SET007 Axioms: SET007+320.ax


%------------------------------------------------------------------------------
% File     : SET007+320 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : The Topological Space E^2_T
% 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    : topreal1 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   69 (  16 unit)
%            Number of atoms       :  335 (  66 equality)
%            Maximal formula depth :   16 (   7 average)
%            Number of connectives :  289 (  23 ~  ;   3  |; 122  &)
%                                         (  17 <=>; 124 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   30 (   1 propositional; 0-4 arity)
%            Number of functors    :   44 (  17 constant; 0-4 arity)
%            Number of variables   :  147 (   0 singleton; 131 !;  16 ?)
%            Maximal term depth    :    5 (   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(fc1_topreal1,axiom,(
    ! [A,B,C] :
      ( ( m1_subset_1(A,k5_numbers)
        & m1_subset_1(B,u1_struct_0(k15_euclid(A)))
        & m1_subset_1(C,u1_struct_0(k15_euclid(A))) )
     => ~ v1_xboole_0(k1_topreal1(A,B,C)) ) )).

fof(fc2_topreal1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k5_numbers)
     => ( ~ v3_struct_0(k15_euclid(A))
        & v1_pre_topc(k15_euclid(A))
        & v2_pre_topc(k15_euclid(A))
        & v3_compts_1(k15_euclid(A)) ) ) )).

fof(fc3_topreal1,axiom,(
    ~ v1_xboole_0(k2_topreal1) )).

fof(cc1_topreal1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
     => ( v5_topreal1(A)
       => ~ v1_xboole_0(A) ) ) )).

fof(d1_topreal1,axiom,(
    $true )).

fof(d2_topreal1,axiom,(
    ! [A] :
      ( ( v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ! [D] :
                  ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
                 => ( r1_topreal1(A,B,C,D)
                  <=> ? [E] :
                        ( v1_funct_1(E)
                        & v1_funct_2(E,u1_struct_0(k5_topmetr),u1_struct_0(k3_pre_topc(A,D)))
                        & m2_relset_1(E,u1_struct_0(k5_topmetr),u1_struct_0(k3_pre_topc(A,D)))
                        & v3_tops_2(E,k5_topmetr,k3_pre_topc(A,D))
                        & k1_funct_1(E,np__0) = B
                        & k1_funct_1(E,np__1) = C ) ) ) ) ) ) )).

fof(t1_topreal1,axiom,(
    $true )).

fof(t2_topreal1,axiom,(
    $true )).

fof(t3_topreal1,axiom,(
    $true )).

fof(t4_topreal1,axiom,(
    ! [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))
             => ! [D] :
                  ( m1_subset_1(D,u1_struct_0(A))
                 => ( r1_topreal1(A,C,D,B)
                   => ( r2_hidden(C,B)
                      & r2_hidden(D,B) ) ) ) ) ) ) )).

fof(t5_topreal1,axiom,(
    ! [A] :
      ( ( v2_pre_topc(A)
        & v3_compts_1(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] :
                  ( m1_subset_1(D,u1_struct_0(A))
                 => ! [E] :
                      ( m1_subset_1(E,u1_struct_0(A))
                     => ! [F] :
                          ( m1_subset_1(F,u1_struct_0(A))
                         => ( ( r1_topreal1(A,D,E,B)
                              & r1_topreal1(A,E,F,C)
                              & k5_subset_1(u1_struct_0(A),B,C) = k1_tarski(E) )
                           => r1_topreal1(A,D,F,k4_subset_1(u1_struct_0(A),B,C)) ) ) ) ) ) ) ) )).

fof(d4_topreal1,axiom,(
    k2_topreal1 = k4_subset_1(u1_struct_0(k15_euclid(np__2)),k4_subset_1(u1_struct_0(k15_euclid(np__2)),k1_topreal1(np__2,k23_euclid(np__0,np__0),k23_euclid(np__0,np__1)),k1_topreal1(np__2,k23_euclid(np__0,np__1),k23_euclid(np__1,np__1))),k4_subset_1(u1_struct_0(k15_euclid(np__2)),k1_topreal1(np__2,k23_euclid(np__1,np__1),k23_euclid(np__1,np__0)),k1_topreal1(np__2,k23_euclid(np__1,np__0),k23_euclid(np__0,np__0)))) )).

