SET007 Axioms: SET007+403.ax


%------------------------------------------------------------------------------
% File     : SET007+403 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Extremal Properties of Vertices on Special Polygons, Part I
% 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    : sppol_1 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   74 (  10 unit)
%            Number of atoms       :  448 (  70 equality)
%            Maximal formula depth :   23 (   9 average)
%            Number of connectives :  445 (  71 ~  ;   6  |; 161  &)
%                                         (  11 <=>; 196 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   25 (   1 propositional; 0-3 arity)
%            Number of functors    :   46 (   9 constant; 0-4 arity)
%            Number of variables   :  182 (   0 singleton; 178 !;   4 ?)
%            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(cc1_sppol_1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
     => ( ( ~ v1_realset1(A)
          & v1_sppol_1(A) )
       => ~ v2_sppol_1(A) ) ) )).

fof(cc2_sppol_1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
     => ( ( ~ v1_realset1(A)
          & v2_sppol_1(A) )
       => ~ v1_sppol_1(A) ) ) )).

fof(t1_sppol_1,axiom,(
    $true )).

fof(t2_sppol_1,axiom,(
    $true )).

fof(t3_sppol_1,axiom,(
    $true )).

fof(t4_sppol_1,axiom,(
    $true )).

fof(t5_sppol_1,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ( r1_xreal_0(A,B)
           => ( r1_xreal_0(k6_xcmplx_0(A,np__1),B)
              & ~ r1_xreal_0(B,k6_xcmplx_0(A,np__1))
              & r1_xreal_0(A,k2_xcmplx_0(B,np__1))
              & ~ r1_xreal_0(k2_xcmplx_0(B,np__1),A) ) ) ) ) )).

fof(t6_sppol_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ( ~ r1_xreal_0(B,A)
           => r1_xreal_0(A,k5_real_1(B,np__1)) ) ) ) )).

fof(t7_sppol_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => ( ( r1_xreal_0(np__1,k5_real_1(A,B))
                  & r1_xreal_0(k5_real_1(A,B),C) )
               => ( r2_hidden(k5_real_1(A,B),k2_finseq_1(C))
                  & m2_subset_1(k5_real_1(A,B),k1_numbers,k5_numbers) ) ) ) ) ) )).

fof(t8_sppol_1,axiom,(
    $true )).

fof(t9_sppol_1,axiom,(
    $true )).

fof(t10_sppol_1,axiom,(
    $true )).

fof(t11_sppol_1,axiom,(
    $true )).

fof(t12_sppol_1,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ! [C] :
              ( v1_xreal_0(C)
             => ~ ( r1_xreal_0(np__0,A)
                  & r1_xreal_0(A,np__1)
                  & r1_xreal_0(np__0,B)
                  & r1_xreal_0(np__0,C)
                  & k2_xcmplx_0(k3_xcmplx_0(A,B),k3_xcmplx_0(k6_xcmplx_0(np__1,A),C)) = np__0
                  & ~ ( A = np__0
                      & C = np__0 )
                  & ~ ( A = np__1
                      & B = np__0 )
                  & ~ ( B = np__0
                      & C = np__0 ) ) ) ) ) )).

fof(t13_sppol_1,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ! [C] :
              ( v1_xreal_0(C)
             => ~ ( ~ r1_xreal_0(B,A)
                  & ~ r1_xreal_0(C,A)
                  & r1_xreal_0(k1_square_1(B,C),A) ) ) ) ) )).

fof(t14_sppol_1,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ! [C] :
              ( v1_xreal_0(C)
             => ~ ( ~ r1_xreal_0(A,B)
                  & ~ r1_xreal_0(A,C)
                  & r1_xreal_0(A,k2_square_1(B,C)) ) ) ) ) )).

