SET007 Axioms: SET007+532.ax


%------------------------------------------------------------------------------
% File     : SET007+532 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : On the Rectangular Finite Sequences of the Points of the Plane
% 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    : sprect_1 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :  128 (   1 unit)
%            Number of atoms       :  685 ( 103 equality)
%            Maximal formula depth :   19 (   6 average)
%            Number of connectives :  700 ( 143 ~  ;   0  |; 356  &)
%                                         (   9 <=>; 192 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   44 (   1 propositional; 0-3 arity)
%            Number of functors    :   76 (  12 constant; 0-4 arity)
%            Number of variables   :  188 (   0 singleton; 176 !;  12 ?)
%            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_sprect_1,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & ~ v5_seqm_3(A) )
     => ( ~ v1_xboole_0(A)
        & v1_relat_1(A)
        & v1_funct_1(A)
        & ~ v1_realset1(A) ) ) )).

fof(cc2_sprect_1,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_realset1(A) )
     => ( v1_relat_1(A)
        & v1_funct_1(A)
        & v5_seqm_3(A) ) ) )).

fof(fc1_sprect_1,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v5_seqm_3(A) )
     => v1_realset1(k2_relat_1(A)) ) )).

fof(rc1_sprect_1,axiom,(
    ? [A] :
      ( ~ v1_xboole_0(A)
      & v1_relat_1(A)
      & v1_funct_1(A)
      & v1_finset_1(A)
      & v1_finseq_1(A)
      & v5_seqm_3(A) ) )).

fof(fc2_sprect_1,axiom,(
    ! [A,B] :
      ( ( m1_subset_1(A,k5_numbers)
        & ~ v1_realset1(B)
        & m1_finseq_1(B,u1_struct_0(k15_euclid(A))) )
     => ~ v1_xboole_0(k5_topreal1(A,B)) ) )).

fof(fc3_sprect_1,axiom,(
    ! [A] :
      ( m1_finseq_1(A,u1_struct_0(k15_euclid(np__2)))
     => v6_compts_1(k5_topreal1(np__2,A),k15_euclid(np__2)) ) )).

fof(fc4_sprect_1,axiom,
    ( ~ v1_xboole_0(k2_topreal1)
    & ~ v1_sppol_1(k2_topreal1)
    & ~ v2_sppol_1(k2_topreal1)
    & v1_topreal4(k2_topreal1)
    & ~ v1_realset1(k2_topreal1) )).

fof(rc2_sprect_1,axiom,(
    ? [A] :
      ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
      & ~ v1_xboole_0(A)
      & ~ v1_sppol_1(A)
      & ~ v2_sppol_1(A)
      & v6_compts_1(A,k15_euclid(np__2)) ) )).

fof(fc5_sprect_1,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & ~ v2_sppol_1(A)
        & v6_compts_1(A,k15_euclid(np__2))
        & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
     => ( ~ v1_xboole_0(k1_sprect_1(A))
        & v1_relat_1(k1_sprect_1(A))
        & v1_funct_1(k1_sprect_1(A))
        & v1_finset_1(k1_sprect_1(A))
        & v1_finseq_1(k1_sprect_1(A))
        & ~ v1_realset1(k1_sprect_1(A))
        & ~ v5_seqm_3(k1_sprect_1(A)) ) ) )).

fof(fc6_sprect_1,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & ~ v1_sppol_1(A)
        & v6_compts_1(A,k15_euclid(np__2))
        & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
     => ( ~ v1_xboole_0(k1_sprect_1(A))
        & v1_relat_1(k1_sprect_1(A))
        & v1_funct_1(k1_sprect_1(A))
        & v1_finset_1(k1_sprect_1(A))
        & v1_finseq_1(k1_sprect_1(A))
        & ~ v1_realset1(k1_sprect_1(A))
        & ~ v5_seqm_3(k1_sprect_1(A)) ) ) )).

fof(fc7_sprect_1,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & ~ v1_sppol_1(A)
        & ~ v2_sppol_1(A)
        & v6_compts_1(A,k15_euclid(np__2))
        & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
     => ( ~ v1_xboole_0(k1_sprect_1(A))
        & v1_relat_1(k1_sprect_1(A))
        & v1_funct_1(k1_sprect_1(A))
        & v1_finset_1(k1_sprect_1(A))
        & v1_finseq_1(k1_sprect_1(A))
        & ~ v1_realset1(k1_sprect_1(A))
        & ~ v5_seqm_3(k1_sprect_1(A))
        & v1_topreal1(k1_sprect_1(A))
        & v2_topreal1(k1_sprect_1(A))
        & v1_finseq_6(k1_sprect_1(A),u1_struct_0(k15_euclid(np__2)))
        & v1_goboard5(k1_sprect_1(A))
        & v2_goboard5(k1_sprect_1(A)) ) ) )).

fof(rc3_sprect_1,axiom,(
    ? [A] :
      ( m1_subset_1(A,k1_zfmisc_1(k1_numbers))
      & ~ v1_xboole_0(A)
      & v1_membered(A)
      & v2_membered(A)
      & v1_seq_4(A)
      & v2_seq_4(A) ) )).

