SET007 Axioms: SET007+348.ax


%------------------------------------------------------------------------------
% File     : SET007+348 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Connectedness Conditions Using Polygonal Arcs
% 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    : topreal4 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   35 (   1 unit)
%            Number of atoms       :  375 (  99 equality)
%            Maximal formula depth :   26 (  14 average)
%            Number of connectives :  370 (  30 ~  ;  24  |; 170  &)
%                                         (   5 <=>; 141 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   18 (   1 propositional; 0-4 arity)
%            Number of functors    :   33 (   4 constant; 0-4 arity)
%            Number of variables   :  130 (   0 singleton; 123 !;   7 ?)
%            Maximal term depth    :    9 (   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(d1_topreal4,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_zfmisc_1(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_topreal4(A,B,C)
              <=> ? [D] :
                    ( m2_finseq_1(D,u1_struct_0(k15_euclid(np__2)))
                    & v4_topreal1(D)
                    & A = k5_topreal1(np__2,D)
                    & B = k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),D,np__1)
                    & C = k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),D,k3_finseq_1(D)) ) ) ) ) ) )).

fof(d2_topreal4,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
     => ( v1_topreal4(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)))
                & ? [D] :
                    ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
                    & ? [E] :
                        ( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
                        & B != C
                        & r2_hidden(B,A)
                        & r2_hidden(C,A)
                        & r1_topreal4(D,B,C)
                        & r1_topreal4(E,B,C)
                        & k5_subset_1(u1_struct_0(k15_euclid(np__2)),D,E) = k2_struct_0(k15_euclid(np__2),B,C)
                        & A = k4_subset_1(u1_struct_0(k15_euclid(np__2)),D,E) ) ) ) ) ) ) )).

fof(d3_topreal4,axiom,(
    ! [A] :
      ( l1_pre_topc(A)
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => ( v2_topreal4(B,A)
          <=> ( v3_pre_topc(B,A)
              & v2_connsp_1(B,A) ) ) ) ) )).

fof(t1_topreal4,axiom,(
    $true )).

fof(t2_topreal4,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_zfmisc_1(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_topreal4(A,B,C)
               => v5_topreal1(A) ) ) ) ) )).

fof(t3_topreal4,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_zfmisc_1(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_topreal4(A,B,C)
               => r1_topreal1(k15_euclid(np__2),B,C,A) ) ) ) ) )).

fof(t4_topreal4,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_zfmisc_1(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_topreal4(A,B,C)
               => ( r2_hidden(B,A)
                  & r2_hidden(C,A) ) ) ) ) ) )).

fof(t5_topreal4,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_zfmisc_1(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_topreal4(A,B,C)
                  & B = C ) ) ) ) )).

fof(t6_topreal4,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
     => ( v1_topreal4(A)
       => v1_topreal2(A) ) ) )).

fof(t7_topreal4,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] :
              ( m2_finseq_1(C,u1_struct_0(k15_euclid(np__2)))
             => ! [D] :
                  ( m1_subset_1(D,k1_numbers)
                 => ! [E] :
                      ( m1_subset_1(E,u1_struct_0(k14_euclid(np__2)))
                     => ( ( k21_euclid(A) = k21_euclid(B)
                          & r2_hidden(A,k9_metric_1(k14_euclid(np__2),E,D))
                          & r2_hidden(B,k9_metric_1(k14_euclid(np__2),E,D))
                          & C = k3_finseq_4(u1_struct_0(k15_euclid(np__2)),A,k23_euclid(k21_euclid(A),k6_real_1(k3_real_1(k22_euclid(A),k22_euclid(B)),np__2)),B) )
                       => ( k22_euclid(A) = k22_euclid(B)
                          | ( v4_topreal1(C)
                            & k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),C,np__1) = A
                            & k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),C,k3_finseq_1(C)) = B
                            & r1_topreal4(k5_topreal1(np__2,C),A,B)
                            & r1_tarski(k5_topreal1(np__2,C),k9_metric_1(k14_euclid(np__2),E,D)) ) ) ) ) ) ) ) ) )).