fof(t6_topreal1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(k15_euclid(A)))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(k15_euclid(A)))
             => ( r2_hidden(B,k1_topreal1(A,B,C))
                & r2_hidden(C,k1_topreal1(A,B,C)) ) ) ) ) )).

fof(t7_topreal1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(k15_euclid(A)))
         => k1_topreal1(A,B,B) = k1_struct_0(k15_euclid(A),B) ) ) )).

fof(t8_topreal1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(k15_euclid(A)))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(k15_euclid(A)))
             => k1_topreal1(A,B,C) = k1_topreal1(A,C,B) ) ) ) )).

fof(t9_topreal1,axiom,(
    ! [A] :
      ( m1_subset_1(A,u1_struct_0(k15_euclid(np__2)))
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(k15_euclid(np__2)))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(k15_euclid(np__2)))
             => ( ( r1_xreal_0(k21_euclid(A),k21_euclid(B))
                  & r2_hidden(C,k3_topreal1(np__2,A,B)) )
               => ( r1_xreal_0(k21_euclid(A),k21_euclid(C))
                  & r1_xreal_0(k21_euclid(C),k21_euclid(B)) ) ) ) ) ) )).

fof(t10_topreal1,axiom,(
    ! [A] :
      ( m1_subset_1(A,u1_struct_0(k15_euclid(np__2)))
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(k15_euclid(np__2)))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(k15_euclid(np__2)))
             => ( ( r1_xreal_0(k22_euclid(A),k22_euclid(B))
                  & r2_hidden(C,k3_topreal1(np__2,A,B)) )
               => ( r1_xreal_0(k22_euclid(A),k22_euclid(C))
                  & r1_xreal_0(k22_euclid(C),k22_euclid(B)) ) ) ) ) ) )).

fof(t11_topreal1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(k15_euclid(A)))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(k15_euclid(A)))
             => ! [D] :
                  ( m1_subset_1(D,u1_struct_0(k15_euclid(A)))
                 => ( r2_hidden(B,k3_topreal1(A,C,D))
                   => k3_topreal1(A,C,D) = k4_subset_1(u1_struct_0(k15_euclid(A)),k3_topreal1(A,C,B),k3_topreal1(A,B,D)) ) ) ) ) ) )).

fof(t12_topreal1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(k15_euclid(A)))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(k15_euclid(A)))
             => ! [D] :
                  ( m1_subset_1(D,u1_struct_0(k15_euclid(A)))
                 => ! [E] :
                      ( m1_subset_1(E,u1_struct_0(k15_euclid(A)))
                     => ( ( r2_hidden(D,k3_topreal1(A,B,C))
                          & r2_hidden(E,k3_topreal1(A,B,C)) )
                       => r1_tarski(k3_topreal1(A,D,E),k3_topreal1(A,B,C)) ) ) ) ) ) ) )).

fof(t13_topreal1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(k15_euclid(A)))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(k15_euclid(A)))
             => ! [D] :
                  ( m1_subset_1(D,u1_struct_0(k15_euclid(A)))
                 => ! [E] :
                      ( m1_subset_1(E,u1_struct_0(k15_euclid(A)))
                     => ( ( r2_hidden(B,k3_topreal1(A,D,E))
                          & r2_hidden(C,k3_topreal1(A,D,E)) )
                       => k3_topreal1(A,D,E) = k4_subset_1(u1_struct_0(k15_euclid(A)),k4_subset_1(u1_struct_0(k15_euclid(A)),k3_topreal1(A,D,B),k3_topreal1(A,B,C)),k3_topreal1(A,C,E)) ) ) ) ) ) ) )).

fof(t14_topreal1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(k15_euclid(A)))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(k15_euclid(A)))
             => ! [D] :
                  ( m1_subset_1(D,u1_struct_0(k15_euclid(A)))
                 => ( r2_hidden(B,k3_topreal1(A,C,D))
                   => k5_subset_1(u1_struct_0(k15_euclid(A)),k3_topreal1(A,C,B),k3_topreal1(A,B,D)) = k1_struct_0(k15_euclid(A),B) ) ) ) ) ) )).

fof(t15_topreal1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(k15_euclid(A)))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(k15_euclid(A)))
             => ( B != C
               => r1_topreal1(k15_euclid(A),B,C,k3_topreal1(A,B,C)) ) ) ) ) )).