fof(fc8_sprect_1,axiom,
    ( v1_xboole_0(k1_subset_1(k1_numbers))
    & v1_relat_1(k1_subset_1(k1_numbers))
    & v1_membered(k1_subset_1(k1_numbers))
    & v2_membered(k1_subset_1(k1_numbers))
    & v3_membered(k1_subset_1(k1_numbers))
    & v4_membered(k1_subset_1(k1_numbers))
    & v5_membered(k1_subset_1(k1_numbers))
    & v3_seq_4(k1_subset_1(k1_numbers))
    & v1_realset1(k1_subset_1(k1_numbers)) )).

fof(fc9_sprect_1,axiom,(
    ! [A,B] :
      ( ( v1_xreal_0(A)
        & v1_xreal_0(B) )
     => ( v1_membered(k1_rcomp_1(A,B))
        & v2_membered(k1_rcomp_1(A,B))
        & v3_seq_4(k1_rcomp_1(A,B)) ) ) )).

fof(cc3_sprect_1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_zfmisc_1(k1_numbers))
     => ( v3_seq_4(A)
       => ( v1_membered(A)
          & v2_membered(A)
          & v1_seq_4(A)
          & v2_seq_4(A) ) ) ) )).

fof(cc4_sprect_1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_zfmisc_1(k1_numbers))
     => ( ( v1_seq_4(A)
          & v2_seq_4(A) )
       => ( v1_membered(A)
          & v2_membered(A)
          & v3_seq_4(A) ) ) ) )).

fof(fc10_sprect_1,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & ~ v1_sppol_1(A)
        & ~ v2_sppol_1(A)
        & v6_compts_1(A,k15_euclid(np__2))
        & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
     => ( ~ v1_xboole_0(k1_sprect_1(A))
        & v1_relat_1(k1_sprect_1(A))
        & v1_funct_1(k1_sprect_1(A))
        & v1_finset_1(k1_sprect_1(A))
        & v1_finseq_1(k1_sprect_1(A))
        & ~ v1_realset1(k1_sprect_1(A))
        & ~ v5_seqm_3(k1_sprect_1(A))
        & v1_topreal1(k1_sprect_1(A))
        & v2_topreal1(k1_sprect_1(A))
        & v1_finseq_6(k1_sprect_1(A),u1_struct_0(k15_euclid(np__2)))
        & v1_goboard5(k1_sprect_1(A))
        & v2_goboard5(k1_sprect_1(A))
        & v1_sprect_1(k1_sprect_1(A)) ) ) )).

fof(rc4_sprect_1,axiom,(
    ? [A] :
      ( m1_finseq_1(A,u1_struct_0(k15_euclid(np__2)))
      & v1_relat_1(A)
      & v1_funct_1(A)
      & v1_finset_1(A)
      & v1_finseq_1(A)
      & v1_sprect_1(A) ) )).

fof(cc5_sprect_1,axiom,(
    ! [A] :
      ( m1_finseq_1(A,u1_struct_0(k15_euclid(np__2)))
     => ( v1_sprect_1(A)
       => ~ v5_seqm_3(A) ) ) )).

fof(cc6_sprect_1,axiom,(
    ! [A] :
      ( m1_finseq_1(A,u1_struct_0(k15_euclid(np__2)))
     => ( ( ~ v1_xboole_0(A)
          & v1_sprect_1(A) )
       => ( ~ v1_xboole_0(A)
          & v1_topreal1(A)
          & v2_topreal1(A)
          & v1_finseq_6(A,u1_struct_0(k15_euclid(np__2)))
          & v1_goboard5(A)
          & v2_goboard5(A) ) ) ) )).

fof(fc11_sprect_1,axiom,(
    ! [A] :
      ( ( v1_sprect_1(A)
        & m1_finseq_1(A,u1_struct_0(k15_euclid(np__2))) )
     => ( ~ v1_xboole_0(k5_topreal1(np__2,A))
        & v2_jordan1(k5_topreal1(np__2,A))
        & v6_compts_1(k5_topreal1(np__2,A),k15_euclid(np__2)) ) ) )).

fof(fc12_sprect_1,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & ~ v5_seqm_3(A)
        & v1_topreal1(A)
        & v2_topreal1(A)
        & v1_finseq_6(A,u1_struct_0(k15_euclid(np__2)))
        & v1_goboard5(A)
        & v2_goboard5(A)
        & m1_finseq_1(A,u1_struct_0(k15_euclid(np__2))) )
     => ~ v1_xboole_0(k2_goboard9(A)) ) )).

fof(fc13_sprect_1,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & ~ v5_seqm_3(A)
        & v1_topreal1(A)
        & v2_topreal1(A)
        & v1_finseq_6(A,u1_struct_0(k15_euclid(np__2)))
        & v1_goboard5(A)
        & v2_goboard5(A)
        & m1_finseq_1(A,u1_struct_0(k15_euclid(np__2))) )
     => ~ v1_xboole_0(k3_goboard9(A)) ) )).

fof(t1_sprect_1,axiom,(
    ! [A] :
      ( v1_realset1(A)
     => ! [B] :
          ( r1_tarski(B,A)
         => v1_realset1(B) ) ) )).

fof(t2_sprect_1,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ( v1_realset1(k2_relat_1(A))
       => v5_seqm_3(A) ) ) )).

fof(t3_sprect_1,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_finseq_1(A) )
     => ! [B] :
          ( ( v1_relat_1(B)
            & v1_funct_1(B)
            & v1_finseq_1(B) )
         => ( v5_seqm_3(k7_finseq_1(A,B))
           => ( v5_seqm_3(A)
              & v5_seqm_3(B) ) ) ) ) )).