fof(t15_sppol_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_euclid(A))
         => ! [C] :
              ( m1_subset_1(C,k1_euclid(A))
             => ! [D] :
                  ( m1_subset_1(D,k1_euclid(A))
                 => r1_xreal_0(k5_real_1(k12_euclid(k8_euclid(A,B,C)),k12_euclid(k8_euclid(A,C,D))),k12_euclid(k8_euclid(A,B,D))) ) ) ) ) )).

fof(t16_sppol_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_euclid(A))
         => ! [C] :
              ( m1_subset_1(C,k1_euclid(A))
             => ! [D] :
                  ( m1_subset_1(D,k1_euclid(A))
                 => r1_xreal_0(k5_real_1(k12_euclid(k8_euclid(A,C,B)),k12_euclid(k8_euclid(A,C,D))),k12_euclid(k8_euclid(A,D,B))) ) ) ) ) )).

fof(t17_sppol_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(k15_euclid(A)))
         => ( m1_subset_1(B,k1_euclid(A))
            & m1_subset_1(B,u1_struct_0(k14_euclid(A))) ) ) ) )).

fof(t18_sppol_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(k14_euclid(A)))
         => ( m1_subset_1(B,k1_euclid(A))
            & m1_subset_1(B,u1_struct_0(k15_euclid(A))) ) ) ) )).

fof(t19_sppol_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_euclid(A))
         => ( m1_subset_1(B,u1_struct_0(k14_euclid(A)))
            & m1_subset_1(B,u1_struct_0(k15_euclid(A))) ) ) ) )).

fof(t20_sppol_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(k14_euclid(A)))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(k14_euclid(A)))
             => ! [D] :
                  ( m1_subset_1(D,k1_euclid(A))
                 => ! [E] :
                      ( m1_subset_1(E,k1_euclid(A))
                     => ( ( D = B
                          & E = C )
                       => k4_metric_1(k14_euclid(A),B,C) = k12_euclid(k8_euclid(A,D,E)) ) ) ) ) ) ) )).

fof(t21_sppol_1,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))
                      & ! [E] :
                          ( m1_subset_1(E,k1_numbers)
                         => ~ ( r1_xreal_0(np__0,E)
                              & r1_xreal_0(E,np__1)
                              & B = k17_euclid(A,k18_euclid(k5_real_1(np__1,E),A,C),k18_euclid(E,A,D)) ) ) ) ) ) ) ) )).

fof(t22_sppol_1,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,k1_numbers)
                 => ( ( r1_xreal_0(np__0,D)
                      & r1_xreal_0(D,np__1) )
                   => r2_hidden(k17_euclid(A,k18_euclid(k5_real_1(np__1,D),A,B),k18_euclid(D,A,C)),k3_topreal1(A,B,C)) ) ) ) ) ) )).

fof(t23_sppol_1,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] :
                  ( ( ~ v1_xboole_0(D)
                    & m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k15_euclid(A)))) )
                 => ~ ( v4_pre_topc(D,k15_euclid(A))
                      & r1_tarski(D,k3_topreal1(A,B,C))
                      & ! [E] :
                          ( m1_subset_1(E,k1_numbers)
                         => ~ ( r2_hidden(k17_euclid(A,k18_euclid(k5_real_1(np__1,E),A,B),k18_euclid(E,A,C)),D)
                              & ! [F] :
                                  ( m1_subset_1(F,k1_numbers)
                                 => ( ( r1_xreal_0(np__0,F)
                                      & r1_xreal_0(F,np__1)
                                      & r2_hidden(k17_euclid(A,k18_euclid(k5_real_1(np__1,F),A,B),k18_euclid(F,A,C)),D) )
                                   => r1_xreal_0(E,F) ) ) ) ) ) ) ) ) ) )).

fof(t24_sppol_1,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)))
                     => ~ ( r1_tarski(k3_topreal1(A,D,E),k3_topreal1(A,B,C))
                          & r2_hidden(B,k3_topreal1(A,D,E))
                          & B != D
                          & B != E ) ) ) ) ) ) )).