fof(t8_topreal4,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] :
              ( m2_finseq_1(C,u1_struct_0(k15_euclid(np__2)))
             => ! [D] :
                  ( m1_subset_1(D,k1_numbers)
                 => ! [E] :
                      ( m1_subset_1(E,u1_struct_0(k14_euclid(np__2)))
                     => ( ( k22_euclid(A) = k22_euclid(B)
                          & r2_hidden(A,k9_metric_1(k14_euclid(np__2),E,D))
                          & r2_hidden(B,k9_metric_1(k14_euclid(np__2),E,D))
                          & C = k3_finseq_4(u1_struct_0(k15_euclid(np__2)),A,k23_euclid(k6_real_1(k3_real_1(k21_euclid(A),k21_euclid(B)),np__2),k22_euclid(A)),B) )
                       => ( k21_euclid(A) = k21_euclid(B)
                          | ( v4_topreal1(C)
                            & k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),C,np__1) = A
                            & k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),C,k3_finseq_1(C)) = B
                            & r1_topreal4(k5_topreal1(np__2,C),A,B)
                            & r1_tarski(k5_topreal1(np__2,C),k9_metric_1(k14_euclid(np__2),E,D)) ) ) ) ) ) ) ) ) )).

fof(t9_topreal4,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] :
              ( m2_finseq_1(C,u1_struct_0(k15_euclid(np__2)))
             => ! [D] :
                  ( m1_subset_1(D,k1_numbers)
                 => ! [E] :
                      ( m1_subset_1(E,u1_struct_0(k14_euclid(np__2)))
                     => ( ( r2_hidden(A,k9_metric_1(k14_euclid(np__2),E,D))
                          & r2_hidden(B,k9_metric_1(k14_euclid(np__2),E,D))
                          & r2_hidden(k23_euclid(k21_euclid(A),k22_euclid(B)),k9_metric_1(k14_euclid(np__2),E,D))
                          & C = k3_finseq_4(u1_struct_0(k15_euclid(np__2)),A,k23_euclid(k21_euclid(A),k22_euclid(B)),B) )
                       => ( k21_euclid(A) = k21_euclid(B)
                          | k22_euclid(A) = k22_euclid(B)
                          | ( v4_topreal1(C)
                            & k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),C,np__1) = A
                            & k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),C,k3_finseq_1(C)) = B
                            & r1_topreal4(k5_topreal1(np__2,C),A,B)
                            & r1_tarski(k5_topreal1(np__2,C),k9_metric_1(k14_euclid(np__2),E,D)) ) ) ) ) ) ) ) ) )).

fof(t10_topreal4,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] :
              ( m2_finseq_1(C,u1_struct_0(k15_euclid(np__2)))
             => ! [D] :
                  ( m1_subset_1(D,k1_numbers)
                 => ! [E] :
                      ( m1_subset_1(E,u1_struct_0(k14_euclid(np__2)))
                     => ( ( r2_hidden(A,k9_metric_1(k14_euclid(np__2),E,D))
                          & r2_hidden(B,k9_metric_1(k14_euclid(np__2),E,D))
                          & r2_hidden(k23_euclid(k21_euclid(B),k22_euclid(A)),k9_metric_1(k14_euclid(np__2),E,D))
                          & C = k3_finseq_4(u1_struct_0(k15_euclid(np__2)),A,k23_euclid(k21_euclid(B),k22_euclid(A)),B) )
                       => ( k21_euclid(A) = k21_euclid(B)
                          | k22_euclid(A) = k22_euclid(B)
                          | ( v4_topreal1(C)
                            & k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),C,np__1) = A
                            & k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),C,k3_finseq_1(C)) = B
                            & r1_topreal4(k5_topreal1(np__2,C),A,B)
                            & r1_tarski(k5_topreal1(np__2,C),k9_metric_1(k14_euclid(np__2),E,D)) ) ) ) ) ) ) ) ) )).