fof(t4_sprect_1,axiom,(
    ! [A,B] :
      ( v5_seqm_3(k10_finseq_1(A,B))
     => A = B ) )).

fof(t5_sprect_1,axiom,(
    ! [A,B,C] :
      ( v5_seqm_3(k11_finseq_1(A,B,C))
     => ( A = B
        & B = C
        & C = A ) ) )).

fof(t6_sprect_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => ! [C] :
              ( ( ~ v1_xboole_0(C)
                & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
             => ~ ( r4_connsp_1(A,C,B)
                  & B = k1_xboole_0 ) ) ) ) )).

fof(t7_sprect_1,axiom,(
    ! [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)))
             => ( r4_connsp_1(A,C,B)
               => r1_tarski(B,C) ) ) ) ) )).

fof(t8_sprect_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ! [B] :
          ( ( ~ v1_xboole_0(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,k1_zfmisc_1(u1_struct_0(A)))
                 => ! [E] :
                      ( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(A)))
                     => ~ ( r4_connsp_1(A,B,C)
                          & r4_connsp_1(A,B,D)
                          & r4_connsp_1(A,B,E)
                          & k4_subset_1(u1_struct_0(A),C,D) = B
                          & E != C
                          & E != D ) ) ) ) ) ) )).

fof(t9_sprect_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ! [B] :
          ( ( ~ v1_xboole_0(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,k1_zfmisc_1(u1_struct_0(A)))
                 => ! [E] :
                      ( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(A)))
                     => ! [F] :
                          ( m1_subset_1(F,k1_zfmisc_1(u1_struct_0(A)))
                         => ( ( r4_connsp_1(A,B,C)
                              & r4_connsp_1(A,B,D)
                              & r4_connsp_1(A,B,E)
                              & r4_connsp_1(A,B,F)
                              & k4_subset_1(u1_struct_0(A),C,D) = B
                              & k4_subset_1(u1_struct_0(A),E,F) = B )
                           => k2_tarski(C,D) = k2_tarski(E,F) ) ) ) ) ) ) ) )).

fof(t10_sprect_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)))
             => k5_topreal1(np__2,k3_finseq_4(u1_struct_0(k15_euclid(np__2)),A,B,C)) = k4_subset_1(u1_struct_0(k15_euclid(np__2)),k3_topreal1(np__2,A,B),k3_topreal1(np__2,B,C)) ) ) ) )).

fof(t11_sprect_1,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))))
         => ( ( r1_tarski(A,B)
              & v1_sppol_1(B) )
           => v1_sppol_1(A) ) ) ) )).

fof(t12_sprect_1,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))))
         => ( ( r1_tarski(A,B)
              & v2_sppol_1(B) )
           => v2_sppol_1(A) ) ) ) )).

fof(t13_sprect_1,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v6_compts_1(A,k15_euclid(np__2))
        & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
     => ( r2_hidden(k32_pscomp_1(A),A)
        & r2_hidden(k33_pscomp_1(A),A) ) ) )).

fof(t14_sprect_1,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v6_compts_1(A,k15_euclid(np__2))
        & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
     => ( r2_hidden(k37_pscomp_1(A),A)
        & r2_hidden(k36_pscomp_1(A),A) ) ) )).

fof(t15_sprect_1,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v6_compts_1(A,k15_euclid(np__2))
        & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
     => ( r2_hidden(k30_pscomp_1(A),A)
        & r2_hidden(k31_pscomp_1(A),A) ) ) )).

fof(t16_sprect_1,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v6_compts_1(A,k15_euclid(np__2))
        & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
     => ( r2_hidden(k35_pscomp_1(A),A)
        & r2_hidden(k34_pscomp_1(A),A) ) ) )).

fof(t17_sprect_1,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v6_compts_1(A,k15_euclid(np__2))
        & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
     => ( v2_sppol_1(A)
      <=> k18_pscomp_1(A) = k20_pscomp_1(A) ) ) )).

fof(t18_sprect_1,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v6_compts_1(A,k15_euclid(np__2))
        & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
     => ( v1_sppol_1(A)
      <=> k21_pscomp_1(A) = k19_pscomp_1(A) ) ) )).

fof(t19_sprect_1,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v6_compts_1(A,k15_euclid(np__2))
        & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
     => ( k23_pscomp_1(A) = k24_pscomp_1(A)
       => v2_sppol_1(A) ) ) )).

fof(t20_sprect_1,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v6_compts_1(A,k15_euclid(np__2))
        & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
     => ( k22_pscomp_1(A) = k25_pscomp_1(A)
       => v2_sppol_1(A) ) ) )).

fof(t21_sprect_1,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v6_compts_1(A,k15_euclid(np__2))
        & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
     => ( k23_pscomp_1(A) = k22_pscomp_1(A)
       => v1_sppol_1(A) ) ) )).

fof(t22_sprect_1,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v6_compts_1(A,k15_euclid(np__2))
        & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
     => ( k24_pscomp_1(A) = k25_pscomp_1(A)
       => v1_sppol_1(A) ) ) )).

fof(t23_sprect_1,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v6_compts_1(A,k15_euclid(np__2))
        & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
     => r1_xreal_0(k18_pscomp_1(A),k20_pscomp_1(A)) ) )).

fof(t24_sprect_1,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v6_compts_1(A,k15_euclid(np__2))
        & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
     => r1_xreal_0(k21_pscomp_1(A),k19_pscomp_1(A)) ) )).

