SET007 Axioms: SET007+684.ax


%------------------------------------------------------------------------------
% File     : SET007+684 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : The Urysohn Lemma
% 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    : urysohn3 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   49 (   1 unit)
%            Number of atoms       :  598 (  63 equality)
%            Maximal formula depth :   24 (  14 average)
%            Number of connectives :  621 (  72 ~  ;  17  |; 287  &)
%                                         (   8 <=>; 237 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   28 (   1 propositional; 0-5 arity)
%            Number of functors    :   37 (  13 constant; 0-5 arity)
%            Number of variables   :  238 (   0 singleton; 230 !;   8 ?)
%            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(cc1_urysohn3,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & m1_subset_1(A,k1_zfmisc_1(k6_supinf_1)) )
     => ! [B] :
          ( m1_subset_1(B,A)
         => ( v1_xcmplx_0(B)
            & v1_xreal_0(B) ) ) ) )).

fof(t1_urysohn3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & v5_compts_1(A)
        & l1_pre_topc(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 = k6_measure5
                  | ! [D] :
                      ( m2_subset_1(D,k1_numbers,k5_numbers)
                     => ? [E] :
                          ( v1_funct_1(E)
                          & v1_funct_2(E,k3_urysohn1(D),k1_pcomps_1(u1_struct_0(A)))
                          & m2_relset_1(E,k3_urysohn1(D),k1_pcomps_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)) ) ) ) ) ) ) ) ) ) ) ) )).

fof(d1_urysohn3,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)))
             => ! [D] :
                  ( m2_subset_1(D,k1_numbers,k5_numbers)
                 => ( ( v5_compts_1(A)
                      & v4_pre_topc(B,A)
                      & v4_pre_topc(C,A)
                      & r1_xboole_0(B,C) )
                   => ( B = k6_measure5
                      | ! [E] :
                          ( ( v1_funct_1(E)
                            & v1_funct_2(E,k3_urysohn1(D),k1_pcomps_1(u1_struct_0(A)))
                            & m2_relset_1(E,k3_urysohn1(D),k1_pcomps_1(u1_struct_0(A))) )
                         => ( m1_urysohn3(E,A,B,C,D)
                          <=> ( 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)) ) ) ) ) ) ) ) ) ) ) ) ) ) )).

fof(t2_urysohn3,axiom,(
    $true )).

fof(t3_urysohn3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & v5_compts_1(A)
        & l1_pre_topc(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 = k6_measure5
                  | ! [D] :
                      ( m2_subset_1(D,k1_numbers,k5_numbers)
                     => ! [E] :
                          ( m1_urysohn3(E,A,B,C,D)
                         => ? [F] :
                              ( m1_urysohn3(F,A,B,C,k1_nat_1(D,np__1))
                              & ! [G] :
                                  ( m2_subset_1(G,k1_numbers,k3_urysohn1(k1_nat_1(D,np__1)))
                                 => ( r2_hidden(G,k3_urysohn1(D))
                                   => k6_urysohn1(A,k3_urysohn1(k1_nat_1(D,np__1)),F,G) = k1_funct_1(E,G) ) ) ) ) ) ) ) ) ) ) )).

fof(t4_urysohn3,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)))
             => ! [D] :
                  ( m2_subset_1(D,k1_numbers,k5_numbers)
                 => ! [E] :
                      ( m1_urysohn3(E,A,B,C,D)
                     => m1_subset_1(E,k4_partfun1(k4_urysohn1,k1_pcomps_1(u1_struct_0(A)))) ) ) ) ) ) )).

