SET007 Axioms: SET007+528.ax


%------------------------------------------------------------------------------
% File     : SET007+528 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Projections in n-Dimensional Euclidean Space to Each Coordinates
% 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    : jordan2b [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   48 (   1 unit)
%            Number of atoms       :  259 (  54 equality)
%            Maximal formula depth :   16 (   8 average)
%            Number of connectives :  226 (  15 ~  ;   3  |;  62  &)
%                                         (   9 <=>; 137 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   25 (   0 propositional; 1-3 arity)
%            Number of functors    :   60 (   9 constant; 0-4 arity)
%            Number of variables   :  144 (   0 singleton; 133 !;  11 ?)
%            Maximal term depth    :    5 (   1 average)
% SPC      : 

% Comments : The individual reference can be found in [Mat90] by looking for
%            the name provided by [Urb08].
%          : Translated by MPTP from the Mizar Mathematical Library 4.48.930.
%          : These set theory axioms are used in encodings of problems in
%            various domains, including ALG, CAT, GRP, LAT, SET, and TOP.
%------------------------------------------------------------------------------
fof(d1_jordan2b,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(k15_euclid(A)))
             => ! [D] :
                  ( m1_subset_1(D,k1_numbers)
                 => ( D = k1_jordan2b(A,B,C)
                  <=> ! [E] :
                        ( m2_finseq_1(E,k1_numbers)
                       => ( E = C
                         => D = k4_finseq_4(k5_numbers,k1_numbers,E,B) ) ) ) ) ) ) ) )).

fof(t1_jordan2b,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ? [C] :
              ( v1_funct_1(C)
              & v1_funct_2(C,u1_struct_0(k15_euclid(A)),u1_struct_0(k3_topmetr))
              & m2_relset_1(C,u1_struct_0(k15_euclid(A)),u1_struct_0(k3_topmetr))
              & ! [D] :
                  ( m1_subset_1(D,u1_struct_0(k15_euclid(A)))
                 => k8_funct_2(u1_struct_0(k15_euclid(A)),u1_struct_0(k3_topmetr),C,D) = k1_jordan2b(A,B,D) ) ) ) ) )).

fof(t2_jordan2b,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ( r2_hidden(B,k2_finseq_1(A))
           => k1_funct_1(k5_euclid(A),B) = np__0 ) ) ) )).

fof(t3_jordan2b,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ( r2_hidden(B,k2_finseq_1(A))
           => k1_jordan2b(A,B,k16_euclid(A)) = np__0 ) ) ) )).

fof(t4_jordan2b,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( v1_xreal_0(B)
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(k15_euclid(A)))
             => ! [D] :
                  ( m2_subset_1(D,k1_numbers,k5_numbers)
                 => ( r2_hidden(D,k2_finseq_1(A))
                   => k1_jordan2b(A,D,k18_euclid(B,A,C)) = k3_xcmplx_0(B,k1_jordan2b(A,D,C)) ) ) ) ) ) )).

fof(t5_jordan2b,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(k15_euclid(A)))
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => ( r2_hidden(C,k2_finseq_1(A))
               => k1_jordan2b(A,C,k19_euclid(A,B)) = k1_real_1(k1_jordan2b(A,C,B)) ) ) ) ) )).

fof(t6_jordan2b,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] :
                  ( m2_subset_1(D,k1_numbers,k5_numbers)
                 => ( r2_hidden(D,k2_finseq_1(A))
                   => k1_jordan2b(A,D,k17_euclid(A,B,C)) = k3_real_1(k1_jordan2b(A,D,B),k1_jordan2b(A,D,C)) ) ) ) ) ) )).

fof(t7_jordan2b,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] :
                  ( m2_subset_1(D,k1_numbers,k5_numbers)
                 => ( r2_hidden(D,k2_finseq_1(A))
                   => k1_jordan2b(A,D,k20_euclid(A,B,C)) = k5_real_1(k1_jordan2b(A,D,B),k1_jordan2b(A,D,C)) ) ) ) ) ) )).