fof(t11_topreal4,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,k1_numbers)
             => ! [D] :
                  ( m1_subset_1(D,u1_struct_0(k14_euclid(np__2)))
                 => ~ ( A != B
                      & r2_hidden(A,k9_metric_1(k14_euclid(np__2),D,C))
                      & r2_hidden(B,k9_metric_1(k14_euclid(np__2),D,C))
                      & ! [E] :
                          ( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
                         => ~ ( r1_topreal4(E,A,B)
                              & r1_tarski(E,k9_metric_1(k14_euclid(np__2),D,C)) ) ) ) ) ) ) ) )).

fof(t12_topreal4,axiom,(
    ! [A] :
      ( m1_subset_1(A,u1_struct_0(k15_euclid(np__2)))
     => ! [B] :
          ( m2_finseq_1(B,u1_struct_0(k15_euclid(np__2)))
         => ! [C] :
              ( m2_finseq_1(C,u1_struct_0(k15_euclid(np__2)))
             => ( ( k22_euclid(k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,np__1)) = k22_euclid(A)
                  & v4_topreal1(B)
                  & r2_hidden(A,k4_topreal1(np__2,B,np__1))
                  & C = k3_finseq_4(u1_struct_0(k15_euclid(np__2)),k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,np__1),k23_euclid(k6_real_1(k3_real_1(k21_euclid(k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,np__1)),k21_euclid(A)),np__2),k22_euclid(k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,np__1))),A) )
               => ( A = k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,np__1)
                  | ( v4_topreal1(C)
                    & k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),C,np__1) = k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,np__1)
                    & k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),C,k3_finseq_1(C)) = A
                    & r1_topreal4(k5_topreal1(np__2,C),k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,np__1),A)
                    & r1_tarski(k5_topreal1(np__2,C),k5_topreal1(np__2,B))
                    & k5_topreal1(np__2,C) = k4_subset_1(u1_struct_0(k15_euclid(np__2)),k5_topreal1(np__2,k16_finseq_1(u1_struct_0(k15_euclid(np__2)),B,np__1)),k3_topreal1(np__2,k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,np__1),A)) ) ) ) ) ) ) )).

fof(t13_topreal4,axiom,(
    ! [A] :
      ( m1_subset_1(A,u1_struct_0(k15_euclid(np__2)))
     => ! [B] :
          ( m2_finseq_1(B,u1_struct_0(k15_euclid(np__2)))
         => ! [C] :
              ( m2_finseq_1(C,u1_struct_0(k15_euclid(np__2)))
             => ( ( k21_euclid(k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,np__1)) = k21_euclid(A)
                  & v4_topreal1(B)
                  & r2_hidden(A,k4_topreal1(np__2,B,np__1))
                  & C = k3_finseq_4(u1_struct_0(k15_euclid(np__2)),k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,np__1),k23_euclid(k21_euclid(k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,np__1)),k6_real_1(k3_real_1(k22_euclid(k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,np__1)),k22_euclid(A)),np__2)),A) )
               => ( A = k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,np__1)
                  | ( v4_topreal1(C)
                    & k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),C,np__1) = k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,np__1)
                    & k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),C,k3_finseq_1(C)) = A
                    & r1_topreal4(k5_topreal1(np__2,C),k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,np__1),A)
                    & r1_tarski(k5_topreal1(np__2,C),k5_topreal1(np__2,B))
                    & k5_topreal1(np__2,C) = k4_subset_1(u1_struct_0(k15_euclid(np__2)),k5_topreal1(np__2,k16_finseq_1(u1_struct_0(k15_euclid(np__2)),B,np__1)),k3_topreal1(np__2,k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,np__1),A)) ) ) ) ) ) ) )).