fof(t16_topreal1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(k15_euclid(A)))
             => ! [D] :
                  ( m1_subset_1(D,u1_struct_0(k15_euclid(A)))
                 => ! [E] :
                      ( m1_subset_1(E,u1_struct_0(k15_euclid(A)))
                     => ( ( r1_topreal1(k15_euclid(A),C,D,B)
                          & k5_subset_1(u1_struct_0(k15_euclid(A)),B,k3_topreal1(A,D,E)) = k1_struct_0(k15_euclid(A),D) )
                       => r1_topreal1(k15_euclid(A),C,E,k4_subset_1(u1_struct_0(k15_euclid(A)),B,k3_topreal1(A,D,E))) ) ) ) ) ) ) )).

fof(t17_topreal1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(k15_euclid(A)))
             => ! [D] :
                  ( m1_subset_1(D,u1_struct_0(k15_euclid(A)))
                 => ! [E] :
                      ( m1_subset_1(E,u1_struct_0(k15_euclid(A)))
                     => ( ( r1_topreal1(k15_euclid(A),D,C,B)
                          & k5_subset_1(u1_struct_0(k15_euclid(A)),k3_topreal1(A,E,D),B) = k1_struct_0(k15_euclid(A),D) )
                       => r1_topreal1(k15_euclid(A),E,C,k4_subset_1(u1_struct_0(k15_euclid(A)),k3_topreal1(A,E,D),B)) ) ) ) ) ) ) )).

fof(t18_topreal1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(k15_euclid(A)))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(k15_euclid(A)))
             => ! [D] :
                  ( m1_subset_1(D,u1_struct_0(k15_euclid(A)))
                 => ( k5_subset_1(u1_struct_0(k15_euclid(A)),k3_topreal1(A,B,C),k3_topreal1(A,C,D)) = k1_struct_0(k15_euclid(A),C)
                   => ( ( B = C
                        & C = D )
                      | r1_topreal1(k15_euclid(A),B,D,k4_subset_1(u1_struct_0(k15_euclid(A)),k3_topreal1(A,B,C),k3_topreal1(A,C,D))) ) ) ) ) ) ) )).

fof(t21_topreal1,axiom,(
    k5_subset_1(u1_struct_0(k15_euclid(np__2)),k3_topreal1(np__2,k23_euclid(np__0,np__0),k23_euclid(np__0,np__1)),k3_topreal1(np__2,k23_euclid(np__0,np__1),k23_euclid(np__1,np__1))) = k1_struct_0(k15_euclid(np__2),k23_euclid(np__0,np__1)) )).

fof(t22_topreal1,axiom,(
    k5_subset_1(u1_struct_0(k15_euclid(np__2)),k3_topreal1(np__2,k23_euclid(np__0,np__0),k23_euclid(np__1,np__0)),k3_topreal1(np__2,k23_euclid(np__1,np__0),k23_euclid(np__1,np__1))) = k1_struct_0(k15_euclid(np__2),k23_euclid(np__1,np__0)) )).

fof(t23_topreal1,axiom,(
    k5_subset_1(u1_struct_0(k15_euclid(np__2)),k3_topreal1(np__2,k23_euclid(np__0,np__0),k23_euclid(np__0,np__1)),k3_topreal1(np__2,k23_euclid(np__0,np__0),k23_euclid(np__1,np__0))) = k1_struct_0(k15_euclid(np__2),k23_euclid(np__0,np__0)) )).

fof(t24_topreal1,axiom,(
    k5_subset_1(u1_struct_0(k15_euclid(np__2)),k3_topreal1(np__2,k23_euclid(np__0,np__1),k23_euclid(np__1,np__1)),k3_topreal1(np__2,k23_euclid(np__1,np__0),k23_euclid(np__1,np__1))) = k1_struct_0(k15_euclid(np__2),k23_euclid(np__1,np__1)) )).

fof(t25_topreal1,axiom,(
    r1_subset_1(k3_topreal1(np__2,k23_euclid(np__0,np__0),k23_euclid(np__1,np__0)),k3_topreal1(np__2,k23_euclid(np__0,np__1),k23_euclid(np__1,np__1))) )).