fof(t29_sprect_1,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v6_compts_1(A,k15_euclid(np__2))
        & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
     => k5_subset_1(u1_struct_0(k15_euclid(np__2)),k3_topreal1(np__2,k22_pscomp_1(A),k23_pscomp_1(A)),k3_topreal1(np__2,k23_pscomp_1(A),k24_pscomp_1(A))) = k1_struct_0(k15_euclid(np__2),k23_pscomp_1(A)) ) )).

fof(t30_sprect_1,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v6_compts_1(A,k15_euclid(np__2))
        & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
     => k5_subset_1(u1_struct_0(k15_euclid(np__2)),k3_topreal1(np__2,k23_pscomp_1(A),k24_pscomp_1(A)),k3_topreal1(np__2,k24_pscomp_1(A),k25_pscomp_1(A))) = k1_struct_0(k15_euclid(np__2),k24_pscomp_1(A)) ) )).

fof(t31_sprect_1,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v6_compts_1(A,k15_euclid(np__2))
        & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
     => k5_subset_1(u1_struct_0(k15_euclid(np__2)),k3_topreal1(np__2,k25_pscomp_1(A),k24_pscomp_1(A)),k3_topreal1(np__2,k22_pscomp_1(A),k25_pscomp_1(A))) = k1_struct_0(k15_euclid(np__2),k25_pscomp_1(A)) ) )).

fof(t32_sprect_1,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v6_compts_1(A,k15_euclid(np__2))
        & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
     => k5_subset_1(u1_struct_0(k15_euclid(np__2)),k3_topreal1(np__2,k23_pscomp_1(A),k22_pscomp_1(A)),k3_topreal1(np__2,k22_pscomp_1(A),k25_pscomp_1(A))) = k1_struct_0(k15_euclid(np__2),k22_pscomp_1(A)) ) )).

fof(t33_sprect_1,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & ~ v2_sppol_1(A)
        & v6_compts_1(A,k15_euclid(np__2))
        & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
     => ~ r1_xreal_0(k20_pscomp_1(A),k18_pscomp_1(A)) ) )).

fof(t34_sprect_1,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & ~ v1_sppol_1(A)
        & v6_compts_1(A,k15_euclid(np__2))
        & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
     => ~ r1_xreal_0(k19_pscomp_1(A),k21_pscomp_1(A)) ) )).

fof(t35_sprect_1,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & ~ v2_sppol_1(A)
        & v6_compts_1(A,k15_euclid(np__2))
        & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
     => r1_subset_1(k3_topreal1(np__2,k22_pscomp_1(A),k23_pscomp_1(A)),k3_topreal1(np__2,k25_pscomp_1(A),k24_pscomp_1(A))) ) )).

fof(t36_sprect_1,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & ~ v1_sppol_1(A)
        & v6_compts_1(A,k15_euclid(np__2))
        & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
     => r1_subset_1(k3_topreal1(np__2,k22_pscomp_1(A),k25_pscomp_1(A)),k3_topreal1(np__2,k23_pscomp_1(A),k24_pscomp_1(A))) ) )).

fof(d1_sprect_1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
     => k1_sprect_1(A) = k8_finseq_1(u1_struct_0(k15_euclid(np__2)),k3_finseq_4(u1_struct_0(k15_euclid(np__2)),k23_pscomp_1(A),k24_pscomp_1(A),k25_pscomp_1(A)),k2_finseq_4(u1_struct_0(k15_euclid(np__2)),k22_pscomp_1(A),k23_pscomp_1(A))) ) )).

fof(t37_sprect_1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
     => k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),k1_sprect_1(A),np__1) = k23_pscomp_1(A) ) )).

fof(t38_sprect_1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
     => k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),k1_sprect_1(A),np__2) = k24_pscomp_1(A) ) )).

fof(t39_sprect_1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
     => k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),k1_sprect_1(A),np__3) = k25_pscomp_1(A) ) )).

fof(t40_sprect_1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
     => k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),k1_sprect_1(A),np__4) = k22_pscomp_1(A) ) )).

fof(t41_sprect_1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
     => k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),k1_sprect_1(A),np__5) = k23_pscomp_1(A) ) )).

fof(t42_sprect_1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
     => k3_finseq_1(k1_sprect_1(A)) = np__5 ) )).

fof(t43_sprect_1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
     => k5_topreal1(np__2,k1_sprect_1(A)) = k4_subset_1(u1_struct_0(k15_euclid(np__2)),k4_subset_1(u1_struct_0(k15_euclid(np__2)),k3_topreal1(np__2,k23_pscomp_1(A),k24_pscomp_1(A)),k3_topreal1(np__2,k24_pscomp_1(A),k25_pscomp_1(A))),k4_subset_1(u1_struct_0(k15_euclid(np__2)),k3_topreal1(np__2,k25_pscomp_1(A),k22_pscomp_1(A)),k3_topreal1(np__2,k22_pscomp_1(A),k23_pscomp_1(A)))) ) )).

fof(t44_sprect_1,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & ~ v1_sppol_1(A)
        & ~ v2_sppol_1(A)
        & v6_compts_1(A,k15_euclid(np__2))
        & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
     => k5_topreal1(np__2,k1_sprect_1(A)) = k1_sppol_2(k18_pscomp_1(A),k20_pscomp_1(A),k21_pscomp_1(A),k19_pscomp_1(A)) ) )).