fof(t25_sppol_1,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)))
                     => ~ ( k3_topreal1(A,B,C) = k3_topreal1(A,D,E)
                          & ~ ( B = D
                              & C = E )
                          & ~ ( B = E
                              & C = D ) ) ) ) ) ) ) )).

fof(t26_sppol_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => v3_compts_1(k15_euclid(A)) ) )).

fof(t27_sppol_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(k15_euclid(A)))
         => v4_pre_topc(k1_struct_0(k15_euclid(A),B),k15_euclid(A)) ) ) )).

fof(t28_sppol_1,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)))
             => v6_compts_1(k3_topreal1(A,B,C),k15_euclid(A)) ) ) ) )).

fof(t29_sppol_1,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)))
             => v4_pre_topc(k3_topreal1(A,B,C),k15_euclid(A)) ) ) ) )).

fof(d1_sppol_1,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,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
             => ( r1_sppol_1(A,B,C)
              <=> ( r2_hidden(B,C)
                  & ! [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))
                              & r1_tarski(k3_topreal1(A,D,E),C)
                              & B != D
                              & B != E ) ) ) ) ) ) ) ) )).

fof(t30_sppol_1,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,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
             => ! [D] :
                  ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
                 => ( ( r1_sppol_1(A,B,C)
                      & r1_tarski(D,C)
                      & r2_hidden(B,D) )
                   => r1_sppol_1(A,B,D) ) ) ) ) ) )).

fof(t31_sppol_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(k15_euclid(A)))
         => r1_sppol_1(A,B,k1_struct_0(k15_euclid(A),B)) ) ) )).

fof(t32_sppol_1,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)))
             => r1_sppol_1(A,B,k3_topreal1(A,B,C)) ) ) ) )).

fof(t33_sppol_1,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)))
             => r1_sppol_1(A,B,k3_topreal1(A,C,B)) ) ) ) )).

fof(t34_sppol_1,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)))
                 => ~ ( r1_sppol_1(A,B,k3_topreal1(A,C,D))
                      & B != C
                      & B != D ) ) ) ) ) )).

fof(t35_sppol_1,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)))
         => ~ ( k21_euclid(A) != k21_euclid(B)
              & k22_euclid(A) != k22_euclid(B)
              & ! [C] :
                  ( m1_subset_1(C,u1_struct_0(k15_euclid(np__2)))
                 => ~ ( r2_hidden(C,k3_topreal1(np__2,A,B))
                      & k21_euclid(C) != k21_euclid(A)
                      & k21_euclid(C) != k21_euclid(B)
                      & k22_euclid(C) != k22_euclid(A)
                      & k22_euclid(C) != k22_euclid(B) ) ) ) ) ) )).

fof(d2_sppol_1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
     => ( v1_sppol_1(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)))
               => ( ( r2_hidden(B,A)
                    & r2_hidden(C,A) )
                 => k22_euclid(B) = k22_euclid(C) ) ) ) ) ) )).

fof(d3_sppol_1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
     => ( v2_sppol_1(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)))
               => ( ( r2_hidden(B,A)
                    & r2_hidden(C,A) )
                 => k21_euclid(B) = k21_euclid(C) ) ) ) ) ) )).

fof(t36_sppol_1,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)))
         => ( k22_euclid(A) = k22_euclid(B)
          <=> v1_sppol_1(k3_topreal1(np__2,A,B)) ) ) ) )).

fof(t37_sppol_1,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)))
         => ( k21_euclid(A) = k21_euclid(B)
          <=> v2_sppol_1(k3_topreal1(np__2,A,B)) ) ) ) )).

fof(t38_sppol_1,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)))
             => ! [D] :
                  ( m1_subset_1(D,u1_struct_0(k15_euclid(np__2)))
                 => ( ( r2_hidden(A,k3_topreal1(np__2,B,C))
                      & r2_hidden(D,k3_topreal1(np__2,B,C))
                      & k22_euclid(A) = k22_euclid(D) )
                   => ( k21_euclid(A) = k21_euclid(D)
                      | v1_sppol_1(k3_topreal1(np__2,B,C)) ) ) ) ) ) ) )).