fof(t5_urysohn3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & v5_compts_1(A)
        & l1_pre_topc(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))) )
             => ~ ( B != k6_measure5
                  & r1_xboole_0(B,C)
                  & ! [D] :
                      ( m1_seqfunc(D,k4_urysohn1,k1_pcomps_1(u1_struct_0(A)))
                     => ~ ! [E] :
                            ( m2_subset_1(E,k1_numbers,k5_numbers)
                           => ( m1_urysohn3(k1_seqfunc(k4_urysohn1,k1_pcomps_1(u1_struct_0(A)),D,E),A,B,C,E)
                              & ! [F] :
                                  ( m1_subset_1(F,k1_relat_1(k1_seqfunc(k4_urysohn1,k1_pcomps_1(u1_struct_0(A)),D,E)))
                                 => k1_funct_1(k1_seqfunc(k4_urysohn1,k1_pcomps_1(u1_struct_0(A)),D,E),F) = k1_funct_1(k1_seqfunc(k4_urysohn1,k1_pcomps_1(u1_struct_0(A)),D,k1_nat_1(E,np__1)),F) ) ) ) ) ) ) ) ) )).

fof(d2_urysohn3,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)
                  & v4_pre_topc(B,A)
                  & v4_pre_topc(C,A)
                  & r1_xboole_0(B,C) )
               => ( B = k6_measure5
                  | ! [D] :
                      ( m1_seqfunc(D,k4_urysohn1,k1_pcomps_1(u1_struct_0(A)))
                     => ( m2_urysohn3(D,A,B,C)
                      <=> ! [E] :
                            ( m2_subset_1(E,k1_numbers,k5_numbers)
                           => ( m1_urysohn3(k1_seqfunc(k4_urysohn1,k1_pcomps_1(u1_struct_0(A)),D,E),A,B,C,E)
                              & ! [F] :
                                  ( m1_subset_1(F,k1_relat_1(k1_seqfunc(k4_urysohn1,k1_pcomps_1(u1_struct_0(A)),D,E)))
                                 => k1_funct_1(k1_seqfunc(k4_urysohn1,k1_pcomps_1(u1_struct_0(A)),D,E),F) = k1_funct_1(k1_seqfunc(k4_urysohn1,k1_pcomps_1(u1_struct_0(A)),D,k1_nat_1(E,np__1)),F) ) ) ) ) ) ) ) ) ) ) )).

fof(d3_urysohn3,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ( r2_hidden(A,k4_urysohn1)
       => ! [B] :
            ( m2_subset_1(B,k1_numbers,k5_numbers)
           => ( B = k3_urysohn3(A)
            <=> ( ( r2_hidden(A,k3_urysohn1(np__0))
                 => B = np__0 )
                & ( B = np__0
                 => r2_hidden(A,k3_urysohn1(np__0)) )
                & ! [C] :
                    ( m2_subset_1(C,k1_numbers,k5_numbers)
                   => ( r2_hidden(A,k3_urysohn1(k1_nat_1(C,np__1)))
                     => ( r2_hidden(A,k3_urysohn1(C))
                        | B = k1_nat_1(C,np__1) ) ) ) ) ) ) ) ) )).

fof(t6_urysohn3,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ( r2_hidden(A,k4_urysohn1)
       => r2_hidden(A,k3_urysohn1(k3_urysohn3(A))) ) ) )).

fof(t7_urysohn3,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ( r2_hidden(A,k4_urysohn1)
       => ! [B] :
            ( m2_subset_1(B,k1_numbers,k5_numbers)
           => ( r1_xreal_0(k3_urysohn3(A),B)
             => r2_hidden(A,k3_urysohn1(B)) ) ) ) ) )).

fof(t8_urysohn3,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ( r2_hidden(A,k3_urysohn1(B))
           => r1_xreal_0(k3_urysohn3(A),B) ) ) ) )).

fof(t9_urysohn3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & v5_compts_1(A)
        & l1_pre_topc(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 = k6_measure5
                  | ! [D] :
                      ( m2_urysohn3(D,A,B,C)
                     => ! [E] :
                          ( m1_subset_1(E,k1_numbers)
                         => ( r2_hidden(E,k4_urysohn1)
                           => ! [F] :
                                ( m2_subset_1(F,k1_numbers,k5_numbers)
                               => k1_funct_1(k1_seqfunc(k4_urysohn1,k1_pcomps_1(u1_struct_0(A)),D,k3_urysohn3(E)),E) = k1_funct_1(k1_seqfunc(k4_urysohn1,k1_pcomps_1(u1_struct_0(A)),D,k1_nat_1(k3_urysohn3(E),F)),E) ) ) ) ) ) ) ) ) ) )).