fof(t14_topreal4,axiom,(
    ! [A] :
      ( m1_subset_1(A,u1_struct_0(k15_euclid(np__2)))
     => ! [B] :
          ( m2_finseq_1(B,u1_struct_0(k15_euclid(np__2)))
         => ! [C] :
              ( m2_finseq_1(C,u1_struct_0(k15_euclid(np__2)))
             => ! [D] :
                  ( m2_subset_1(D,k1_numbers,k5_numbers)
                 => ( ( v4_topreal1(B)
                      & r2_hidden(D,k4_finseq_1(B))
                      & r2_hidden(k1_nat_1(D,np__1),k4_finseq_1(B))
                      & r2_hidden(A,k4_topreal1(np__2,B,D))
                      & C = k8_finseq_1(u1_struct_0(k15_euclid(np__2)),k16_finseq_1(u1_struct_0(k15_euclid(np__2)),B,D),k12_finseq_1(u1_struct_0(k15_euclid(np__2)),A)) )
                   => ( A = k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,np__1)
                      | r1_xreal_0(D,np__1)
                      | A = k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,D)
                      | A = k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,k1_nat_1(D,np__1))
                      | ( v4_topreal1(C)
                        & k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),C,np__1) = k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,np__1)
                        & k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),C,k3_finseq_1(C)) = A
                        & r1_topreal4(k5_topreal1(np__2,C),k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,np__1),A)
                        & r1_tarski(k5_topreal1(np__2,C),k5_topreal1(np__2,B))
                        & k5_topreal1(np__2,C) = k4_subset_1(u1_struct_0(k15_euclid(np__2)),k5_topreal1(np__2,k16_finseq_1(u1_struct_0(k15_euclid(np__2)),B,D)),k3_topreal1(np__2,k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,D),A)) ) ) ) ) ) ) ) )).

fof(t15_topreal4,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)
              & k22_euclid(k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),A,np__2)) = k22_euclid(k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),A,np__1))
              & B = k3_finseq_4(u1_struct_0(k15_euclid(np__2)),k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),A,np__1),k23_euclid(k6_real_1(k3_real_1(k21_euclid(k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),A,np__1)),k21_euclid(k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),A,np__2))),np__2),k22_euclid(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,np__2)) )
           => ( k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),A,np__2) = k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),A,np__1)
              | ( v4_topreal1(B)
                & k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,np__1) = 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)),B,k3_finseq_1(B)) = k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),A,np__2)
                & r1_topreal4(k5_topreal1(np__2,B),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,np__2))
                & r1_tarski(k5_topreal1(np__2,B),k5_topreal1(np__2,A))
                & k5_topreal1(np__2,B) = k4_subset_1(u1_struct_0(k15_euclid(np__2)),k5_topreal1(np__2,k16_finseq_1(u1_struct_0(k15_euclid(np__2)),A,np__1)),k3_topreal1(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,np__2)))
                & k5_topreal1(np__2,B) = k4_subset_1(u1_struct_0(k15_euclid(np__2)),k5_topreal1(np__2,k16_finseq_1(u1_struct_0(k15_euclid(np__2)),A,np__2)),k3_topreal1(np__2,k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),A,np__2),k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),A,np__2))) ) ) ) ) ) )).

fof(t16_topreal4,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)
              & k21_euclid(k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),A,np__2)) = k21_euclid(k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),A,np__1))
              & B = k3_finseq_4(u1_struct_0(k15_euclid(np__2)),k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),A,np__1),k23_euclid(k21_euclid(k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),A,np__1)),k6_real_1(k3_real_1(k22_euclid(k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),A,np__1)),k22_euclid(k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),A,np__2))),np__2)),k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),A,np__2)) )
           => ( k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),A,np__2) = k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),A,np__1)
              | ( v4_topreal1(B)
                & k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,np__1) = 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)),B,k3_finseq_1(B)) = k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),A,np__2)
                & r1_topreal4(k5_topreal1(np__2,B),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,np__2))
                & r1_tarski(k5_topreal1(np__2,B),k5_topreal1(np__2,A))
                & k5_topreal1(np__2,B) = k4_subset_1(u1_struct_0(k15_euclid(np__2)),k5_topreal1(np__2,k16_finseq_1(u1_struct_0(k15_euclid(np__2)),A,np__1)),k3_topreal1(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,np__2)))
                & k5_topreal1(np__2,B) = k4_subset_1(u1_struct_0(k15_euclid(np__2)),k5_topreal1(np__2,k16_finseq_1(u1_struct_0(k15_euclid(np__2)),A,np__2)),k3_topreal1(np__2,k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),A,np__2),k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),A,np__2))) ) ) ) ) ) )).