fof(t39_sppol_1,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)))
             => ! [D] :
                  ( m1_subset_1(D,u1_struct_0(k15_euclid(np__2)))
                 => ( ( r2_hidden(A,k3_topreal1(np__2,B,C))
                      & r2_hidden(D,k3_topreal1(np__2,B,C))
                      & k21_euclid(A) = k21_euclid(D) )
                   => ( k22_euclid(A) = k22_euclid(D)
                      | v2_sppol_1(k3_topreal1(np__2,B,C)) ) ) ) ) ) ) )).

fof(t40_sppol_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_finseq_1(B,u1_struct_0(k15_euclid(np__2)))
         => v4_pre_topc(k4_topreal1(np__2,B,A),k15_euclid(np__2)) ) ) )).

fof(t41_sppol_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_finseq_1(B,u1_struct_0(k15_euclid(np__2)))
         => ~ ( v1_topreal1(B)
              & ~ v2_sppol_1(k4_topreal1(np__2,B,A))
              & ~ v1_sppol_1(k4_topreal1(np__2,B,A)) ) ) ) )).

fof(t42_sppol_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_finseq_1(B,u1_struct_0(k15_euclid(np__2)))
         => ~ ( v2_funct_1(B)
              & r1_xreal_0(np__1,A)
              & r1_xreal_0(k1_nat_1(A,np__1),k3_finseq_1(B))
              & v1_realset1(k4_topreal1(np__2,B,A)) ) ) ) )).

fof(t43_sppol_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_finseq_1(B,u1_struct_0(k15_euclid(np__2)))
         => ~ ( v2_funct_1(B)
              & r1_xreal_0(np__1,A)
              & r1_xreal_0(k1_nat_1(A,np__1),k3_finseq_1(B))
              & v2_sppol_1(k4_topreal1(np__2,B,A))
              & v1_sppol_1(k4_topreal1(np__2,B,A)) ) ) ) )).

fof(t49_sppol_1,axiom,(
    ! [A] :
      ( m2_finseq_1(A,u1_struct_0(k15_euclid(np__2)))
     => v4_pre_topc(k5_topreal1(np__2,A),k15_euclid(np__2)) ) )).

fof(d4_sppol_1,axiom,(
    ! [A] :
      ( m2_finseq_1(A,u1_struct_0(k15_euclid(np__2)))
     => ( v3_sppol_1(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)) )
             => ( 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__2)))
                & 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__2))) ) ) ) ) ) )).

fof(t50_sppol_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_finseq_1(B,u1_struct_0(k15_euclid(np__2)))
         => ( ( v1_topreal1(B)
              & v3_sppol_1(B)
              & r1_xreal_0(np__1,A)
              & r1_xreal_0(k1_nat_1(A,np__2),k3_finseq_1(B))
              & k21_euclid(k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,A)) = k21_euclid(k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,k1_nat_1(A,np__1))) )
           => k22_euclid(k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,k1_nat_1(A,np__1))) = k22_euclid(k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,k1_nat_1(A,np__2))) ) ) ) )).

fof(t51_sppol_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_finseq_1(B,u1_struct_0(k15_euclid(np__2)))
         => ( ( v1_topreal1(B)
              & v3_sppol_1(B)
              & r1_xreal_0(np__1,A)
              & r1_xreal_0(k1_nat_1(A,np__2),k3_finseq_1(B))
              & k22_euclid(k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,A)) = k22_euclid(k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,k1_nat_1(A,np__1))) )
           => k21_euclid(k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,k1_nat_1(A,np__1))) = k21_euclid(k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,k1_nat_1(A,np__2))) ) ) ) )).