fof(t45_sprect_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_pre_topc(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,u1_struct_0(A),k1_numbers)
                & m2_relset_1(C,u1_struct_0(A),k1_numbers) )
             => k1_pscomp_1(u1_struct_0(k3_pre_topc(A,B)),k1_numbers,k15_pscomp_1(A,B,C)) = k2_funct_2(u1_struct_0(A),k1_numbers,C,B) ) ) ) )).

fof(t46_sprect_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & v6_compts_1(B,A)
            & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,u1_struct_0(A),k1_numbers)
                & v9_pscomp_1(C,A)
                & m2_relset_1(C,u1_struct_0(A),k1_numbers) )
             => v2_seq_4(k2_funct_2(u1_struct_0(A),k1_numbers,C,B)) ) ) ) )).

fof(t47_sprect_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & v6_compts_1(B,A)
            & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,u1_struct_0(A),k1_numbers)
                & v9_pscomp_1(C,A)
                & m2_relset_1(C,u1_struct_0(A),k1_numbers) )
             => v1_seq_4(k2_funct_2(u1_struct_0(A),k1_numbers,C,B)) ) ) ) )).

fof(t48_sprect_1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
     => k18_pscomp_1(A) = k4_pscomp_1(k2_funct_2(u1_struct_0(k15_euclid(np__2)),k1_numbers,k16_pscomp_1,A)) ) )).

fof(t49_sprect_1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
     => k21_pscomp_1(A) = k4_pscomp_1(k2_funct_2(u1_struct_0(k15_euclid(np__2)),k1_numbers,k17_pscomp_1,A)) ) )).

fof(t50_sprect_1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
     => k19_pscomp_1(A) = k3_pscomp_1(k2_funct_2(u1_struct_0(k15_euclid(np__2)),k1_numbers,k17_pscomp_1,A)) ) )).

fof(t51_sprect_1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
     => k20_pscomp_1(A) = k3_pscomp_1(k2_funct_2(u1_struct_0(k15_euclid(np__2)),k1_numbers,k16_pscomp_1,A)) ) )).

fof(t52_sprect_1,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v2_seq_4(A)
        & m1_subset_1(A,k1_zfmisc_1(k1_numbers)) )
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & v2_seq_4(B)
            & m1_subset_1(B,k1_zfmisc_1(k1_numbers)) )
         => k4_pscomp_1(k4_subset_1(k1_numbers,A,B)) = k3_square_1(k4_pscomp_1(A),k4_pscomp_1(B)) ) ) )).

fof(t53_sprect_1,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_seq_4(A)
        & m1_subset_1(A,k1_zfmisc_1(k1_numbers)) )
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & v1_seq_4(B)
            & m1_subset_1(B,k1_zfmisc_1(k1_numbers)) )
         => k3_pscomp_1(k4_subset_1(k1_numbers,A,B)) = k4_square_1(k3_pscomp_1(A),k3_pscomp_1(B)) ) ) )).

fof(t54_sprect_1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & v6_compts_1(B,k15_euclid(np__2))
            & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
         => ! [C] :
              ( ( ~ v1_xboole_0(C)
                & v6_compts_1(C,k15_euclid(np__2))
                & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
             => ( A = k4_subset_1(u1_struct_0(k15_euclid(np__2)),B,C)
               => k18_pscomp_1(A) = k3_square_1(k18_pscomp_1(B),k18_pscomp_1(C)) ) ) ) ) )).

fof(t55_sprect_1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & v6_compts_1(B,k15_euclid(np__2))
            & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
         => ! [C] :
              ( ( ~ v1_xboole_0(C)
                & v6_compts_1(C,k15_euclid(np__2))
                & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
             => ( A = k4_subset_1(u1_struct_0(k15_euclid(np__2)),B,C)
               => k21_pscomp_1(A) = k3_square_1(k21_pscomp_1(B),k21_pscomp_1(C)) ) ) ) ) )).

fof(t56_sprect_1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & v6_compts_1(B,k15_euclid(np__2))
            & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
         => ! [C] :
              ( ( ~ v1_xboole_0(C)
                & v6_compts_1(C,k15_euclid(np__2))
                & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
             => ( A = k4_subset_1(u1_struct_0(k15_euclid(np__2)),B,C)
               => k19_pscomp_1(A) = k4_square_1(k19_pscomp_1(B),k19_pscomp_1(C)) ) ) ) ) )).

fof(t57_sprect_1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & v6_compts_1(B,k15_euclid(np__2))
            & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
         => ! [C] :
              ( ( ~ v1_xboole_0(C)
                & v6_compts_1(C,k15_euclid(np__2))
                & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
             => ( A = k4_subset_1(u1_struct_0(k15_euclid(np__2)),B,C)
               => k20_pscomp_1(A) = k4_square_1(k20_pscomp_1(B),k20_pscomp_1(C)) ) ) ) ) )).

fof(t58_sprect_1,axiom,(
    $true )).

fof(t59_sprect_1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ! [C] :
              ( m1_subset_1(C,k1_numbers)
             => ( r1_xreal_0(A,B)
               => ( r2_hidden(C,k1_rcomp_1(A,B))
                <=> ? [D] :
                      ( m1_subset_1(D,k1_numbers)
                      & r1_xreal_0(np__0,D)
                      & r1_xreal_0(D,np__1)
                      & C = k3_real_1(k4_real_1(D,A),k4_real_1(k5_real_1(np__1,D),B)) ) ) ) ) ) ) )).