fof(t10_urysohn3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & v5_compts_1(A)
        & l1_pre_topc(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 = k6_measure5
                  | ! [D] :
                      ( m2_urysohn3(D,A,B,C)
                     => ! [E] :
                          ( m1_subset_1(E,k1_numbers)
                         => ~ ( r2_hidden(E,k4_urysohn1)
                              & ! [F] :
                                  ( m1_subset_1(F,k1_zfmisc_1(u1_struct_0(A)))
                                 => ? [G] :
                                      ( m2_subset_1(G,k1_numbers,k5_numbers)
                                      & r2_hidden(E,k3_urysohn1(G))
                                      & F != k1_funct_1(k1_seqfunc(k4_urysohn1,k1_pcomps_1(u1_struct_0(A)),D,G),E) ) ) ) ) ) ) ) ) ) ) )).

fof(t11_urysohn3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & v5_compts_1(A)
        & l1_pre_topc(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 = k6_measure5
                  | ! [D] :
                      ( m2_urysohn3(D,A,B,C)
                     => ? [E] :
                          ( v1_funct_1(E)
                          & v1_funct_2(E,k5_urysohn1,k1_pcomps_1(u1_struct_0(A)))
                          & m2_relset_1(E,k5_urysohn1,k1_pcomps_1(u1_struct_0(A)))
                          & ! [F] :
                              ( m1_subset_1(F,k1_numbers)
                             => ( r2_hidden(F,k5_urysohn1)
                               => ( ( r2_hidden(F,k1_urysohn1)
                                   => k1_funct_1(E,F) = k6_measure5 )
                                  & ( r2_hidden(F,k2_urysohn1)
                                   => k1_funct_1(E,F) = u1_struct_0(A) )
                                  & ( r2_hidden(F,k4_urysohn1)
                                   => ! [G] :
                                        ( m2_subset_1(G,k1_numbers,k5_numbers)
                                       => ( r2_hidden(F,k3_urysohn1(G))
                                         => k1_funct_1(E,F) = k1_funct_1(k1_seqfunc(k4_urysohn1,k1_pcomps_1(u1_struct_0(A)),D,G),F) ) ) ) ) ) ) ) ) ) ) ) ) ) )).

fof(d4_urysohn3,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)
                  & v4_pre_topc(B,A)
                  & v4_pre_topc(C,A)
                  & r1_xboole_0(B,C) )
               => ( B = k6_measure5
                  | ! [D] :
                      ( m2_urysohn3(D,A,B,C)
                     => ! [E] :
                          ( ( v1_funct_1(E)
                            & v1_funct_2(E,k5_urysohn1,k1_pcomps_1(u1_struct_0(A)))
                            & m2_relset_1(E,k5_urysohn1,k1_pcomps_1(u1_struct_0(A))) )
                         => ( E = k4_urysohn3(A,B,C,D)
                          <=> ! [F] :
                                ( m1_subset_1(F,k1_numbers)
                               => ( r2_hidden(F,k5_urysohn1)
                                 => ( ( r2_hidden(F,k1_urysohn1)
                                     => k1_funct_1(E,F) = k6_measure5 )
                                    & ( r2_hidden(F,k2_urysohn1)
                                     => k1_funct_1(E,F) = u1_struct_0(A) )
                                    & ( r2_hidden(F,k4_urysohn1)
                                     => ! [G] :
                                          ( m2_subset_1(G,k1_numbers,k5_numbers)
                                         => ( r2_hidden(F,k3_urysohn1(G))
                                           => k1_funct_1(E,F) = k1_funct_1(k1_seqfunc(k4_urysohn1,k1_pcomps_1(u1_struct_0(A)),D,G),F) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )).