fof(t8_jordan2b,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => k3_finseq_1(k5_euclid(A)) = A ) )).

fof(t9_jordan2b,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ( r1_xreal_0(A,B)
           => k16_finseq_1(k1_numbers,k5_euclid(B),A) = k5_euclid(A) ) ) ) )).

fof(t10_jordan2b,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => k1_rfinseq(k1_numbers,k5_euclid(A),B) = k5_euclid(k5_binarith(A,B)) ) ) )).

fof(t11_jordan2b,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => k15_rvsum_1(k5_euclid(A)) = np__0 ) )).

fof(t12_jordan2b,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_finseq_1(A) )
     => ! [B,C] :
          ( m2_subset_1(C,k1_numbers,k5_numbers)
         => k3_finseq_1(k2_funct_7(A,C,B)) = k3_finseq_1(A) ) ) )).

fof(t13_jordan2b,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ~ v1_xboole_0(B)
         => ! [C] :
              ( m2_finseq_1(C,B)
             => ! [D] :
                  ( m1_subset_1(D,B)
                 => ( r2_hidden(A,k4_finseq_1(C))
                   => k3_funct_7(B,C,A,D) = k8_finseq_1(B,k8_finseq_1(B,k16_finseq_1(B,C,k5_binarith(A,np__1)),k13_binarith(B,D)),k1_rfinseq(B,C,A)) ) ) ) ) ) )).

fof(t14_jordan2b,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ! [C] :
              ( m1_subset_1(C,k1_numbers)
             => ( r2_hidden(A,k2_finseq_1(B))
               => k15_rvsum_1(k3_funct_7(k1_numbers,k5_euclid(B),A,C)) = C ) ) ) ) )).

fof(t15_jordan2b,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_finseq_2(B,k1_numbers,k1_euclid(A))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(k15_euclid(A)))
             => ! [D] :
                  ( m2_subset_1(D,k1_numbers,k5_numbers)
                 => ( ( r2_hidden(D,k2_finseq_1(A))
                      & B = C )
                   => ( r1_xreal_0(k1_jordan2b(A,D,C),k12_euclid(B))
                      & r1_xreal_0(k7_square_1(k1_jordan2b(A,D,C)),k7_square_1(k12_euclid(B))) ) ) ) ) ) ) )).

fof(t20_jordan2b,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & v6_metric_1(B)
            & v7_metric_1(B)
            & v8_metric_1(B)
            & v9_metric_1(B)
            & l1_metric_1(B) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,u1_struct_0(A),u1_struct_0(k5_pcomps_1(B)))
                & m2_relset_1(C,u1_struct_0(A),u1_struct_0(k5_pcomps_1(B))) )
             => ( ! [D] :
                    ( v1_xreal_0(D)
                   => ! [E] :
                        ( m1_subset_1(E,u1_struct_0(B))
                       => ! [F] :
                            ( m1_subset_1(F,k1_zfmisc_1(u1_struct_0(k5_pcomps_1(B))))
                           => ( F = k9_metric_1(B,E,D)
                             => ( r1_xreal_0(D,np__0)
                                | v3_pre_topc(k5_pre_topc(A,k5_pcomps_1(B),C,F),A) ) ) ) ) )
               => v5_pre_topc(C,A,k5_pcomps_1(B)) ) ) ) ) )).

fof(t22_jordan2b,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,u1_struct_0(k15_euclid(A)),u1_struct_0(k3_topmetr))
            & m2_relset_1(B,u1_struct_0(k15_euclid(A)),u1_struct_0(k3_topmetr)) )
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => ( ( r2_hidden(C,k2_finseq_1(A))
                  & ! [D] :
                      ( m1_subset_1(D,u1_struct_0(k15_euclid(A)))
                     => k8_funct_2(u1_struct_0(k15_euclid(A)),u1_struct_0(k3_topmetr),B,D) = k1_jordan2b(A,C,D) ) )
               => v5_pre_topc(B,k15_euclid(A),k3_topmetr) ) ) ) ) )).