fof(t60_sprect_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)))
         => ( r1_xreal_0(k21_euclid(A),k21_euclid(B))
           => k2_funct_2(u1_struct_0(k15_euclid(np__2)),k1_numbers,k16_pscomp_1,k3_topreal1(np__2,A,B)) = k1_rcomp_1(k21_euclid(A),k21_euclid(B)) ) ) ) )).

fof(t61_sprect_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)))
         => ( r1_xreal_0(k22_euclid(A),k22_euclid(B))
           => k2_funct_2(u1_struct_0(k15_euclid(np__2)),k1_numbers,k17_pscomp_1,k3_topreal1(np__2,A,B)) = k1_rcomp_1(k22_euclid(A),k22_euclid(B)) ) ) ) )).

fof(t62_sprect_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)))
         => ( r1_xreal_0(k21_euclid(A),k21_euclid(B))
           => k18_pscomp_1(k3_topreal1(np__2,A,B)) = k21_euclid(A) ) ) ) )).

fof(t63_sprect_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)))
         => ( r1_xreal_0(k22_euclid(A),k22_euclid(B))
           => k21_pscomp_1(k3_topreal1(np__2,A,B)) = k22_euclid(A) ) ) ) )).

fof(t64_sprect_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)))
         => ( r1_xreal_0(k22_euclid(A),k22_euclid(B))
           => k19_pscomp_1(k3_topreal1(np__2,A,B)) = k22_euclid(B) ) ) ) )).

fof(t65_sprect_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)))
         => ( r1_xreal_0(k21_euclid(A),k21_euclid(B))
           => k20_pscomp_1(k3_topreal1(np__2,A,B)) = k21_euclid(B) ) ) ) )).

fof(t66_sprect_1,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v6_compts_1(A,k15_euclid(np__2))
        & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
     => k18_pscomp_1(k5_topreal1(np__2,k1_sprect_1(A))) = k18_pscomp_1(A) ) )).

fof(t67_sprect_1,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v6_compts_1(A,k15_euclid(np__2))
        & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
     => k21_pscomp_1(k5_topreal1(np__2,k1_sprect_1(A))) = k21_pscomp_1(A) ) )).

fof(t68_sprect_1,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v6_compts_1(A,k15_euclid(np__2))
        & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
     => k19_pscomp_1(k5_topreal1(np__2,k1_sprect_1(A))) = k19_pscomp_1(A) ) )).

fof(t69_sprect_1,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v6_compts_1(A,k15_euclid(np__2))
        & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
     => k20_pscomp_1(k5_topreal1(np__2,k1_sprect_1(A))) = k20_pscomp_1(A) ) )).

fof(t70_sprect_1,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v6_compts_1(A,k15_euclid(np__2))
        & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
     => k23_pscomp_1(k5_topreal1(np__2,k1_sprect_1(A))) = k23_pscomp_1(A) ) )).

fof(t71_sprect_1,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v6_compts_1(A,k15_euclid(np__2))
        & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
     => k24_pscomp_1(k5_topreal1(np__2,k1_sprect_1(A))) = k24_pscomp_1(A) ) )).

fof(t72_sprect_1,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v6_compts_1(A,k15_euclid(np__2))
        & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
     => k22_pscomp_1(k5_topreal1(np__2,k1_sprect_1(A))) = k22_pscomp_1(A) ) )).

fof(t73_sprect_1,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v6_compts_1(A,k15_euclid(np__2))
        & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
     => k25_pscomp_1(k5_topreal1(np__2,k1_sprect_1(A))) = k25_pscomp_1(A) ) )).

fof(t74_sprect_1,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v6_compts_1(A,k15_euclid(np__2))
        & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
     => k26_pscomp_1(k5_topreal1(np__2,k1_sprect_1(A))) = k3_topreal1(np__2,k22_pscomp_1(A),k23_pscomp_1(A)) ) )).

fof(t75_sprect_1,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v6_compts_1(A,k15_euclid(np__2))
        & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
     => k27_pscomp_1(k5_topreal1(np__2,k1_sprect_1(A))) = k3_topreal1(np__2,k23_pscomp_1(A),k24_pscomp_1(A)) ) )).

fof(t76_sprect_1,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v6_compts_1(A,k15_euclid(np__2))
        & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
     => k29_pscomp_1(k5_topreal1(np__2,k1_sprect_1(A))) = k3_topreal1(np__2,k22_pscomp_1(A),k25_pscomp_1(A)) ) )).

fof(t77_sprect_1,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v6_compts_1(A,k15_euclid(np__2))
        & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
     => k28_pscomp_1(k5_topreal1(np__2,k1_sprect_1(A))) = k3_topreal1(np__2,k25_pscomp_1(A),k24_pscomp_1(A)) ) )).

fof(t78_sprect_1,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v6_compts_1(A,k15_euclid(np__2))
        & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
     => k2_funct_2(u1_struct_0(k15_euclid(np__2)),k1_numbers,k17_pscomp_1,k3_topreal1(np__2,k22_pscomp_1(A),k23_pscomp_1(A))) = k1_rcomp_1(k21_pscomp_1(A),k19_pscomp_1(A)) ) )).