fof(t12_urysohn3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & v5_compts_1(A)
        & l1_pre_topc(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 = k6_measure5
                  | ! [D] :
                      ( m2_urysohn3(D,A,B,C)
                     => ! [E] :
                          ( m1_subset_1(E,k1_numbers)
                         => ! [F] :
                              ( m1_subset_1(F,k1_zfmisc_1(u1_struct_0(A)))
                             => ( ( F = k1_funct_1(k4_urysohn3(A,B,C,D),E)
                                  & r2_hidden(E,k5_urysohn1) )
                               => v3_pre_topc(F,A) ) ) ) ) ) ) ) ) ) )).

fof(t13_urysohn3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & v5_compts_1(A)
        & l1_pre_topc(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 = k6_measure5
                  | ! [D] :
                      ( m2_urysohn3(D,A,B,C)
                     => ! [E] :
                          ( m1_subset_1(E,k1_numbers)
                         => ! [F] :
                              ( m1_subset_1(F,k1_numbers)
                             => ( ( r2_hidden(E,k5_urysohn1)
                                  & r2_hidden(F,k5_urysohn1) )
                               => ( r1_xreal_0(F,E)
                                  | ! [G] :
                                      ( m1_subset_1(G,k1_zfmisc_1(u1_struct_0(A)))
                                     => ( G = k1_funct_1(k4_urysohn3(A,B,C,D),E)
                                       => r1_tarski(k6_pre_topc(A,G),k1_funct_1(k4_urysohn3(A,B,C,D),F)) ) ) ) ) ) ) ) ) ) ) ) ) )).

fof(t14_urysohn3,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)))
             => ! [D] :
                  ( m2_urysohn3(D,A,B,C)
                 => ! [E] :
                      ( m1_subset_1(E,u1_struct_0(A))
                     => ? [F] :
                          ( m1_subset_1(F,k1_zfmisc_1(k3_supinf_1))
                          & ! [G] :
                              ( r2_hidden(G,F)
                            <=> ( r2_hidden(G,k4_urysohn1)
                                & ! [H] :
                                    ( m1_subset_1(H,k1_numbers)
                                   => ~ ( H = G
                                        & r2_hidden(E,k1_funct_1(k4_urysohn3(A,B,C,D),H)) ) ) ) ) ) ) ) ) ) ) )).

fof(d5_urysohn3,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)))
             => ! [D] :
                  ( m2_urysohn3(D,A,B,C)
                 => ! [E] :
                      ( m1_subset_1(E,u1_struct_0(A))
                     => ! [F] :
                          ( m1_subset_1(F,k1_zfmisc_1(k3_supinf_1))
                         => ( F = k6_urysohn3(A,B,C,D,E)
                          <=> ! [G] :
                                ( r2_hidden(G,F)
                              <=> ( r2_hidden(G,k4_urysohn1)
                                  & ! [H] :
                                      ( m1_subset_1(H,k1_numbers)
                                     => ~ ( H = G
                                          & r2_hidden(E,k1_funct_1(k4_urysohn3(A,B,C,D),H)) ) ) ) ) ) ) ) ) ) ) ) )).

fof(t15_urysohn3,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)))
             => ! [D] :
                  ( m2_urysohn3(D,A,B,C)
                 => ! [E] :
                      ( m1_subset_1(E,u1_struct_0(A))
                     => r1_tarski(k6_urysohn3(A,B,C,D,E),k4_urysohn1) ) ) ) ) ) )).

