SET007 Axioms: SET007+615.ax


%------------------------------------------------------------------------------
% File     : SET007+615 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Properties of the External Approximation of Jordan's Curve
% 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    : jordan10 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   35 (   2 unit)
%            Number of atoms       :  256 (  18 equality)
%            Maximal formula depth :   18 (   9 average)
%            Number of connectives :  297 (  76 ~  ;   0  |; 155  &)
%                                         (   2 <=>;  64 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   18 (   1 propositional; 0-3 arity)
%            Number of functors    :   39 (   4 constant; 0-4 arity)
%            Number of variables   :   69 (   0 singleton;  66 !;   3 ?)
%            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(rc1_jordan10,axiom,(
    ? [A] :
      ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
      & ~ v1_xboole_0(A)
      & v4_pre_topc(A,k15_euclid(np__2))
      & v2_connsp_1(A,k15_euclid(np__2))
      & v1_jordan2c(A,np__2)
      & v6_compts_1(A,k15_euclid(np__2))
      & ~ v1_sppol_1(A)
      & ~ v2_sppol_1(A) ) )).

fof(fc1_jordan10,axiom,(
    ! [A,B] :
      ( ( v2_connsp_1(A,k15_euclid(np__2))
        & v6_compts_1(A,k15_euclid(np__2))
        & ~ v1_sppol_1(A)
        & ~ v2_sppol_1(A)
        & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
        & m1_subset_1(B,k5_numbers) )
     => ( v4_pre_topc(k6_pre_topc(k15_euclid(np__2),k3_goboard9(k1_jordan9(A,B))),k15_euclid(np__2))
        & v1_jordan2c(k6_pre_topc(k15_euclid(np__2),k3_goboard9(k1_jordan9(A,B))),np__2)
        & v6_compts_1(k6_pre_topc(k15_euclid(np__2),k3_goboard9(k1_jordan9(A,B))),k15_euclid(np__2)) ) ) )).

fof(fc2_jordan10,axiom,(
    ! [A] :
      ( ( v6_compts_1(A,k15_euclid(np__2))
        & ~ v1_sppol_1(A)
        & ~ v2_sppol_1(A)
        & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
     => ~ v1_xboole_0(k1_jordan10(A)) ) )).

fof(fc3_jordan10,axiom,(
    ! [A] :
      ( ( v6_compts_1(A,k15_euclid(np__2))
        & ~ v1_sppol_1(A)
        & ~ v2_sppol_1(A)
        & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
     => ~ v1_xboole_0(k2_jordan10(A)) ) )).

fof(t1_jordan10,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)
             => ! [D] :
                  ( m2_subset_1(D,k1_numbers,k5_numbers)
                 => ! [E] :
                      ( ( v2_connsp_1(E,k15_euclid(np__2))
                        & v6_compts_1(E,k15_euclid(np__2))
                        & ~ v1_sppol_1(E)
                        & ~ v2_sppol_1(E)
                        & m1_subset_1(E,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
                     => ~ ( r1_xreal_0(np__1,A)
                          & r1_xreal_0(k1_nat_1(A,np__1),k3_finseq_1(k1_jordan9(E,B)))
                          & r2_hidden(k4_tarski(C,D),k2_matrix_1(k1_jordan8(E,B)))
                          & r2_hidden(k4_tarski(C,k1_nat_1(D,np__1)),k2_matrix_1(k1_jordan8(E,B)))
                          & k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),k1_jordan9(E,B),A) = k3_matrix_1(u1_struct_0(k15_euclid(np__2)),k1_jordan8(E,B),C,D)
                          & k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),k1_jordan9(E,B),k1_nat_1(A,np__1)) = k3_matrix_1(u1_struct_0(k15_euclid(np__2)),k1_jordan8(E,B),C,k1_nat_1(D,np__1))
                          & r1_xreal_0(k3_finseq_1(k1_jordan8(E,B)),C) ) ) ) ) ) ) )).