fof(t17_topreal4,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)))
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => ( ( v4_topreal1(A)
                  & r2_hidden(C,k4_finseq_1(A))
                  & B = k16_finseq_1(u1_struct_0(k15_euclid(np__2)),A,C) )
               => ( k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),A,C) = k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),A,np__1)
                  | r1_xreal_0(C,np__2)
                  | ( v4_topreal1(B)
                    & k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,np__1) = 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)),B,k3_finseq_1(B)) = k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),A,C)
                    & r1_topreal4(k5_topreal1(np__2,B),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,C))
                    & r1_tarski(k5_topreal1(np__2,B),k5_topreal1(np__2,A))
                    & k5_topreal1(np__2,B) = k4_subset_1(u1_struct_0(k15_euclid(np__2)),k5_topreal1(np__2,k16_finseq_1(u1_struct_0(k15_euclid(np__2)),A,C)),k3_topreal1(np__2,k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),A,C),k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),A,C))) ) ) ) ) ) ) )).

fof(t18_topreal4,axiom,(
    ! [A] :
      ( m1_subset_1(A,u1_struct_0(k15_euclid(np__2)))
     => ! [B] :
          ( m2_finseq_1(B,u1_struct_0(k15_euclid(np__2)))
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => ~ ( A != k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,np__1)
                  & v4_topreal1(B)
                  & r2_hidden(A,k4_topreal1(np__2,B,C))
                  & ! [D] :
                      ( m2_finseq_1(D,u1_struct_0(k15_euclid(np__2)))
                     => ~ ( v4_topreal1(D)
                          & k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),D,np__1) = k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,np__1)
                          & k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),D,k3_finseq_1(D)) = A
                          & r1_topreal4(k5_topreal1(np__2,D),k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,np__1),A)
                          & r1_tarski(k5_topreal1(np__2,D),k5_topreal1(np__2,B))
                          & k5_topreal1(np__2,D) = k4_subset_1(u1_struct_0(k15_euclid(np__2)),k5_topreal1(np__2,k16_finseq_1(u1_struct_0(k15_euclid(np__2)),B,C)),k3_topreal1(np__2,k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,C),A)) ) ) ) ) ) ) )).

fof(t19_topreal4,axiom,(
    ! [A] :
      ( m1_subset_1(A,u1_struct_0(k15_euclid(np__2)))
     => ! [B] :
          ( m2_finseq_1(B,u1_struct_0(k15_euclid(np__2)))
         => ~ ( A != k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,np__1)
              & v4_topreal1(B)
              & r2_hidden(A,k5_topreal1(np__2,B))
              & ! [C] :
                  ( m2_finseq_1(C,u1_struct_0(k15_euclid(np__2)))
                 => ~ ( v4_topreal1(C)
                      & k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),C,np__1) = k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,np__1)
                      & k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),C,k3_finseq_1(C)) = A
                      & r1_topreal4(k5_topreal1(np__2,C),k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,np__1),A)
                      & r1_tarski(k5_topreal1(np__2,C),k5_topreal1(np__2,B)) ) ) ) ) ) )).