fof(t23_jordan2b,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => k3_euclid(k13_binarith(k1_numbers,A)) = k13_binarith(k1_numbers,k18_complex1(A)) ) )).

fof(t24_jordan2b,axiom,(
    ! [A] :
      ( m1_subset_1(A,u1_struct_0(k15_euclid(np__1)))
     => ? [B] :
          ( m1_subset_1(B,k1_numbers)
          & A = k13_binarith(k1_numbers,B) ) ) )).

fof(t25_jordan2b,axiom,(
    ! [A] :
      ( m1_subset_1(A,u1_struct_0(k14_euclid(np__1)))
     => ? [B] :
          ( m1_subset_1(B,k1_numbers)
          & A = k13_binarith(k1_numbers,B) ) ) )).

fof(d2_jordan2b,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => k2_jordan2b(A) = k9_finseq_1(A) ) )).

fof(t26_jordan2b,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => k18_euclid(B,np__1,k2_jordan2b(A)) = k2_jordan2b(k3_xcmplx_0(B,A)) ) ) )).

fof(t27_jordan2b,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => k2_jordan2b(k2_xcmplx_0(A,B)) = k17_euclid(np__1,k2_jordan2b(A),k2_jordan2b(B)) ) ) )).

fof(t28_jordan2b,axiom,(
    k2_jordan2b(np__0) = k16_euclid(np__1) )).

fof(t29_jordan2b,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ( k2_jordan2b(A) = k2_jordan2b(B)
           => A = B ) ) ) )).

fof(t34_jordan2b,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,u1_struct_0(k15_euclid(np__1)),u1_struct_0(k3_topmetr))
        & m2_relset_1(A,u1_struct_0(k15_euclid(np__1)),u1_struct_0(k3_topmetr)) )
     => ( ! [B] :
            ( m1_subset_1(B,u1_struct_0(k15_euclid(np__1)))
           => k8_funct_2(u1_struct_0(k15_euclid(np__1)),u1_struct_0(k3_topmetr),A,B) = k1_jordan2b(np__1,np__1,B) )
       => v3_tops_2(A,k15_euclid(np__1),k3_topmetr) ) ) )).

fof(t35_jordan2b,axiom,(
    ! [A] :
      ( m1_subset_1(A,u1_struct_0(k15_euclid(np__2)))
     => ( k1_jordan2b(np__2,np__1,A) = k21_euclid(A)
        & k1_jordan2b(np__2,np__2,A) = k22_euclid(A) ) ) )).

fof(t36_jordan2b,axiom,(
    ! [A] :
      ( m1_subset_1(A,u1_struct_0(k15_euclid(np__2)))
     => ( k1_jordan2b(np__2,np__1,A) = k8_funct_2(u1_struct_0(k15_euclid(np__2)),k1_numbers,k16_pscomp_1,A)
        & k1_jordan2b(np__2,np__2,A) = k8_funct_2(u1_struct_0(k15_euclid(np__2)),k1_numbers,k17_pscomp_1,A) ) ) )).

fof(dt_k1_jordan2b,axiom,(
    ! [A,B,C] :
      ( ( m1_subset_1(A,k5_numbers)
        & m1_subset_1(B,k5_numbers)
        & m1_subset_1(C,u1_struct_0(k15_euclid(A))) )
     => m1_subset_1(k1_jordan2b(A,B,C),k1_numbers) ) )).

fof(dt_k2_jordan2b,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => m1_subset_1(k2_jordan2b(A),u1_struct_0(k15_euclid(np__1))) ) )).

fof(t16_jordan2b,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( v1_xreal_0(B)
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
             => ! [D] :
                  ( m2_subset_1(D,k1_numbers,k5_numbers)
                 => ( ( C = a_3_0_jordan2b(A,B,D)
                      & r2_hidden(D,k2_finseq_1(A)) )
                   => v3_pre_topc(C,k15_euclid(A)) ) ) ) ) ) )).