fof(t52_sppol_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_finseq_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,u1_struct_0(k15_euclid(np__2)))
                     => ~ ( v1_topreal1(B)
                          & v3_sppol_1(B)
                          & r1_xreal_0(np__1,A)
                          & r1_xreal_0(k1_nat_1(A,np__2),k3_finseq_1(B))
                          & C = k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,A)
                          & D = k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,k1_nat_1(A,np__1))
                          & E = k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,k1_nat_1(A,np__2))
                          & ~ ( k21_euclid(C) = k21_euclid(D)
                              & k21_euclid(E) != k21_euclid(D) )
                          & ~ ( k22_euclid(C) = k22_euclid(D)
                              & k22_euclid(E) != k22_euclid(D) ) ) ) ) ) ) ) )).

fof(t53_sppol_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_finseq_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,u1_struct_0(k15_euclid(np__2)))
                     => ~ ( v1_topreal1(B)
                          & v3_sppol_1(B)
                          & r1_xreal_0(np__1,A)
                          & r1_xreal_0(k1_nat_1(A,np__2),k3_finseq_1(B))
                          & C = k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,A)
                          & D = k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,k1_nat_1(A,np__1))
                          & E = k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,k1_nat_1(A,np__2))
                          & ~ ( k21_euclid(D) = k21_euclid(E)
                              & k21_euclid(C) != k21_euclid(D) )
                          & ~ ( k22_euclid(D) = k22_euclid(E)
                              & k22_euclid(C) != k22_euclid(D) ) ) ) ) ) ) ) )).

fof(t54_sppol_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_finseq_1(B,u1_struct_0(k15_euclid(np__2)))
         => ~ ( v1_topreal1(B)
              & v3_sppol_1(B)
              & r1_xreal_0(np__1,A)
              & r1_xreal_0(k1_nat_1(A,np__2),k3_finseq_1(B))
              & r1_tarski(k3_topreal1(np__2,k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,A),k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,k1_nat_1(A,np__2))),k4_subset_1(u1_struct_0(k15_euclid(np__2)),k4_topreal1(np__2,B,A),k4_topreal1(np__2,B,k1_nat_1(A,np__1)))) ) ) ) )).

fof(t55_sppol_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_finseq_1(B,u1_struct_0(k15_euclid(np__2)))
         => ( ( v1_topreal1(B)
              & v3_sppol_1(B)
              & r1_xreal_0(np__1,A)
              & r1_xreal_0(k1_nat_1(A,np__2),k3_finseq_1(B))
              & v2_sppol_1(k4_topreal1(np__2,B,A)) )
           => v1_sppol_1(k4_topreal1(np__2,B,k1_nat_1(A,np__1))) ) ) ) )).

fof(t56_sppol_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_finseq_1(B,u1_struct_0(k15_euclid(np__2)))
         => ( ( v1_topreal1(B)
              & v3_sppol_1(B)
              & r1_xreal_0(np__1,A)
              & r1_xreal_0(k1_nat_1(A,np__2),k3_finseq_1(B))
              & v1_sppol_1(k4_topreal1(np__2,B,A)) )
           => v2_sppol_1(k4_topreal1(np__2,B,k1_nat_1(A,np__1))) ) ) ) )).

fof(t57_sppol_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_finseq_1(B,u1_struct_0(k15_euclid(np__2)))
         => ~ ( v1_topreal1(B)
              & v3_sppol_1(B)
              & r1_xreal_0(np__1,A)
              & r1_xreal_0(k1_nat_1(A,np__2),k3_finseq_1(B))
              & ~ ( v2_sppol_1(k4_topreal1(np__2,B,A))
                  & v1_sppol_1(k4_topreal1(np__2,B,k1_nat_1(A,np__1))) )
              & ~ ( v1_sppol_1(k4_topreal1(np__2,B,A))
                  & v2_sppol_1(k4_topreal1(np__2,B,k1_nat_1(A,np__1))) ) ) ) ) )).