fof(t20_topreal4,axiom,(
    ! [A] :
      ( m1_subset_1(A,u1_struct_0(k15_euclid(np__2)))
     => ! [B] :
          ( m2_finseq_1(B,u1_struct_0(k15_euclid(np__2)))
         => ! [C] :
              ( m2_finseq_1(C,u1_struct_0(k15_euclid(np__2)))
             => ! [D] :
                  ( m1_subset_1(D,k1_numbers)
                 => ! [E] :
                      ( m1_subset_1(E,u1_struct_0(k14_euclid(np__2)))
                     => ( ( r2_hidden(k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,k3_finseq_1(B)),k9_metric_1(k14_euclid(np__2),E,D))
                          & r2_hidden(A,k9_metric_1(k14_euclid(np__2),E,D))
                          & v4_topreal1(B)
                          & k5_subset_1(u1_struct_0(k15_euclid(np__2)),k3_topreal1(np__2,k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,k3_finseq_1(B)),A),k5_topreal1(np__2,B)) = k1_struct_0(k15_euclid(np__2),k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,k3_finseq_1(B)))
                          & C = k8_finseq_1(u1_struct_0(k15_euclid(np__2)),B,k12_finseq_1(u1_struct_0(k15_euclid(np__2)),A)) )
                       => ( ( ~ ( k21_euclid(A) = k21_euclid(k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,k3_finseq_1(B)))
                                & k22_euclid(A) != k22_euclid(k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,k3_finseq_1(B))) )
                            & ~ ( k21_euclid(A) != k21_euclid(k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,k3_finseq_1(B)))
                                & k22_euclid(A) = k22_euclid(k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,k3_finseq_1(B))) ) )
                          | r2_hidden(k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,np__1),k9_metric_1(k14_euclid(np__2),E,D))
                          | ( v4_topreal1(C)
                            & r1_topreal4(k5_topreal1(np__2,C),k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,np__1),A)
                            & r1_tarski(k5_topreal1(np__2,C),k2_xboole_0(k5_topreal1(np__2,B),k9_metric_1(k14_euclid(np__2),E,D))) ) ) ) ) ) ) ) ) )).

fof(t21_topreal4,axiom,(
    ! [A] :
      ( m1_subset_1(A,u1_struct_0(k15_euclid(np__2)))
     => ! [B] :
          ( m2_finseq_1(B,u1_struct_0(k15_euclid(np__2)))
         => ! [C] :
              ( m2_finseq_1(C,u1_struct_0(k15_euclid(np__2)))
             => ! [D] :
                  ( m1_subset_1(D,k1_numbers)
                 => ! [E] :
                      ( m1_subset_1(E,u1_struct_0(k14_euclid(np__2)))
                     => ( ( r2_hidden(k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,k3_finseq_1(B)),k9_metric_1(k14_euclid(np__2),E,D))
                          & r2_hidden(A,k9_metric_1(k14_euclid(np__2),E,D))
                          & r2_hidden(k23_euclid(k21_euclid(A),k22_euclid(k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,k3_finseq_1(B)))),k9_metric_1(k14_euclid(np__2),E,D))
                          & v4_topreal1(B)
                          & C = k8_finseq_1(u1_struct_0(k15_euclid(np__2)),B,k2_finseq_4(u1_struct_0(k15_euclid(np__2)),k23_euclid(k21_euclid(A),k22_euclid(k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,k3_finseq_1(B)))),A))
                          & k5_subset_1(u1_struct_0(k15_euclid(np__2)),k4_subset_1(u1_struct_0(k15_euclid(np__2)),k3_topreal1(np__2,k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,k3_finseq_1(B)),k23_euclid(k21_euclid(A),k22_euclid(k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,k3_finseq_1(B))))),k3_topreal1(np__2,k23_euclid(k21_euclid(A),k22_euclid(k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,k3_finseq_1(B)))),A)),k5_topreal1(np__2,B)) = k1_struct_0(k15_euclid(np__2),k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,k3_finseq_1(B))) )
                       => ( r2_hidden(k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,np__1),k9_metric_1(k14_euclid(np__2),E,D))
                          | k21_euclid(A) = k21_euclid(k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,k3_finseq_1(B)))
                          | k22_euclid(A) = k22_euclid(k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,k3_finseq_1(B)))
                          | ( r1_topreal4(k5_topreal1(np__2,C),k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,np__1),A)
                            & r1_tarski(k5_topreal1(np__2,C),k2_xboole_0(k5_topreal1(np__2,B),k9_metric_1(k14_euclid(np__2),E,D))) ) ) ) ) ) ) ) ) )).