fof(t17_jordan2b,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( v1_xreal_0(B)
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
             => ! [D] :
                  ( m2_subset_1(D,k1_numbers,k5_numbers)
                 => ( ( C = a_3_1_jordan2b(A,B,D)
                      & r2_hidden(D,k2_finseq_1(A)) )
                   => v3_pre_topc(C,k15_euclid(A)) ) ) ) ) ) )).

fof(t18_jordan2b,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
         => ! [C] :
              ( v1_xreal_0(C)
             => ! [D] :
                  ( v1_xreal_0(D)
                 => ! [E] :
                      ( m2_subset_1(E,k1_numbers,k5_numbers)
                     => ( ( B = a_4_0_jordan2b(A,C,D,E)
                          & r2_hidden(E,k2_finseq_1(A)) )
                       => v3_pre_topc(B,k15_euclid(A)) ) ) ) ) ) ) )).

fof(t19_jordan2b,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( v1_xreal_0(B)
         => ! [C] :
              ( v1_xreal_0(C)
             => ! [D] :
                  ( ( v1_funct_1(D)
                    & v1_funct_2(D,u1_struct_0(k15_euclid(A)),u1_struct_0(k3_topmetr))
                    & m2_relset_1(D,u1_struct_0(k15_euclid(A)),u1_struct_0(k3_topmetr)) )
                 => ! [E] :
                      ( m2_subset_1(E,k1_numbers,k5_numbers)
                     => ( ! [F] :
                            ( m1_subset_1(F,u1_struct_0(k15_euclid(A)))
                           => k8_funct_2(u1_struct_0(k15_euclid(A)),u1_struct_0(k3_topmetr),D,F) = k1_jordan2b(A,E,F) )
                       => k5_pre_topc(k15_euclid(A),k3_topmetr,D,a_2_0_jordan2b(B,C)) = a_4_0_jordan2b(A,B,C,E) ) ) ) ) ) ) )).

fof(t21_jordan2b,axiom,(
    ! [A] :
      ( m1_subset_1(A,u1_struct_0(k8_metric_1))
     => ! [B] :
          ( v1_xreal_0(B)
         => ! [C] :
              ( v1_xreal_0(C)
             => ( C = A
               => ( r1_xreal_0(B,np__0)
                  | k9_metric_1(k8_metric_1,A,B) = a_2_1_jordan2b(B,C) ) ) ) ) ) )).

fof(t30_jordan2b,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k3_topmetr)))
     => ! [B] :
          ( v1_xreal_0(B)
         => ( A = a_1_0_jordan2b(B)
           => v3_pre_topc(A,k3_topmetr) ) ) ) )).

fof(t31_jordan2b,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k3_topmetr)))
     => ! [B] :
          ( v1_xreal_0(B)
         => ( A = a_1_1_jordan2b(B)
           => v3_pre_topc(A,k3_topmetr) ) ) ) )).

fof(t32_jordan2b,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k3_topmetr)))
     => ! [B] :
          ( v1_xreal_0(B)
         => ! [C] :
              ( v1_xreal_0(C)
             => ( A = a_2_0_jordan2b(B,C)
               => v3_pre_topc(A,k3_topmetr) ) ) ) ) )).

fof(t33_jordan2b,axiom,(
    ! [A] :
      ( m1_subset_1(A,u1_struct_0(k14_euclid(np__1)))
     => ! [B] :
          ( v1_xreal_0(B)
         => ! [C] :
              ( v1_xreal_0(C)
             => ( k9_finseq_1(C) = A
               => ( r1_xreal_0(B,np__0)
                  | k9_metric_1(k14_euclid(np__1),A,B) = a_2_2_jordan2b(B,C) ) ) ) ) ) )).