fof(t26_topreal1,axiom,(
    r1_subset_1(k3_topreal1(np__2,k23_euclid(np__0,np__0),k23_euclid(np__0,np__1)),k3_topreal1(np__2,k23_euclid(np__1,np__0),k23_euclid(np__1,np__1))) )).

fof(d5_topreal1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_finseq_1(B,u1_struct_0(k15_euclid(A)))
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => ( ( ( r1_xreal_0(np__1,C)
                    & r1_xreal_0(k1_nat_1(C,np__1),k3_finseq_1(B)) )
                 => k4_topreal1(A,B,C) = k3_topreal1(A,k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(A)),B,C),k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(A)),B,k1_nat_1(C,np__1))) )
                & ( ~ ( r1_xreal_0(np__1,C)
                      & r1_xreal_0(k1_nat_1(C,np__1),k3_finseq_1(B)) )
                 => k4_topreal1(A,B,C) = k1_xboole_0 ) ) ) ) ) )).

fof(t27_topreal1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ! [C] :
              ( m2_finseq_1(C,u1_struct_0(k15_euclid(A)))
             => ( ( r1_xreal_0(np__1,B)
                  & r1_xreal_0(k1_nat_1(B,np__1),k3_finseq_1(C)) )
               => ( r2_hidden(k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(A)),C,B),k4_topreal1(A,C,B))
                  & r2_hidden(k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(A)),C,k1_nat_1(B,np__1)),k4_topreal1(A,C,B)) ) ) ) ) ) )).

fof(t28_topreal1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_finseq_1(B,u1_struct_0(k15_euclid(A)))
         => ( ( k3_finseq_1(B) = np__0
              | k3_finseq_1(B) = np__1 )
          <=> k5_topreal1(A,B) = k1_xboole_0 ) ) ) )).

fof(t29_topreal1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_finseq_1(B,u1_struct_0(k15_euclid(A)))
         => ~ ( r1_xreal_0(np__2,k3_finseq_1(B))
              & k5_topreal1(A,B) = k1_xboole_0 ) ) ) )).

fof(d7_topreal1,axiom,(
    ! [A] :
      ( m2_finseq_1(A,u1_struct_0(k15_euclid(np__2)))
     => ( v1_topreal1(A)
      <=> ! [B] :
            ( m2_subset_1(B,k1_numbers,k5_numbers)
           => ~ ( r1_xreal_0(np__1,B)
                & r1_xreal_0(k1_nat_1(B,np__1),k3_finseq_1(A))
                & k21_euclid(k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),A,B)) != k21_euclid(k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),A,k1_nat_1(B,np__1)))
                & k22_euclid(k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),A,B)) != k22_euclid(k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),A,k1_nat_1(B,np__1))) ) ) ) ) )).

fof(d8_topreal1,axiom,(
    ! [A] :
      ( m2_finseq_1(A,u1_struct_0(k15_euclid(np__2)))
     => ( v2_topreal1(A)
      <=> ! [B] :
            ( m2_subset_1(B,k1_numbers,k5_numbers)
           => ( ( r1_xreal_0(np__1,B)
                & r1_xreal_0(k1_nat_1(B,np__2),k3_finseq_1(A)) )
             => k5_subset_1(u1_struct_0(k15_euclid(np__2)),k4_topreal1(np__2,A,B),k4_topreal1(np__2,A,k1_nat_1(B,np__1))) = k1_struct_0(k15_euclid(np__2),k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),A,k1_nat_1(B,np__1))) ) ) ) ) )).

fof(d9_topreal1,axiom,(
    ! [A] :
      ( m2_finseq_1(A,u1_struct_0(k15_euclid(np__2)))
     => ( v3_topreal1(A)
      <=> ! [B] :
            ( m2_subset_1(B,k1_numbers,k5_numbers)
           => ! [C] :
                ( m2_subset_1(C,k1_numbers,k5_numbers)
               => ( ~ r1_xreal_0(C,k1_nat_1(B,np__1))
                 => r1_xboole_0(k4_topreal1(np__2,A,B),k4_topreal1(np__2,A,C)) ) ) ) ) ) )).

fof(d10_topreal1,axiom,(
    ! [A] :
      ( m2_finseq_1(A,u1_struct_0(k15_euclid(np__2)))
     => ( v4_topreal1(A)
      <=> ( v2_funct_1(A)
          & r1_xreal_0(np__2,k3_finseq_1(A))
          & v2_topreal1(A)
          & v3_topreal1(A)
          & v1_topreal1(A) ) ) ) )).