fof(t22_topreal4,axiom,(
    ! [A] :
      ( m1_subset_1(A,u1_struct_0(k15_euclid(np__2)))
     => ! [B] :
          ( m2_finseq_1(B,u1_struct_0(k15_euclid(np__2)))
         => ! [C] :
              ( m2_finseq_1(C,u1_struct_0(k15_euclid(np__2)))
             => ! [D] :
                  ( m1_subset_1(D,k1_numbers)
                 => ! [E] :
                      ( m1_subset_1(E,u1_struct_0(k14_euclid(np__2)))
                     => ( ( r2_hidden(k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,k3_finseq_1(B)),k9_metric_1(k14_euclid(np__2),E,D))
                          & r2_hidden(A,k9_metric_1(k14_euclid(np__2),E,D))
                          & r2_hidden(k23_euclid(k21_euclid(k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,k3_finseq_1(B))),k22_euclid(A)),k9_metric_1(k14_euclid(np__2),E,D))
                          & v4_topreal1(B)
                          & C = k8_finseq_1(u1_struct_0(k15_euclid(np__2)),B,k2_finseq_4(u1_struct_0(k15_euclid(np__2)),k23_euclid(k21_euclid(k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,k3_finseq_1(B))),k22_euclid(A)),A))
                          & k5_subset_1(u1_struct_0(k15_euclid(np__2)),k4_subset_1(u1_struct_0(k15_euclid(np__2)),k3_topreal1(np__2,k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,k3_finseq_1(B)),k23_euclid(k21_euclid(k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,k3_finseq_1(B))),k22_euclid(A))),k3_topreal1(np__2,k23_euclid(k21_euclid(k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,k3_finseq_1(B))),k22_euclid(A)),A)),k5_topreal1(np__2,B)) = k1_struct_0(k15_euclid(np__2),k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,k3_finseq_1(B))) )
                       => ( r2_hidden(k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,np__1),k9_metric_1(k14_euclid(np__2),E,D))
                          | k21_euclid(A) = k21_euclid(k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,k3_finseq_1(B)))
                          | k22_euclid(A) = k22_euclid(k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,k3_finseq_1(B)))
                          | ( r1_topreal4(k5_topreal1(np__2,C),k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,np__1),A)
                            & r1_tarski(k5_topreal1(np__2,C),k2_xboole_0(k5_topreal1(np__2,B),k9_metric_1(k14_euclid(np__2),E,D))) ) ) ) ) ) ) ) ) )).

fof(t23_topreal4,axiom,(
    ! [A] :
      ( m1_subset_1(A,u1_struct_0(k15_euclid(np__2)))
     => ! [B] :
          ( m2_finseq_1(B,u1_struct_0(k15_euclid(np__2)))
         => ! [C] :
              ( m1_subset_1(C,k1_numbers)
             => ! [D] :
                  ( m1_subset_1(D,u1_struct_0(k14_euclid(np__2)))
                 => ~ ( ~ r2_hidden(k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,np__1),k9_metric_1(k14_euclid(np__2),D,C))
                      & r2_hidden(k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,k3_finseq_1(B)),k9_metric_1(k14_euclid(np__2),D,C))
                      & r2_hidden(A,k9_metric_1(k14_euclid(np__2),D,C))
                      & v4_topreal1(B)
                      & ~ r2_hidden(A,k5_topreal1(np__2,B))
                      & ! [E] :
                          ( m2_finseq_1(E,u1_struct_0(k15_euclid(np__2)))
                         => ~ ( r1_topreal4(k5_topreal1(np__2,E),k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,np__1),A)
                              & r1_tarski(k5_topreal1(np__2,E),k2_xboole_0(k5_topreal1(np__2,B),k9_metric_1(k14_euclid(np__2),D,C))) ) ) ) ) ) ) ) )).