fof(t16_urysohn3,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)))
             => ! [D] :
                  ( m2_urysohn3(D,A,B,C)
                 => ? [E] :
                      ( v1_funct_1(E)
                      & v1_funct_2(E,u1_struct_0(A),u1_struct_0(k3_topmetr))
                      & m2_relset_1(E,u1_struct_0(A),u1_struct_0(k3_topmetr))
                      & ! [F] :
                          ( m1_subset_1(F,u1_struct_0(A))
                         => ( ( k6_urysohn3(A,B,C,D,F) = k6_measure5
                             => k7_urysohn3(A,k3_topmetr,E,F) = np__0 )
                            & ! [G] :
                                ( ( ~ v1_xboole_0(G)
                                  & m1_subset_1(G,k1_zfmisc_1(k3_supinf_1)) )
                               => ( G = k6_urysohn3(A,B,C,D,F)
                                 => k7_urysohn3(A,k3_topmetr,E,F) = k9_supinf_1(G) ) ) ) ) ) ) ) ) ) )).

fof(d6_urysohn3,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)))
             => ! [D] :
                  ( m2_urysohn3(D,A,B,C)
                 => ! [E] :
                      ( ( v1_funct_1(E)
                        & v1_funct_2(E,u1_struct_0(A),u1_struct_0(k3_topmetr))
                        & m2_relset_1(E,u1_struct_0(A),u1_struct_0(k3_topmetr)) )
                     => ( E = k8_urysohn3(A,B,C,D)
                      <=> ! [F] :
                            ( m1_subset_1(F,u1_struct_0(A))
                           => ( ( k6_urysohn3(A,B,C,D,F) = k6_measure5
                               => k7_urysohn3(A,k3_topmetr,E,F) = np__0 )
                              & ! [G] :
                                  ( ( ~ v1_xboole_0(G)
                                    & m1_subset_1(G,k1_zfmisc_1(k3_supinf_1)) )
                                 => ( G = k6_urysohn3(A,B,C,D,F)
                                   => k7_urysohn3(A,k3_topmetr,E,F) = k9_supinf_1(G) ) ) ) ) ) ) ) ) ) ) )).

fof(t17_urysohn3,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)))
             => ! [D] :
                  ( m2_urysohn3(D,A,B,C)
                 => ! [E] :
                      ( m1_subset_1(E,u1_struct_0(A))
                     => ! [F] :
                          ( ( ~ v1_xboole_0(F)
                            & m1_subset_1(F,k1_zfmisc_1(k3_supinf_1)) )
                         => ( F = k6_urysohn3(A,B,C,D,E)
                           => ! [G] :
                                ( m1_subset_1(G,k3_supinf_1)
                               => ( G = np__1
                                 => ( r1_supinf_1(k1_supinf_2,k9_supinf_1(F))
                                    & r1_supinf_1(k9_supinf_1(F),G) ) ) ) ) ) ) ) ) ) ) )).

fof(t18_urysohn3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & v5_compts_1(A)
        & l1_pre_topc(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 = k6_measure5
                  | ! [D] :
                      ( m2_urysohn3(D,A,B,C)
                     => ! [E] :
                          ( m2_subset_1(E,k1_numbers,k5_urysohn1)
                         => ! [F] :
                              ( m1_subset_1(F,u1_struct_0(A))
                             => ( ~ r1_xreal_0(E,k9_urysohn3(A,k8_urysohn3(A,B,C,D),F))
                               => r2_hidden(F,k5_urysohn3(k5_urysohn1,A,k4_urysohn3(A,B,C,D),E)) ) ) ) ) ) ) ) ) ) )).

fof(t19_urysohn3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & v5_compts_1(A)
        & l1_pre_topc(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 = k6_measure5
                  | ! [D] :
                      ( m2_urysohn3(D,A,B,C)
                     => ! [E] :
                          ( m1_subset_1(E,k1_numbers)
                         => ( r2_hidden(E,k4_subset_1(k1_numbers,k4_urysohn1,k2_urysohn1))
                           => ( r1_xreal_0(E,np__0)
                              | ! [F] :
                                  ( m1_subset_1(F,u1_struct_0(A))
                                 => ( r2_hidden(F,k1_funct_1(k4_urysohn3(A,B,C,D),E))
                                   => r1_xreal_0(k9_urysohn3(A,k8_urysohn3(A,B,C,D),F),E) ) ) ) ) ) ) ) ) ) ) ) )).