fof(t30_topreal1,axiom,(
    ? [A] :
      ( m2_finseq_1(A,u1_struct_0(k15_euclid(np__2)))
      & ? [B] :
          ( m2_finseq_1(B,u1_struct_0(k15_euclid(np__2)))
          & v4_topreal1(A)
          & v4_topreal1(B)
          & k2_topreal1 = k4_subset_1(u1_struct_0(k15_euclid(np__2)),k5_topreal1(np__2,A),k5_topreal1(np__2,B))
          & k5_subset_1(u1_struct_0(k15_euclid(np__2)),k5_topreal1(np__2,A),k5_topreal1(np__2,B)) = k2_struct_0(k15_euclid(np__2),k23_euclid(np__0,np__0),k23_euclid(np__1,np__1))
          & k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),A,np__1) = k23_euclid(np__0,np__0)
          & k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),A,k3_finseq_1(A)) = k23_euclid(np__1,np__1)
          & k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,np__1) = k23_euclid(np__0,np__0)
          & k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,k3_finseq_1(B)) = k23_euclid(np__1,np__1) ) ) )).

fof(t31_topreal1,axiom,(
    ! [A] :
      ( m2_finseq_1(A,u1_struct_0(k15_euclid(np__2)))
     => ( v4_topreal1(A)
       => r1_topreal1(k15_euclid(np__2),k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),A,np__1),k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),A,k3_finseq_1(A)),k5_topreal1(np__2,A)) ) ) )).

fof(d11_topreal1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
     => ( v5_topreal1(A)
      <=> ? [B] :
            ( m2_finseq_1(B,u1_struct_0(k15_euclid(np__2)))
            & v4_topreal1(B)
            & A = k5_topreal1(np__2,B) ) ) ) )).

fof(t32_topreal1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
     => ~ ( v5_topreal1(A)
          & A = k1_xboole_0 ) ) )).

fof(t33_topreal1,axiom,(
    $true )).

fof(t34_topreal1,axiom,(
    ? [A] :
      ( ~ v1_xboole_0(A)
      & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
      & ? [B] :
          ( ~ v1_xboole_0(B)
          & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
          & v5_topreal1(A)
          & v5_topreal1(B)
          & k2_topreal1 = k4_subset_1(u1_struct_0(k15_euclid(np__2)),A,B)
          & k5_subset_1(u1_struct_0(k15_euclid(np__2)),A,B) = k2_struct_0(k15_euclid(np__2),k23_euclid(np__0,np__0),k23_euclid(np__1,np__1)) ) ) )).

fof(t35_topreal1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
     => ~ ( v5_topreal1(A)
          & ! [B] :
              ( m1_subset_1(B,u1_struct_0(k15_euclid(np__2)))
             => ! [C] :
                  ( m1_subset_1(C,u1_struct_0(k15_euclid(np__2)))
                 => ~ r1_topreal1(k15_euclid(np__2),B,C,A) ) ) ) ) )).

fof(t36_topreal1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
     => ~ ( v5_topreal1(A)
          & ! [B] :
              ( ( v1_funct_1(B)
                & v1_funct_2(B,u1_struct_0(k5_topmetr),u1_struct_0(k3_pre_topc(k15_euclid(np__2),A)))
                & m2_relset_1(B,u1_struct_0(k5_topmetr),u1_struct_0(k3_pre_topc(k15_euclid(np__2),A))) )
             => ~ v3_tops_2(B,k5_topmetr,k3_pre_topc(k15_euclid(np__2),A)) ) ) ) )).

fof(dt_k1_topreal1,axiom,(
    ! [A,B,C] :
      ( ( m1_subset_1(A,k5_numbers)
        & m1_subset_1(B,u1_struct_0(k15_euclid(A)))
        & m1_subset_1(C,u1_struct_0(k15_euclid(A))) )
     => m1_subset_1(k1_topreal1(A,B,C),k1_zfmisc_1(u1_struct_0(k15_euclid(A)))) ) )).

fof(dt_k2_topreal1,axiom,(
    m1_subset_1(k2_topreal1,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )).

