xboole_0(B)
        & m1_subset_1(B,k1_zfmisc_1(k1_numbers))
        & v1_funct_1(C)
        & v1_funct_2(C,B,k1_zfmisc_1(u1_struct_0(A)))
        & m1_relset_1(C,B,k1_zfmisc_1(u1_struct_0(A)))
        & m1_subset_1(D,B) )
     => m1_subset_1(k6_urysohn1(A,B,C,D),k1_zfmisc_1(u1_struct_0(A))) ) )).

fof(redefinition_k6_urysohn1,axiom,(
    ! [A,B,C,D] :
      ( ( v2_pre_topc(A)
        & l1_pre_topc(A)
        & ~ v1_xboole_0(B)
        & m1_subset_1(B,k1_zfmisc_1(k1_numbers))
        & v1_funct_1(C)
        & v1_funct_2(C,B,k1_zfmisc_1(u1_struct_0(A)))
        & m1_relset_1(C,B,k1_zfmisc_1(u1_struct_0(A)))
        & m1_subset_1(D,B) )
     => k6_urysohn1(A,B,C,D) = k1_funct_1(C,D) ) )).

fof(dt_k7_urysohn1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k5_numbers)
     => ( v1_relat_1(k7_urysohn1(A))
        & v1_funct_1(k7_urysohn1(A))
        & v1_finseq_1(k7_urysohn1(A)) ) ) )).

fof(dt_k8_urysohn1,axiom,(
    ! [A,B] :
      ( ( m1_subset_1(A,k5_numbers)
        & m1_subset_1(B,k3_urysohn1(A)) )
     => m2_subset_1(k8_urysohn1(A,B),k1_numbers,k5_numbers) ) )).

fof(dt_k9_urysohn1,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),u1_struct_0(k3_topmetr))
        & m1_relset_1(B,u1_struct_0(A),u1_struct_0(k3_topmetr))
        & m1_subset_1(C,u1_struct_0(A)) )
     => m1_subset_1(k9_urysohn1(A,B,C),k1_numbers) ) )).

fof(redefinition_k9_urysohn1,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),u1_struct_0(k3_topmetr))
        & m1_relset_1(B,u1_struct_0(A),u1_struct_0(k3_topmetr))
        & m1_subset_1(C,u1_struct_0(A)) )
     => k9_urysohn1(A,B,C) = k1_funct_1(B,C) ) )).
%------------------------------------------------------------------------------