fof(t24_topreal4,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_zfmisc_1(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)))
             => ! [D] :
                  ( m1_subset_1(D,u1_struct_0(k15_euclid(np__2)))
                 => ! [E] :
                      ( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
                     => ! [F] :
                          ( m1_subset_1(F,k1_numbers)
                         => ! [G] :
                              ( m1_subset_1(G,u1_struct_0(k14_euclid(np__2)))
                             => ~ ( B != C
                                  & r1_topreal4(E,C,D)
                                  & r1_tarski(E,A)
                                  & r2_hidden(B,k9_metric_1(k14_euclid(np__2),G,F))
                                  & r2_hidden(D,k9_metric_1(k14_euclid(np__2),G,F))
                                  & r1_tarski(k9_metric_1(k14_euclid(np__2),G,F),A)
                                  & ! [H] :
                                      ( m1_subset_1(H,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
                                     => ~ ( r1_topreal4(H,C,B)
                                          & r1_tarski(H,A) ) ) ) ) ) ) ) ) ) ) )).

fof(t30_topreal4,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,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
             => ~ ( v2_topreal4(C,k15_euclid(np__2))
                  & r2_hidden(A,C)
                  & r2_hidden(B,C)
                  & A != B
                  & ! [D] :
                      ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
                     => ~ ( r1_topreal4(D,A,B)
                          & r1_tarski(D,C) ) ) ) ) ) ) )).

fof(t25_topreal4,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(k15_euclid(np__2)))
             => ( ( v2_topreal4(A,k15_euclid(np__2))
                  & B = a_2_0_topreal4(A,C) )
               => v3_pre_topc(B,k15_euclid(np__2)) ) ) ) ) )).

fof(t26_topreal4,axiom,(
    ! [A] :
      ( m1_subset_1(A,u1_struct_0(k15_euclid(np__2)))
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
             => ( ( v2_topreal4(B,k15_euclid(np__2))
                  & r2_hidden(A,B)
                  & C = a_2_1_topreal4(A,B) )
               => v3_pre_topc(C,k15_euclid(np__2)) ) ) ) ) )).

fof(t27_topreal4,axiom,(
    ! [A] :
      ( m1_subset_1(A,u1_struct_0(k15_euclid(np__2)))
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
             => ( ( r2_hidden(A,B)
                  & C = a_2_1_topreal4(A,B) )
               => r1_tarski(C,B) ) ) ) ) )).

fof(t28_topreal4,axiom,(
    ! [A] :
      ( m1_subset_1(A,u1_struct_0(k15_euclid(np__2)))
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
             => ( ( v2_topreal4(B,k15_euclid(np__2))
                  & r2_hidden(A,B)
                  & C = a_2_1_topreal4(A,B) )
               => r1_tarski(B,C) ) ) ) ) )).

fof(t29_topreal4,axiom,(
    ! [A] :
      ( m1_subset_1(A,u1_struct_0(k15_euclid(np__2)))
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
             => ( ( v2_topreal4(B,k15_euclid(np__2))
                  & r2_hidden(A,B)
                  & C = a_2_1_topreal4(A,B) )
               => B = C ) ) ) ) )).

fof(fraenkel_a_2_0_topreal4,axiom,(
    ! [A,B,C] :
      ( ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
        & m1_subset_1(C,u1_struct_0(k15_euclid(np__2))) )
     => ( r2_hidden(A,a_2_0_topreal4(B,C))
      <=> ? [D] :
            ( m1_subset_1(D,u1_struct_0(k15_euclid(np__2)))
            & A = D
            & D != C
            & r2_hidden(D,B)
            & ! [E] :
                ( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
               => ~ ( r1_topreal4(E,C,D)
                    & r1_tarski(E,B) ) ) ) ) ) )).

fof(fraenkel_a_2_1_topreal4,axiom,(
    ! [A,B,C] :
      ( ( m1_subset_1(B,u1_struct_0(k15_euclid(np__2)))
        & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
     => ( r2_hidden(A,a_2_1_topreal4(B,C))
      <=> ? [D] :
            ( m1_subset_1(D,u1_struct_0(k15_euclid(np__2)))
            & A = D
            & ~ ( D != B
                & ! [E] :
                    ( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
                   => ~ ( r1_topreal4(E,B,D)
                        & r1_tarski(E,C) ) ) ) ) ) ) )).
%------------------------------------------------------------------------------