fof(t58_sppol_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_finseq_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)))
                 => ~ ( v1_topreal1(B)
                      & v3_sppol_1(B)
                      & r1_xreal_0(np__1,A)
                      & r1_xreal_0(k1_nat_1(A,np__2),k3_finseq_1(B))
                      & r2_hidden(k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,k1_nat_1(A,np__1)),k3_topreal1(np__2,C,D))
                      & r1_tarski(k3_topreal1(np__2,C,D),k4_subset_1(u1_struct_0(k15_euclid(np__2)),k4_topreal1(np__2,B,A),k4_topreal1(np__2,B,k1_nat_1(A,np__1))))
                      & k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,k1_nat_1(A,np__1)) != C
                      & k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,k1_nat_1(A,np__1)) != D ) ) ) ) ) )).

fof(t59_sppol_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_finseq_1(B,u1_struct_0(k15_euclid(np__2)))
         => ( ( v1_topreal1(B)
              & v3_sppol_1(B)
              & r1_xreal_0(np__1,A)
              & r1_xreal_0(k1_nat_1(A,np__2),k3_finseq_1(B)) )
           => r1_sppol_1(np__2,k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,k1_nat_1(A,np__1)),k4_subset_1(u1_struct_0(k15_euclid(np__2)),k4_topreal1(np__2,B,A),k4_topreal1(np__2,B,k1_nat_1(A,np__1)))) ) ) ) )).

fof(t60_sppol_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_finseq_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,u1_struct_0(k14_euclid(np__2)))
                     => ( ( v1_topreal1(B)
                          & v3_sppol_1(B)
                          & r1_xreal_0(np__1,A)
                          & r1_xreal_0(k1_nat_1(A,np__2),k3_finseq_1(B))
                          & E = k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,k1_nat_1(A,np__1))
                          & r2_hidden(k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,k1_nat_1(A,np__1)),k3_topreal1(np__2,C,D)) )
                       => ( k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,k1_nat_1(A,np__1)) = D
                          | r2_hidden(C,k4_subset_1(u1_struct_0(k15_euclid(np__2)),k4_topreal1(np__2,B,A),k4_topreal1(np__2,B,k1_nat_1(A,np__1))))
                          | ! [F] :
                              ( m1_subset_1(F,k1_numbers)
                             => ~ ( ~ r1_xreal_0(F,np__0)
                                  & ! [G] :
                                      ( m1_subset_1(G,u1_struct_0(k15_euclid(np__2)))
                                     => ~ ( ~ r2_hidden(G,k4_subset_1(u1_struct_0(k15_euclid(np__2)),k4_topreal1(np__2,B,A),k4_topreal1(np__2,B,k1_nat_1(A,np__1))))
                                          & r2_hidden(G,k3_topreal1(np__2,C,D))
                                          & r2_hidden(G,k9_metric_1(k14_euclid(np__2),E,F)) ) ) ) ) ) ) ) ) ) ) ) )).

fof(d5_sppol_1,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] :
              ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
             => ( r2_sppol_1(A,B,C)
              <=> ( v3_sppol_1(A)
                  & v4_topreal1(A)
                  & v3_sppol_1(B)
                  & v4_topreal1(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)),B,np__1)
                  & k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),A,k3_finseq_1(A)) = k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,k3_finseq_1(B))
                  & v3_sppol_1(k3_finseq_4(u1_struct_0(k15_euclid(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),k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,np__2)))
                  & v3_sppol_1(k3_finseq_4(u1_struct_0(k15_euclid(np__2)),k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),A,k5_real_1(k3_finseq_1(A),np__1)),k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),A,k3_finseq_1(A)),k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,k5_real_1(k3_finseq_1(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)),A,k3_finseq_1(A))
                  & 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),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)))
                  & C = k4_subset_1(u1_struct_0(k15_euclid(np__2)),k5_topreal1(np__2,A),k5_topreal1(np__2,B)) ) ) ) ) ) )).