fof(t20_urysohn3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & v5_compts_1(A)
        & l1_pre_topc(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 = k6_measure5
                  | ! [D] :
                      ( m2_urysohn3(D,A,B,C)
                     => ! [E] :
                          ( m2_subset_1(E,k1_numbers,k5_urysohn1)
                         => ( ~ r1_xreal_0(E,np__0)
                           => ! [F] :
                                ( m1_subset_1(F,u1_struct_0(A))
                               => ~ ( ~ r1_xreal_0(k9_urysohn3(A,k8_urysohn3(A,B,C,D),F),E)
                                    & r2_hidden(F,k5_urysohn3(k5_urysohn1,A,k4_urysohn3(A,B,C,D),E)) ) ) ) ) ) ) ) ) ) ) )).

fof(t21_urysohn3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & v5_compts_1(A)
        & l1_pre_topc(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 = k6_measure5
                  | ! [D] :
                      ( m2_urysohn3(D,A,B,C)
                     => ( v5_pre_topc(k8_urysohn3(A,B,C,D),A,k3_topmetr)
                        & ! [E] :
                            ( m1_subset_1(E,u1_struct_0(A))
                           => ( r1_xreal_0(np__0,k9_urysohn3(A,k8_urysohn3(A,B,C,D),E))
                              & r1_xreal_0(k9_urysohn3(A,k8_urysohn3(A,B,C,D),E),np__1)
                              & ( r2_hidden(E,B)
                               => k9_urysohn3(A,k8_urysohn3(A,B,C,D),E) = np__0 )
                              & ( r2_hidden(E,C)
                               => k9_urysohn3(A,k8_urysohn3(A,B,C,D),E) = np__1 ) ) ) ) ) ) ) ) ) ) )).

fof(t22_urysohn3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & v5_compts_1(A)
        & l1_pre_topc(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))) )
             => ~ ( B != k6_measure5
                  & r1_xboole_0(B,C)
                  & ! [D] :
                      ( ( v1_funct_1(D)
                        & v1_funct_2(D,u1_struct_0(A),u1_struct_0(k3_topmetr))
                        & m2_relset_1(D,u1_struct_0(A),u1_struct_0(k3_topmetr)) )
                     => ~ ( v5_pre_topc(D,A,k3_topmetr)
                          & ! [E] :
                              ( m1_subset_1(E,u1_struct_0(A))
                             => ( r1_xreal_0(np__0,k9_urysohn3(A,D,E))
                                & r1_xreal_0(k9_urysohn3(A,D,E),np__1)
                                & ( r2_hidden(E,B)
                                 => k9_urysohn3(A,D,E) = np__0 )
                                & ( r2_hidden(E,C)
                                 => k9_urysohn3(A,D,E) = np__1 ) ) ) ) ) ) ) ) ) )).

fof(t23_urysohn3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & v5_compts_1(A)
        & l1_pre_topc(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)
                  & ! [D] :
                      ( ( v1_funct_1(D)
                        & v1_funct_2(D,u1_struct_0(A),u1_struct_0(k3_topmetr))
                        & m2_relset_1(D,u1_struct_0(A),u1_struct_0(k3_topmetr)) )
                     => ~ ( v5_pre_topc(D,A,k3_topmetr)
                          & ! [E] :
                              ( m1_subset_1(E,u1_struct_0(A))
                             => ( r1_xreal_0(np__0,k9_urysohn3(A,D,E))
                                & r1_xreal_0(k9_urysohn3(A,D,E),np__1)
                                & ( r2_hidden(E,B)
                                 => k9_urysohn3(A,D,E) = np__0 )
                                & ( r2_hidden(E,C)
                                 => k9_urysohn3(A,D,E) = np__1 ) ) ) ) ) ) ) ) ) )).