fof(t2_jordan10,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)
             => ! [D] :
                  ( m2_subset_1(D,k1_numbers,k5_numbers)
                 => ! [E] :
                      ( ( v2_connsp_1(E,k15_euclid(np__2))
                        & v6_compts_1(E,k15_euclid(np__2))
                        & ~ v1_sppol_1(E)
                        & ~ v2_sppol_1(E)
                        & m1_subset_1(E,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
                     => ~ ( r1_xreal_0(np__1,A)
                          & r1_xreal_0(k1_nat_1(A,np__1),k3_finseq_1(k1_jordan9(E,B)))
                          & r2_hidden(k4_tarski(C,D),k2_matrix_1(k1_jordan8(E,B)))
                          & r2_hidden(k4_tarski(C,k1_nat_1(D,np__1)),k2_matrix_1(k1_jordan8(E,B)))
                          & k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),k1_jordan9(E,B),A) = k3_matrix_1(u1_struct_0(k15_euclid(np__2)),k1_jordan8(E,B),C,k1_nat_1(D,np__1))
                          & k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),k1_jordan9(E,B),k1_nat_1(A,np__1)) = k3_matrix_1(u1_struct_0(k15_euclid(np__2)),k1_jordan8(E,B),C,D)
                          & r1_xreal_0(C,np__1) ) ) ) ) ) ) )).

fof(t3_jordan10,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)
             => ! [D] :
                  ( m2_subset_1(D,k1_numbers,k5_numbers)
                 => ! [E] :
                      ( ( v2_connsp_1(E,k15_euclid(np__2))
                        & v6_compts_1(E,k15_euclid(np__2))
                        & ~ v1_sppol_1(E)
                        & ~ v2_sppol_1(E)
                        & m1_subset_1(E,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
                     => ~ ( r1_xreal_0(np__1,A)
                          & r1_xreal_0(k1_nat_1(A,np__1),k3_finseq_1(k1_jordan9(E,B)))
                          & r2_hidden(k4_tarski(C,D),k2_matrix_1(k1_jordan8(E,B)))
                          & r2_hidden(k4_tarski(k1_nat_1(C,np__1),D),k2_matrix_1(k1_jordan8(E,B)))
                          & k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),k1_jordan9(E,B),A) = k3_matrix_1(u1_struct_0(k15_euclid(np__2)),k1_jordan8(E,B),C,D)
                          & k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),k1_jordan9(E,B),k1_nat_1(A,np__1)) = k3_matrix_1(u1_struct_0(k15_euclid(np__2)),k1_jordan8(E,B),k1_nat_1(C,np__1),D)
                          & r1_xreal_0(D,np__1) ) ) ) ) ) ) )).

fof(t4_jordan10,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)
             => ! [D] :
                  ( m2_subset_1(D,k1_numbers,k5_numbers)
                 => ! [E] :
                      ( ( v2_connsp_1(E,k15_euclid(np__2))
                        & v6_compts_1(E,k15_euclid(np__2))
                        & ~ v1_sppol_1(E)
                        & ~ v2_sppol_1(E)
                        & m1_subset_1(E,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
                     => ~ ( r1_xreal_0(np__1,A)
                          & r1_xreal_0(k1_nat_1(A,np__1),k3_finseq_1(k1_jordan9(E,B)))
                          & r2_hidden(k4_tarski(C,D),k2_matrix_1(k1_jordan8(E,B)))
                          & r2_hidden(k4_tarski(k1_nat_1(C,np__1),D),k2_matrix_1(k1_jordan8(E,B)))
                          & k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),k1_jordan9(E,B),A) = k3_matrix_1(u1_struct_0(k15_euclid(np__2)),k1_jordan8(E,B),k1_nat_1(C,np__1),D)
                          & k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),k1_jordan9(E,B),k1_nat_1(A,np__1)) = k3_matrix_1(u1_struct_0(k15_euclid(np__2)),k1_jordan8(E,B),C,D)
                          & r1_xreal_0(k1_matrix_1(k1_jordan8(E,B)),D) ) ) ) ) ) ) )).

fof(t5_jordan10,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ( v2_connsp_1(B,k15_euclid(np__2))
            & v6_compts_1(B,k15_euclid(np__2))
            & ~ v1_sppol_1(B)
            & ~ v2_sppol_1(B)
            & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
         => r1_subset_1(B,k5_topreal1(np__2,k1_jordan9(B,A))) ) ) )).

fof(t6_jordan10,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ( v2_connsp_1(B,k15_euclid(np__2))
            & v6_compts_1(B,k15_euclid(np__2))
            & ~ v1_sppol_1(B)
            & ~ v2_sppol_1(B)
            & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
         => k19_pscomp_1(k5_topreal1(np__2,k1_jordan9(B,A))) = k3_real_1(k19_pscomp_1(B),k6_real_1(k5_real_1(k19_pscomp_1(B),k21_pscomp_1(B)),k3_newton(np__2,A))) ) ) )).