fof(t79_sprect_1,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v6_compts_1(A,k15_euclid(np__2))
        & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
     => k2_funct_2(u1_struct_0(k15_euclid(np__2)),k1_numbers,k16_pscomp_1,k3_topreal1(np__2,k23_pscomp_1(A),k24_pscomp_1(A))) = k1_rcomp_1(k18_pscomp_1(A),k20_pscomp_1(A)) ) )).

fof(t80_sprect_1,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v6_compts_1(A,k15_euclid(np__2))
        & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
     => k2_funct_2(u1_struct_0(k15_euclid(np__2)),k1_numbers,k17_pscomp_1,k3_topreal1(np__2,k24_pscomp_1(A),k25_pscomp_1(A))) = k1_rcomp_1(k21_pscomp_1(A),k19_pscomp_1(A)) ) )).

fof(t81_sprect_1,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v6_compts_1(A,k15_euclid(np__2))
        & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
     => k2_funct_2(u1_struct_0(k15_euclid(np__2)),k1_numbers,k16_pscomp_1,k3_topreal1(np__2,k25_pscomp_1(A),k22_pscomp_1(A))) = k1_rcomp_1(k18_pscomp_1(A),k20_pscomp_1(A)) ) )).

fof(t82_sprect_1,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v6_compts_1(A,k15_euclid(np__2))
        & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
     => k30_pscomp_1(k5_topreal1(np__2,k1_sprect_1(A))) = k22_pscomp_1(A) ) )).

fof(t83_sprect_1,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v6_compts_1(A,k15_euclid(np__2))
        & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
     => k31_pscomp_1(k5_topreal1(np__2,k1_sprect_1(A))) = k23_pscomp_1(A) ) )).

fof(t84_sprect_1,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v6_compts_1(A,k15_euclid(np__2))
        & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
     => k32_pscomp_1(k5_topreal1(np__2,k1_sprect_1(A))) = k23_pscomp_1(A) ) )).

fof(t85_sprect_1,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v6_compts_1(A,k15_euclid(np__2))
        & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
     => k33_pscomp_1(k5_topreal1(np__2,k1_sprect_1(A))) = k24_pscomp_1(A) ) )).

fof(t86_sprect_1,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v6_compts_1(A,k15_euclid(np__2))
        & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
     => k35_pscomp_1(k5_topreal1(np__2,k1_sprect_1(A))) = k25_pscomp_1(A) ) )).

fof(t87_sprect_1,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v6_compts_1(A,k15_euclid(np__2))
        & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
     => k34_pscomp_1(k5_topreal1(np__2,k1_sprect_1(A))) = k24_pscomp_1(A) ) )).

fof(t88_sprect_1,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v6_compts_1(A,k15_euclid(np__2))
        & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
     => k37_pscomp_1(k5_topreal1(np__2,k1_sprect_1(A))) = k22_pscomp_1(A) ) )).

fof(t89_sprect_1,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v6_compts_1(A,k15_euclid(np__2))
        & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
     => k36_pscomp_1(k5_topreal1(np__2,k1_sprect_1(A))) = k25_pscomp_1(A) ) )).

fof(d2_sprect_1,axiom,(
    ! [A] :
      ( m2_finseq_1(A,u1_struct_0(k15_euclid(np__2)))
     => ( v1_sprect_1(A)
      <=> ? [B] :
            ( ~ v1_xboole_0(B)
            & ~ v1_sppol_1(B)
            & ~ v2_sppol_1(B)
            & v6_compts_1(B,k15_euclid(np__2))
            & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
            & A = k1_sprect_1(B) ) ) ) )).

fof(t90_sprect_1,axiom,(
    ! [A] :
      ( ( v1_sprect_1(A)
        & m2_finseq_1(A,u1_struct_0(k15_euclid(np__2))) )
     => k3_finseq_1(A) = np__5 ) )).

fof(t91_sprect_1,axiom,(
    ! [A] :
      ( ( v1_sprect_1(A)
        & m2_finseq_1(A,u1_struct_0(k15_euclid(np__2))) )
     => ( k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),A,np__1) = k32_pscomp_1(k5_topreal1(np__2,A))
        & k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),A,np__1) = k31_pscomp_1(k5_topreal1(np__2,A)) ) ) )).

fof(t92_sprect_1,axiom,(
    ! [A] :
      ( ( v1_sprect_1(A)
        & m2_finseq_1(A,u1_struct_0(k15_euclid(np__2))) )
     => ( k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),A,np__2) = k33_pscomp_1(k5_topreal1(np__2,A))
        & k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),A,np__2) = k34_pscomp_1(k5_topreal1(np__2,A)) ) ) )).

fof(t93_sprect_1,axiom,(
    ! [A] :
      ( ( v1_sprect_1(A)
        & m2_finseq_1(A,u1_struct_0(k15_euclid(np__2))) )
     => ( k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),A,np__3) = k36_pscomp_1(k5_topreal1(np__2,A))
        & k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),A,np__3) = k35_pscomp_1(k5_topreal1(np__2,A)) ) ) )).

fof(t94_sprect_1,axiom,(
    ! [A] :
      ( ( v1_sprect_1(A)
        & m2_finseq_1(A,u1_struct_0(k15_euclid(np__2))) )
     => ( k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),A,np__4) = k37_pscomp_1(k5_topreal1(np__2,A))
        & k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),A,np__4) = k30_pscomp_1(k5_topreal1(np__2,A)) ) ) )).