fof(t24_urysohn3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & v2_compts_1(A)
        & v3_compts_1(A)
        & l1_pre_topc(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)
                  & ! [D] :
                      ( ( v1_funct_1(D)
                        & v1_funct_2(D,u1_struct_0(A),u1_struct_0(k3_topmetr))
                        & m2_relset_1(D,u1_struct_0(A),u1_struct_0(k3_topmetr)) )
                     => ~ ( v5_pre_topc(D,A,k3_topmetr)
                          & ! [E] :
                              ( m1_subset_1(E,u1_struct_0(A))
                             => ( r1_xreal_0(np__0,k9_urysohn3(A,D,E))
                                & r1_xreal_0(k9_urysohn3(A,D,E),np__1)
                                & ( r2_hidden(E,B)
                                 => k9_urysohn3(A,D,E) = np__0 )
                                & ( r2_hidden(E,C)
                                 => k9_urysohn3(A,D,E) = np__1 ) ) ) ) ) ) ) ) ) )).

fof(dt_m1_urysohn3,axiom,(
    ! [A,B,C,D] :
      ( ( ~ 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)))
        & m1_subset_1(D,k5_numbers) )
     => ! [E] :
          ( m1_urysohn3(E,A,B,C,D)
         => ( v1_funct_1(E)
            & v1_funct_2(E,k3_urysohn1(D),k1_pcomps_1(u1_struct_0(A)))
            & m2_relset_1(E,k3_urysohn1(D),k1_pcomps_1(u1_struct_0(A))) ) ) ) )).

fof(existence_m1_urysohn3,axiom,(
    ! [A,B,C,D] :
      ( ( ~ 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)))
        & m1_subset_1(D,k5_numbers) )
     => ? [E] : m1_urysohn3(E,A,B,C,D) ) )).

fof(dt_m2_urysohn3,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] :
          ( m2_urysohn3(D,A,B,C)
         => m1_seqfunc(D,k4_urysohn1,k1_pcomps_1(u1_struct_0(A))) ) ) )).

fof(existence_m2_urysohn3,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] : m2_urysohn3(D,A,B,C) ) )).

fof(dt_k1_urysohn3,axiom,(
    ! [A,B,C,D] :
      ( ( ~ v1_xboole_0(A)
        & ~ v1_xboole_0(B)
        & v1_funct_1(C)
        & v1_funct_2(C,k5_numbers,k4_partfun1(A,B))
        & m1_relset_1(C,k5_numbers,k4_partfun1(A,B))
        & m1_subset_1(D,k5_numbers) )
     => ( v1_funct_1(k1_urysohn3(A,B,C,D))
        & m2_relset_1(k1_urysohn3(A,B,C,D),A,B) ) ) )).

fof(redefinition_k1_urysohn3,axiom,(
    ! [A,B,C,D] :
      ( ( ~ v1_xboole_0(A)
        & ~ v1_xboole_0(B)
        & v1_funct_1(C)
        & v1_funct_2(C,k5_numbers,k4_partfun1(A,B))
        & m1_relset_1(C,k5_numbers,k4_partfun1(A,B))
        & m1_subset_1(D,k5_numbers) )
     => k1_urysohn3(A,B,C,D) = k1_funct_1(C,D) ) )).

fof(dt_k2_urysohn3,axiom,(
    ! [A,B,C,D] :
      ( ( ~ v1_xboole_0(A)
        & ~ v1_xboole_0(B)
        & v1_funct_1(C)
        & v1_funct_2(C,k5_numbers,k4_partfun1(A,B))
        & m1_relset_1(C,k5_numbers,k4_partfun1(A,B))
        & m1_subset_1(D,k5_numbers) )
     => m1_subset_1(k2_urysohn3(A,B,C,D),k4_partfun1(A,B)) ) )).