fof(t61_sppol_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
         => ! [C] :
              ( m2_finseq_1(C,u1_struct_0(k15_euclid(np__2)))
             => ! [D] :
                  ( m2_finseq_1(D,u1_struct_0(k15_euclid(np__2)))
                 => ( r2_sppol_1(C,D,B)
                   => ( r1_xreal_0(A,np__1)
                      | r1_xreal_0(k3_finseq_1(C),A)
                      | r1_sppol_1(np__2,k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),C,A),B) ) ) ) ) ) ) )).

fof(t44_sppol_1,axiom,(
    ! [A] :
      ( m2_finseq_1(A,u1_struct_0(k15_euclid(np__2)))
     => v1_finset_1(a_1_0_sppol_1(A)) ) )).

fof(t45_sppol_1,axiom,(
    ! [A] :
      ( m2_finseq_1(A,u1_struct_0(k15_euclid(np__2)))
     => v1_finset_1(a_1_1_sppol_1(A)) ) )).

fof(t46_sppol_1,axiom,(
    ! [A] :
      ( m2_finseq_1(A,u1_struct_0(k15_euclid(np__2)))
     => m1_subset_1(a_1_0_sppol_1(A),k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))) ) )).

fof(t47_sppol_1,axiom,(
    ! [A] :
      ( m2_finseq_1(A,u1_struct_0(k15_euclid(np__2)))
     => m1_subset_1(a_1_1_sppol_1(A),k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))) ) )).

fof(t48_sppol_1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
     => ! [B] :
          ( m2_finseq_1(B,u1_struct_0(k15_euclid(np__2)))
         => ( A = k3_tarski(a_1_1_sppol_1(B))
           => v4_pre_topc(A,k15_euclid(np__2)) ) ) ) )).

fof(s1_sppol_1,axiom,(
    v1_finset_1(a_0_0_sppol_1) )).

fof(s2_sppol_1,axiom,(
    v1_finset_1(a_0_1_sppol_1) )).

fof(fraenkel_a_1_0_sppol_1,axiom,(
    ! [A,B] :
      ( m2_finseq_1(B,u1_struct_0(k15_euclid(np__2)))
     => ( r2_hidden(A,a_1_0_sppol_1(B))
      <=> ? [C] :
            ( m2_subset_1(C,k1_numbers,k5_numbers)
            & A = k4_topreal1(np__2,B,C)
            & r1_xreal_0(np__1,C)
            & r1_xreal_0(C,k3_finseq_1(B)) ) ) ) )).

fof(fraenkel_a_1_1_sppol_1,axiom,(
    ! [A,B] :
      ( m2_finseq_1(B,u1_struct_0(k15_euclid(np__2)))
     => ( r2_hidden(A,a_1_1_sppol_1(B))
      <=> ? [C] :
            ( m2_subset_1(C,k1_numbers,k5_numbers)
            & A = k4_topreal1(np__2,B,C)
            & r1_xreal_0(np__1,C)
            & r1_xreal_0(k1_nat_1(C,np__1),k3_finseq_1(B)) ) ) ) )).

fof(fraenkel_a_0_0_sppol_1,axiom,(
    ! [A] :
      ( r2_hidden(A,a_0_0_sppol_1)
    <=> ? [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
          & A = f3_s1_sppol_1(f2_s1_sppol_1,B)
          & r2_hidden(B,k4_finseq_1(f2_s1_sppol_1))
          & p1_s1_sppol_1(B) ) ) )).

fof(fraenkel_a_0_1_sppol_1,axiom,(
    ! [A] :
      ( r2_hidden(A,a_0_1_sppol_1)
    <=> ? [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
          & A = f3_s2_sppol_1(f2_s2_sppol_1,B)
          & r1_xreal_0(np__1,B)
          & r1_xreal_0(B,k3_finseq_1(f2_s2_sppol_1))
          & p1_s2_sppol_1(B) ) ) )).
%------------------------------------------------------------------------------