fof(t7_jordan10,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ! [C] :
              ( ( v2_connsp_1(C,k15_euclid(np__2))
                & v6_compts_1(C,k15_euclid(np__2))
                & ~ v1_sppol_1(C)
                & ~ v2_sppol_1(C)
                & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
             => ~ ( ~ r1_xreal_0(B,A)
                  & r1_xreal_0(k19_pscomp_1(k5_topreal1(np__2,k1_jordan9(C,A))),k19_pscomp_1(k5_topreal1(np__2,k1_jordan9(C,B)))) ) ) ) ) )).

fof(t8_jordan10,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ( v2_connsp_1(B,k15_euclid(np__2))
            & v6_compts_1(B,k15_euclid(np__2))
            & ~ v1_sppol_1(B)
            & ~ v2_sppol_1(B)
            & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
         => r2_hidden(k32_pscomp_1(B),k3_goboard9(k1_jordan9(B,A))) ) ) )).

fof(t9_jordan10,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ( v2_connsp_1(B,k15_euclid(np__2))
            & v6_compts_1(B,k15_euclid(np__2))
            & ~ v1_sppol_1(B)
            & ~ v2_sppol_1(B)
            & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
         => ~ r2_subset_1(B,k3_goboard9(k1_jordan9(B,A))) ) ) )).

fof(t10_jordan10,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ( v2_connsp_1(B,k15_euclid(np__2))
            & v6_compts_1(B,k15_euclid(np__2))
            & ~ v1_sppol_1(B)
            & ~ v2_sppol_1(B)
            & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
         => r1_subset_1(B,k2_goboard9(k1_jordan9(B,A))) ) ) )).

fof(t11_jordan10,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ( v2_connsp_1(B,k15_euclid(np__2))
            & v6_compts_1(B,k15_euclid(np__2))
            & ~ v1_sppol_1(B)
            & ~ v2_sppol_1(B)
            & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
         => r1_tarski(B,k3_goboard9(k1_jordan9(B,A))) ) ) )).

fof(t12_jordan10,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ( v2_connsp_1(B,k15_euclid(np__2))
            & v6_compts_1(B,k15_euclid(np__2))
            & ~ v1_sppol_1(B)
            & ~ v2_sppol_1(B)
            & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
         => r1_tarski(B,k1_jordan2c(np__2,k5_topreal1(np__2,k1_jordan9(B,A)))) ) ) )).

fof(t13_jordan10,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ( v2_connsp_1(B,k15_euclid(np__2))
            & v6_compts_1(B,k15_euclid(np__2))
            & ~ v1_sppol_1(B)
            & ~ v2_sppol_1(B)
            & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
         => r1_tarski(k2_jordan2c(np__2,k5_topreal1(np__2,k1_jordan9(B,A))),k2_jordan2c(np__2,B)) ) ) )).

fof(t14_jordan10,axiom,(
    ! [A] :
      ( ( v2_connsp_1(A,k15_euclid(np__2))
        & v6_compts_1(A,k15_euclid(np__2))
        & ~ v1_sppol_1(A)
        & ~ v2_sppol_1(A)
        & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
     => k5_setfam_1(u1_struct_0(k15_euclid(np__2)),k3_jordan10(A)) = k2_jordan2c(np__2,A) ) )).

fof(t15_jordan10,axiom,(
    ! [A] :
      ( ( v2_connsp_1(A,k15_euclid(np__2))
        & v6_compts_1(A,k15_euclid(np__2))
        & ~ v1_sppol_1(A)
        & ~ v2_sppol_1(A)
        & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
     => r1_tarski(A,k6_setfam_1(u1_struct_0(k15_euclid(np__2)),k4_jordan10(A))) ) )).

fof(t16_jordan10,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ( v2_connsp_1(B,k15_euclid(np__2))
            & v6_compts_1(B,k15_euclid(np__2))
            & ~ v1_sppol_1(B)
            & ~ v2_sppol_1(B)
            & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
         => r1_xboole_0(k1_jordan2c(np__2,B),k2_goboard9(k1_jordan9(B,A))) ) ) )).

fof(t17_jordan10,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ( v2_connsp_1(B,k15_euclid(np__2))
            & v6_compts_1(B,k15_euclid(np__2))
            & ~ v1_sppol_1(B)
            & ~ v2_sppol_1(B)
            & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
         => r1_tarski(k1_jordan2c(np__2,B),k3_goboard9(k1_jordan9(B,A))) ) ) )).

fof(t18_jordan10,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] :
              ( ( v2_connsp_1(C,k15_euclid(np__2))
                & v6_compts_1(C,k15_euclid(np__2))
                & ~ v1_sppol_1(C)
                & ~ v2_sppol_1(C)
                & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
             => ( r1_jordan2c(np__2,C,B)
               => r1_xboole_0(B,k5_topreal1(np__2,k1_jordan9(C,A))) ) ) ) ) )).