fof(fraenkel_a_3_0_jordan2b,axiom,(
    ! [A,B,C,D] :
      ( ( m2_subset_1(B,k1_numbers,k5_numbers)
        & v1_xreal_0(C)
        & m2_subset_1(D,k1_numbers,k5_numbers) )
     => ( r2_hidden(A,a_3_0_jordan2b(B,C,D))
      <=> ? [E] :
            ( m1_subset_1(E,u1_struct_0(k15_euclid(B)))
            & A = E
            & ~ r1_xreal_0(C,k1_jordan2b(B,D,E)) ) ) ) )).

fof(fraenkel_a_3_1_jordan2b,axiom,(
    ! [A,B,C,D] :
      ( ( m2_subset_1(B,k1_numbers,k5_numbers)
        & v1_xreal_0(C)
        & m2_subset_1(D,k1_numbers,k5_numbers) )
     => ( r2_hidden(A,a_3_1_jordan2b(B,C,D))
      <=> ? [E] :
            ( m1_subset_1(E,u1_struct_0(k15_euclid(B)))
            & A = E
            & ~ r1_xreal_0(k1_jordan2b(B,D,E),C) ) ) ) )).

fof(fraenkel_a_4_0_jordan2b,axiom,(
    ! [A,B,C,D,E] :
      ( ( m2_subset_1(B,k1_numbers,k5_numbers)
        & v1_xreal_0(C)
        & v1_xreal_0(D)
        & m2_subset_1(E,k1_numbers,k5_numbers) )
     => ( r2_hidden(A,a_4_0_jordan2b(B,C,D,E))
      <=> ? [F] :
            ( m1_subset_1(F,u1_struct_0(k15_euclid(B)))
            & A = F
            & ~ r1_xreal_0(k1_jordan2b(B,E,F),C)
            & ~ r1_xreal_0(D,k1_jordan2b(B,E,F)) ) ) ) )).

fof(fraenkel_a_2_0_jordan2b,axiom,(
    ! [A,B,C] :
      ( ( v1_xreal_0(B)
        & v1_xreal_0(C) )
     => ( r2_hidden(A,a_2_0_jordan2b(B,C))
      <=> ? [D] :
            ( m1_subset_1(D,k1_numbers)
            & A = D
            & ~ r1_xreal_0(D,B)
            & ~ r1_xreal_0(C,D) ) ) ) )).

fof(fraenkel_a_2_1_jordan2b,axiom,(
    ! [A,B,C] :
      ( ( v1_xreal_0(B)
        & v1_xreal_0(C) )
     => ( r2_hidden(A,a_2_1_jordan2b(B,C))
      <=> ? [D] :
            ( m1_subset_1(D,k1_numbers)
            & A = D
            & ~ r1_xreal_0(D,k6_xcmplx_0(C,B))
            & ~ r1_xreal_0(k2_xcmplx_0(C,B),D) ) ) ) )).

fof(fraenkel_a_1_0_jordan2b,axiom,(
    ! [A,B] :
      ( v1_xreal_0(B)
     => ( r2_hidden(A,a_1_0_jordan2b(B))
      <=> ? [C] :
            ( m1_subset_1(C,k1_numbers)
            & A = C
            & ~ r1_xreal_0(B,C) ) ) ) )).

fof(fraenkel_a_1_1_jordan2b,axiom,(
    ! [A,B] :
      ( v1_xreal_0(B)
     => ( r2_hidden(A,a_1_1_jordan2b(B))
      <=> ? [C] :
            ( m1_subset_1(C,k1_numbers)
            & A = C
            & ~ r1_xreal_0(C,B) ) ) ) )).

fof(fraenkel_a_2_2_jordan2b,axiom,(
    ! [A,B,C] :
      ( ( v1_xreal_0(B)
        & v1_xreal_0(C) )
     => ( r2_hidden(A,a_2_2_jordan2b(B,C))
      <=> ? [D] :
            ( m1_subset_1(D,k1_numbers)
            & A = k13_binarith(k1_numbers,D)
            & ~ r1_xreal_0(D,k6_xcmplx_0(C,B))
            & ~ r1_xreal_0(k2_xcmplx_0(C,B),D) ) ) ) )).
%------------------------------------------------------------------------------