fof(t95_sprect_1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ! [C] :
              ( m1_subset_1(C,k1_numbers)
             => ! [D] :
                  ( m1_subset_1(D,k1_numbers)
                 => ~ ( ~ r1_xreal_0(B,A)
                      & ~ r1_xreal_0(D,C)
                      & ~ v2_jordan1(k1_sppol_2(A,B,C,D)) ) ) ) ) ) )).

fof(d3_sprect_1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
     => ( v2_jordan1(A)
      <=> ( k3_subset_1(u1_struct_0(k15_euclid(np__2)),A) != k1_xboole_0
          & ? [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))))
                  & k3_subset_1(u1_struct_0(k15_euclid(np__2)),A) = k4_subset_1(u1_struct_0(k15_euclid(np__2)),B,C)
                  & r1_xboole_0(B,C)
                  & k6_subset_1(u1_struct_0(k15_euclid(np__2)),k6_pre_topc(k15_euclid(np__2),B),B) = k6_subset_1(u1_struct_0(k15_euclid(np__2)),k6_pre_topc(k15_euclid(np__2),C),C)
                  & r4_connsp_1(k15_euclid(np__2),k3_subset_1(u1_struct_0(k15_euclid(np__2)),A),B)
                  & r4_connsp_1(k15_euclid(np__2),k3_subset_1(u1_struct_0(k15_euclid(np__2)),A),C) ) ) ) ) ) )).

fof(t96_sprect_1,axiom,(
    ! [A] :
      ( ( v1_sprect_1(A)
        & m2_finseq_1(A,u1_struct_0(k15_euclid(np__2))) )
     => r1_xboole_0(k2_goboard9(A),k3_goboard9(A)) ) )).

fof(t97_sprect_1,axiom,(
    ! [A] :
      ( ( v1_sprect_1(A)
        & m2_finseq_1(A,u1_struct_0(k15_euclid(np__2))) )
     => k2_goboard9(A) != k3_goboard9(A) ) )).

fof(dt_k1_sprect_1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
     => m2_finseq_1(k1_sprect_1(A),u1_struct_0(k15_euclid(np__2))) ) )).

fof(t25_sprect_1,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v6_compts_1(A,k15_euclid(np__2))
        & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
     => k3_topreal1(np__2,k25_pscomp_1(A),k24_pscomp_1(A)) = a_1_0_sprect_1(A) ) )).

fof(t26_sprect_1,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v6_compts_1(A,k15_euclid(np__2))
        & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
     => k3_topreal1(np__2,k22_pscomp_1(A),k25_pscomp_1(A)) = a_1_1_sprect_1(A) ) )).

fof(t27_sprect_1,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v6_compts_1(A,k15_euclid(np__2))
        & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
     => k3_topreal1(np__2,k23_pscomp_1(A),k24_pscomp_1(A)) = a_1_2_sprect_1(A) ) )).

fof(t28_sprect_1,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v6_compts_1(A,k15_euclid(np__2))
        & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
     => k3_topreal1(np__2,k22_pscomp_1(A),k23_pscomp_1(A)) = a_1_3_sprect_1(A) ) )).

fof(fraenkel_a_1_0_sprect_1,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(B)
        & v6_compts_1(B,k15_euclid(np__2))
        & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
     => ( r2_hidden(A,a_1_0_sprect_1(B))
      <=> ? [C] :
            ( m1_subset_1(C,u1_struct_0(k15_euclid(np__2)))
            & A = C
            & k21_euclid(C) = k20_pscomp_1(B)
            & r1_xreal_0(k22_euclid(C),k19_pscomp_1(B))
            & r1_xreal_0(k21_pscomp_1(B),k22_euclid(C)) ) ) ) )).

fof(fraenkel_a_1_1_sprect_1,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(B)
        & v6_compts_1(B,k15_euclid(np__2))
        & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
     => ( r2_hidden(A,a_1_1_sprect_1(B))
      <=> ? [C] :
            ( m1_subset_1(C,u1_struct_0(k15_euclid(np__2)))
            & A = C
            & r1_xreal_0(k21_euclid(C),k20_pscomp_1(B))
            & r1_xreal_0(k18_pscomp_1(B),k21_euclid(C))
            & k22_euclid(C) = k21_pscomp_1(B) ) ) ) )).

fof(fraenkel_a_1_2_sprect_1,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(B)
        & v6_compts_1(B,k15_euclid(np__2))
        & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
     => ( r2_hidden(A,a_1_2_sprect_1(B))
      <=> ? [C] :
            ( m1_subset_1(C,u1_struct_0(k15_euclid(np__2)))
            & A = C
            & r1_xreal_0(k21_euclid(C),k20_pscomp_1(B))
            & r1_xreal_0(k18_pscomp_1(B),k21_euclid(C))
            & k22_euclid(C) = k19_pscomp_1(B) ) ) ) )).

fof(fraenkel_a_1_3_sprect_1,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(B)
        & v6_compts_1(B,k15_euclid(np__2))
        & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
     => ( r2_hidden(A,a_1_3_sprect_1(B))
      <=> ? [C] :
            ( m1_subset_1(C,u1_struct_0(k15_euclid(np__2)))
            & A = C
            & k21_euclid(C) = k18_pscomp_1(B)
            & r1_xreal_0(k22_euclid(C),k19_pscomp_1(B))
            & r1_xreal_0(k21_pscomp_1(B),k22_euclid(C)) ) ) ) )).
%------------------------------------------------------------------------------