fof(t19_jordan10,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ( v2_connsp_1(B,k15_euclid(np__2))
            & v6_compts_1(B,k15_euclid(np__2))
            & ~ v1_sppol_1(B)
            & ~ v2_sppol_1(B)
            & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
         => r1_xboole_0(k1_jordan2c(np__2,B),k5_topreal1(np__2,k1_jordan9(B,A))) ) ) )).

fof(t20_jordan10,axiom,(
    ! [A] :
      ( ( v2_connsp_1(A,k15_euclid(np__2))
        & v6_compts_1(A,k15_euclid(np__2))
        & ~ v1_sppol_1(A)
        & ~ v2_sppol_1(A)
        & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
     => k7_setfam_1(u1_struct_0(k15_euclid(np__2)),k3_jordan10(A)) = k4_jordan10(A) ) )).

fof(t21_jordan10,axiom,(
    ! [A] :
      ( ( v2_connsp_1(A,k15_euclid(np__2))
        & v6_compts_1(A,k15_euclid(np__2))
        & ~ v1_sppol_1(A)
        & ~ v2_sppol_1(A)
        & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
     => k6_setfam_1(u1_struct_0(k15_euclid(np__2)),k4_jordan10(A)) = k4_subset_1(u1_struct_0(k15_euclid(np__2)),A,k1_jordan2c(np__2,A)) ) )).

fof(dt_k1_jordan10,axiom,(
    $true )).

fof(dt_k2_jordan10,axiom,(
    $true )).

fof(dt_k3_jordan10,axiom,(
    ! [A] :
      ( ( v6_compts_1(A,k15_euclid(np__2))
        & ~ v1_sppol_1(A)
        & ~ v2_sppol_1(A)
        & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
     => m1_subset_1(k3_jordan10(A),k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))) ) )).

fof(redefinition_k3_jordan10,axiom,(
    ! [A] :
      ( ( v6_compts_1(A,k15_euclid(np__2))
        & ~ v1_sppol_1(A)
        & ~ v2_sppol_1(A)
        & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
     => k3_jordan10(A) = k1_jordan10(A) ) )).

fof(dt_k4_jordan10,axiom,(
    ! [A] :
      ( ( v6_compts_1(A,k15_euclid(np__2))
        & ~ v1_sppol_1(A)
        & ~ v2_sppol_1(A)
        & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
     => m1_subset_1(k4_jordan10(A),k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))) ) )).

fof(redefinition_k4_jordan10,axiom,(
    ! [A] :
      ( ( v6_compts_1(A,k15_euclid(np__2))
        & ~ v1_sppol_1(A)
        & ~ v2_sppol_1(A)
        & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
     => k4_jordan10(A) = k2_jordan10(A) ) )).

fof(d1_jordan10,axiom,(
    ! [A] :
      ( ( v6_compts_1(A,k15_euclid(np__2))
        & ~ v1_sppol_1(A)
        & ~ v2_sppol_1(A)
        & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
     => k1_jordan10(A) = a_1_0_jordan10(A) ) )).

fof(d2_jordan10,axiom,(
    ! [A] :
      ( ( v6_compts_1(A,k15_euclid(np__2))
        & ~ v1_sppol_1(A)
        & ~ v2_sppol_1(A)
        & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
     => k2_jordan10(A) = a_1_1_jordan10(A) ) )).

fof(fraenkel_a_1_0_jordan10,axiom,(
    ! [A,B] :
      ( ( v6_compts_1(B,k15_euclid(np__2))
        & ~ v1_sppol_1(B)
        & ~ v2_sppol_1(B)
        & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
     => ( r2_hidden(A,a_1_0_jordan10(B))
      <=> ? [C] :
            ( m2_subset_1(C,k1_numbers,k5_numbers)
            & A = k2_jordan2c(np__2,k5_topreal1(np__2,k1_jordan9(B,C))) ) ) ) )).

fof(fraenkel_a_1_1_jordan10,axiom,(
    ! [A,B] :
      ( ( v6_compts_1(B,k15_euclid(np__2))
        & ~ v1_sppol_1(B)
        & ~ v2_sppol_1(B)
        & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
     => ( r2_hidden(A,a_1_1_jordan10(B))
      <=> ? [C] :
            ( m2_subset_1(C,k1_numbers,k5_numbers)
            & A = k6_pre_topc(k15_euclid(np__2),k1_jordan2c(np__2,k5_topreal1(np__2,k1_jordan9(B,C)))) ) ) ) )).
%------------------------------------------------------------------------------