fof(redefinition_k2_urysohn3,axiom,(
    ! [A,B,C,D] :
      ( ( ~ v1_xboole_0(A)
        & ~ v1_xboole_0(B)
        & v1_funct_1(C)
        & v1_funct_2(C,k5_numbers,k4_partfun1(A,B))
        & m1_relset_1(C,k5_numbers,k4_partfun1(A,B))
        & m1_subset_1(D,k5_numbers) )
     => k2_urysohn3(A,B,C,D) = k1_funct_1(C,D) ) )).

fof(dt_k3_urysohn3,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => m2_subset_1(k3_urysohn3(A),k1_numbers,k5_numbers) ) )).

fof(dt_k4_urysohn3,axiom,(
    ! [A,B,C,D] :
      ( ( ~ 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)))
        & m2_urysohn3(D,A,B,C) )
     => ( v1_funct_1(k4_urysohn3(A,B,C,D))
        & v1_funct_2(k4_urysohn3(A,B,C,D),k5_urysohn1,k1_pcomps_1(u1_struct_0(A)))
        & m2_relset_1(k4_urysohn3(A,B,C,D),k5_urysohn1,k1_pcomps_1(u1_struct_0(A))) ) ) )).

fof(dt_k5_urysohn3,axiom,(
    ! [A,B,C,D] :
      ( ( ~ v1_xboole_0(A)
        & v2_pre_topc(B)
        & l1_pre_topc(B)
        & v1_funct_1(C)
        & v1_funct_2(C,A,k1_pcomps_1(u1_struct_0(B)))
        & m1_relset_1(C,A,k1_pcomps_1(u1_struct_0(B)))
        & m1_subset_1(D,A) )
     => m1_subset_1(k5_urysohn3(A,B,C,D),k1_zfmisc_1(u1_struct_0(B))) ) )).

fof(redefinition_k5_urysohn3,axiom,(
    ! [A,B,C,D] :
      ( ( ~ v1_xboole_0(A)
        & v2_pre_topc(B)
        & l1_pre_topc(B)
        & v1_funct_1(C)
        & v1_funct_2(C,A,k1_pcomps_1(u1_struct_0(B)))
        & m1_relset_1(C,A,k1_pcomps_1(u1_struct_0(B)))
        & m1_subset_1(D,A) )
     => k5_urysohn3(A,B,C,D) = k1_funct_1(C,D) ) )).

fof(dt_k6_urysohn3,axiom,(
    ! [A,B,C,D,E] :
      ( ( ~ 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)))
        & m2_urysohn3(D,A,B,C)
        & m1_subset_1(E,u1_struct_0(A)) )
     => m1_subset_1(k6_urysohn3(A,B,C,D,E),k1_zfmisc_1(k3_supinf_1)) ) )).

fof(dt_k7_urysohn3,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),u1_struct_0(B))
        & m1_relset_1(C,u1_struct_0(A),u1_struct_0(B))
        & m1_subset_1(D,u1_struct_0(A)) )
     => m1_subset_1(k7_urysohn3(A,B,C,D),u1_struct_0(B)) ) )).

fof(redefinition_k7_urysohn3,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),u1_struct_0(B))
        & m1_relset_1(C,u1_struct_0(A),u1_struct_0(B))
        & m1_subset_1(D,u1_struct_0(A)) )
     => k7_urysohn3(A,B,C,D) = k1_funct_1(C,D) ) )).

fof(dt_k8_urysohn3,axiom,(
    ! [A,B,C,D] :
      ( ( ~ 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)))
        & m2_urysohn3(D,A,B,C) )
     => ( v1_funct_1(k8_urysohn3(A,B,C,D))
        & v1_funct_2(k8_urysohn3(A,B,C,D),u1_struct_0(A),u1_struct_0(k3_topmetr))
        & m2_relset_1(k8_urysohn3(A,B,C,D),u1_struct_0(A),u1_struct_0(k3_topmetr)) ) ) )).

fof(dt_k9_urysohn3,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_urysohn3(A,B,C),k1_numbers) ) )).

fof(redefinition_k9_urysohn3,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_urysohn3(A,B,C) = k1_funct_1(B,C) ) )).
%------------------------------------------------------------------------------