fof(dt_k3_topreal1,axiom,(
    ! [A,B,C] :
      ( ( m1_subset_1(A,k5_numbers)
        & m1_subset_1(B,u1_struct_0(k15_euclid(A)))
        & m1_subset_1(C,u1_struct_0(k15_euclid(A))) )
     => m1_subset_1(k3_topreal1(A,B,C),k1_zfmisc_1(u1_struct_0(k15_euclid(A)))) ) )).

fof(commutativity_k3_topreal1,axiom,(
    ! [A,B,C] :
      ( ( m1_subset_1(A,k5_numbers)
        & m1_subset_1(B,u1_struct_0(k15_euclid(A)))
        & m1_subset_1(C,u1_struct_0(k15_euclid(A))) )
     => k3_topreal1(A,B,C) = k3_topreal1(A,C,B) ) )).

fof(redefinition_k3_topreal1,axiom,(
    ! [A,B,C] :
      ( ( m1_subset_1(A,k5_numbers)
        & m1_subset_1(B,u1_struct_0(k15_euclid(A)))
        & m1_subset_1(C,u1_struct_0(k15_euclid(A))) )
     => k3_topreal1(A,B,C) = k1_topreal1(A,B,C) ) )).

fof(dt_k4_topreal1,axiom,(
    ! [A,B,C] :
      ( ( m1_subset_1(A,k5_numbers)
        & m1_finseq_1(B,u1_struct_0(k15_euclid(A)))
        & m1_subset_1(C,k5_numbers) )
     => m1_subset_1(k4_topreal1(A,B,C),k1_zfmisc_1(u1_struct_0(k15_euclid(A)))) ) )).

fof(dt_k5_topreal1,axiom,(
    ! [A,B] :
      ( ( m1_subset_1(A,k5_numbers)
        & m1_finseq_1(B,u1_struct_0(k15_euclid(A))) )
     => m1_subset_1(k5_topreal1(A,B),k1_zfmisc_1(u1_struct_0(k15_euclid(A)))) ) )).

fof(d3_topreal1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(k15_euclid(A)))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(k15_euclid(A)))
             => k1_topreal1(A,B,C) = a_3_0_topreal1(A,B,C) ) ) ) )).

fof(t19_topreal1,axiom,
    ( k3_topreal1(np__2,k23_euclid(np__0,np__0),k23_euclid(np__0,np__1)) = a_0_0_topreal1
    & k3_topreal1(np__2,k23_euclid(np__0,np__1),k23_euclid(np__1,np__1)) = a_0_1_topreal1
    & k3_topreal1(np__2,k23_euclid(np__0,np__0),k23_euclid(np__1,np__0)) = a_0_2_topreal1
    & k3_topreal1(np__2,k23_euclid(np__1,np__0),k23_euclid(np__1,np__1)) = a_0_3_topreal1 )).

fof(t20_topreal1,axiom,(
    k2_topreal1 = a_0_4_topreal1 )).

fof(d6_topreal1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_finseq_1(B,u1_struct_0(k15_euclid(A)))
         => k5_topreal1(A,B) = k3_tarski(a_2_0_topreal1(A,B)) ) ) )).

fof(s1_topreal1,axiom,(
    a_0_5_topreal1 = k2_xboole_0(a_0_6_topreal1,a_0_7_topreal1) )).

fof(fraenkel_a_3_0_topreal1,axiom,(
    ! [A,B,C,D] :
      ( ( m2_subset_1(B,k1_numbers,k5_numbers)
        & m1_subset_1(C,u1_struct_0(k15_euclid(B)))
        & m1_subset_1(D,u1_struct_0(k15_euclid(B))) )
     => ( r2_hidden(A,a_3_0_topreal1(B,C,D))
      <=> ? [E] :
            ( m1_subset_1(E,k1_numbers)
            & A = k17_euclid(B,k18_euclid(k5_real_1(np__1,E),B,C),k18_euclid(E,B,D))
            & r1_xreal_0(np__0,E)
            & r1_xreal_0(E,np__1) ) ) ) )).

fof(fraenkel_a_0_0_topreal1,axiom,(
    ! [A] :
      ( r2_hidden(A,a_0_0_topreal1)
    <=> ? [B] :
          ( m1_subset_1(B,u1_struct_0(k15_euclid(np__2)))
          & A = B
          & k21_euclid(B) = np__0
          & r1_xreal_0(k22_euclid(B),np__1)
          & r1_xreal_0(np__0,k22_euclid(B)) ) ) )).

fof(fraenkel_a_0_1_topreal1,axiom,(
    ! [A] :
      ( r2_hidden(A,a_0_1_topreal1)
    <=> ? [B] :
          ( m1_subset_1(B,u1_struct_0(k15_euclid(np__2)))
          & A = B
          & r1_xreal_0(k21_euclid(B),np__1)
          & r1_xreal_0(np__0,k21_euclid(B))
          & k22_euclid(B) = np__1 ) ) )).

fof(fraenkel_a_0_2_topreal1,axiom,(
    ! [A] :
      ( r2_hidden(A,a_0_2_topreal1)
    <=> ? [B] :
          ( m1_subset_1(B,u1_struct_0(k15_euclid(np__2)))
          & A = B
          & r1_xreal_0(k21_euclid(B),np__1)
          & r1_xreal_0(np__0,k21_euclid(B))
          & k22_euclid(B) = np__0 ) ) )).

fof(fraenkel_a_0_3_topreal1,axiom,(
    ! [A] :
      ( r2_hidden(A,a_0_3_topreal1)
    <=> ? [B] :
          ( m1_subset_1(B,u1_struct_0(k15_euclid(np__2)))
          & A = B
          & k21_euclid(B) = np__1
          & r1_xreal_0(k22_euclid(B),np__1)
          & r1_xreal_0(np__0,k22_euclid(B)) ) ) )).

fof(fraenkel_a_0_4_topreal1,axiom,(
    ! [A] :
      ( r2_hidden(A,a_0_4_topreal1)
    <=> ? [B] :
          ( m1_subset_1(B,u1_struct_0(k15_euclid(np__2)))
          & A = B
          & ~ ( ~ ( k21_euclid(B) = np__0
                  & r1_xreal_0(k22_euclid(B),np__1)
                  & r1_xreal_0(np__0,k22_euclid(B)) )
              & ~ ( r1_xreal_0(k21_euclid(B),np__1)
                  & r1_xreal_0(np__0,k21_euclid(B))
                  & k22_euclid(B) = np__1 )
              & ~ ( r1_xreal_0(k21_euclid(B),np__1)
                  & r1_xreal_0(np__0,k21_euclid(B))
                  & k22_euclid(B) = np__0 )
              & ~ ( k21_euclid(B) = np__1
                  & r1_xreal_0(k22_euclid(B),np__1)
                  & r1_xreal_0(np__0,k22_euclid(B)) ) ) ) ) )).

fof(fraenkel_a_2_0_topreal1,axiom,(
    ! [A,B,C] :
      ( ( m2_subset_1(B,k1_numbers,k5_numbers)
        & m2_finseq_1(C,u1_struct_0(k15_euclid(B))) )
     => ( r2_hidden(A,a_2_0_topreal1(B,C))
      <=> ? [D] :
            ( m2_subset_1(D,k1_numbers,k5_numbers)
            & A = k4_topreal1(B,C,D)
            & r1_xreal_0(np__1,D)
            & r1_xreal_0(k1_nat_1(D,np__1),k3_finseq_1(C)) ) ) ) )).

fof(fraenkel_a_0_5_topreal1,axiom,(
    ! [A] :
      ( r2_hidden(A,a_0_5_topreal1)
    <=> ? [B] :
          ( m1_subset_1(B,f1_s1_topreal1)
          & A = B
          & ( p1_s1_topreal1(B)
            | p2_s1_topreal1(B) ) ) ) )).

fof(fraenkel_a_0_6_topreal1,axiom,(
    ! [A] :
      ( r2_hidden(A,a_0_6_topreal1)
    <=> ? [B] :
          ( m1_subset_1(B,f1_s1_topreal1)
          & A = B
          & p1_s1_topreal1(B) ) ) )).

fof(fraenkel_a_0_7_topreal1,axiom,(
    ! [A] :
      ( r2_hidden(A,a_0_7_topreal1)
    <=> ? [B] :
          ( m1_subset_1(B,f1_s1_topreal1)
          & A = B
          & p2_s1_topreal1(B) ) ) )).
%------------------------------------------------------------------------------