SET007 Axioms: SET007+730.ax


%------------------------------------------------------------------------------
% File     : SET007+730 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : On the Minimal Distance Between Sets in Euclidean Space
% 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    : jordan1k [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   82 (   0 unit)
%            Number of atoms       :  530 (  50 equality)
%            Maximal formula depth :   16 (   9 average)
%            Number of connectives :  534 (  86 ~  ;   1  |; 214  &)
%                                         (   3 <=>; 230 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   31 (   0 propositional; 1-3 arity)
%            Number of functors    :   64 (   6 constant; 0-5 arity)
%            Number of variables   :  235 (   0 singleton; 227 !;   8 ?)
%            Maximal term depth    :    6 (   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(fc1_jordan1k,axiom,(
    ! [A] :
      ( ( v1_topreal2(A)
        & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
     => ( ~ v1_xboole_0(k1_jordan2c(np__2,A))
        & v3_pre_topc(k1_jordan2c(np__2,A),k15_euclid(np__2)) ) ) )).

fof(t1_jordan1k,axiom,(
    ! [A,B] :
      ( ~ v1_xboole_0(B)
     => ! [C] :
          ( ( v1_funct_1(C)
            & v1_funct_2(C,A,B)
            & m2_relset_1(C,A,B) )
         => ( v2_funct_2(C,A,B)
           => ! [D] :
                ( m1_subset_1(D,B)
               => ? [E] :
                    ( r2_hidden(E,A)
                    & D = k1_funct_1(C,E) ) ) ) ) ) )).

fof(t2_jordan1k,axiom,(
    ! [A,B] :
      ( ~ v1_xboole_0(B)
     => ! [C] :
          ( ( v1_funct_1(C)
            & v1_funct_2(C,A,B)
            & m2_relset_1(C,A,B) )
         => ( v2_funct_2(C,A,B)
           => ! [D] :
                ( m1_subset_1(D,B)
               => ? [E] :
                    ( m1_subset_1(E,A)
                    & D = k1_funct_1(C,E) ) ) ) ) ) )).

fof(t3_jordan1k,axiom,(
    ! [A,B] :
      ( ~ v1_xboole_0(B)
     => ! [C] :
          ( ( v1_funct_1(C)
            & v1_funct_2(C,A,B)
            & m2_relset_1(C,A,B) )
         => ! [D] :
              ( m1_subset_1(D,k1_zfmisc_1(A))
             => ( v2_funct_2(C,A,B)
               => r1_tarski(k3_subset_1(B,k2_funct_2(A,B,C,D)),k2_funct_2(A,B,C,k3_subset_1(A,D))) ) ) ) ) )).

fof(t4_jordan1k,axiom,(
    ! [A,B] :
      ( ~ v1_xboole_0(B)
     => ! [C] :
          ( ( v1_funct_1(C)
            & v1_funct_2(C,A,B)
            & m2_relset_1(C,A,B) )
         => ! [D] :
              ( m1_subset_1(D,k1_zfmisc_1(A))
             => ( v2_funct_1(C)
               => r1_tarski(k2_funct_2(A,B,C,k3_subset_1(A,D)),k3_subset_1(B,k2_funct_2(A,B,C,D))) ) ) ) ) )).

fof(t5_jordan1k,axiom,(
    ! [A,B] :
      ( ~ v1_xboole_0(B)
     => ! [C] :
          ( ( v1_funct_1(C)
            & v1_funct_2(C,A,B)
            & m2_relset_1(C,A,B) )
         => ! [D] :
              ( m1_subset_1(D,k1_zfmisc_1(A))
             => ( v3_funct_2(C,A,B)
               => k3_subset_1(B,k2_funct_2(A,B,C,D)) = k2_funct_2(A,B,C,k3_subset_1(A,D)) ) ) ) ) )).

fof(t6_jordan1k,axiom,(
    ! [A] :
      ( ( v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => ( r4_connsp_1(A,k1_pre_topc(A),B)
          <=> v1_xboole_0(B) ) ) ) )).

fof(t7_jordan1k,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] :
              ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
             => ! [D] :
                  ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
                 => ( ( r1_tarski(B,C)
                      & r4_connsp_1(A,D,B)
                      & r4_connsp_1(A,D,C) )
                   => B = C ) ) ) ) ) )).

fof(t8_jordan1k,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ( r1_xreal_0(np__1,A)
       => ! [B] :
            ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k14_euclid(A))))
           => ~ ( v6_tbsp_1(B,k14_euclid(A))
                & v6_tbsp_1(k3_subset_1(u1_struct_0(k14_euclid(A)),B),k14_euclid(A)) ) ) ) ) )).

fof(t9_jordan1k,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v6_metric_1(A)
        & v7_metric_1(A)
        & v8_metric_1(A)
        & v9_metric_1(A)
        & l1_metric_1(A) )
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k5_pcomps_1(A)))) )
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(k5_pcomps_1(A)))
             => r1_xreal_0(np__0,k8_funct_2(u1_struct_0(k5_pcomps_1(A)),u1_struct_0(k3_topmetr),k8_weierstr(A,B),C)) ) ) ) )).

fof(t10_jordan1k,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [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_xboole_0(C)
                & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k5_pcomps_1(B)))) )
             => ! [D] :
                  ( m1_subset_1(D,u1_struct_0(B))
                 => ( ! [E] :
                        ( m1_subset_1(E,u1_struct_0(B))
                       => ( r2_hidden(E,C)
                         => r1_xreal_0(A,k4_metric_1(B,D,E)) ) )
                   => r1_xreal_0(A,k1_funct_1(k8_weierstr(B,C),D)) ) ) ) ) ) )).

fof(t11_jordan1k,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v6_metric_1(A)
        & v7_metric_1(A)
        & v8_metric_1(A)
        & v9_metric_1(A)
        & l1_metric_1(A) )
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k5_pcomps_1(A)))) )
         => ! [C] :
              ( ( ~ v1_xboole_0(C)
                & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k5_pcomps_1(A)))) )
             => r1_xreal_0(np__0,k9_weierstr(A,B,C)) ) ) ) )).

fof(t12_jordan1k,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v6_metric_1(A)
        & v7_metric_1(A)
        & v8_metric_1(A)
        & v9_metric_1(A)
        & l1_metric_1(A) )
     => ! [B] :
          ( ( v6_compts_1(B,k5_pcomps_1(A))
            & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k5_pcomps_1(A)))) )
         => ! [C] :
              ( ( v6_compts_1(C,k5_pcomps_1(A))
                & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k5_pcomps_1(A)))) )
             => ( ~ r1_xboole_0(B,C)
               => k9_weierstr(A,B,C) = np__0 ) ) ) ) )).

fof(t13_jordan1k,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [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_xboole_0(C)
                & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k5_pcomps_1(B)))) )
             => ! [D] :
                  ( ( ~ v1_xboole_0(D)
                    & m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k5_pcomps_1(B)))) )
                 => ( ! [E] :
                        ( m1_subset_1(E,u1_struct_0(B))
                       => ! [F] :
                            ( m1_subset_1(F,u1_struct_0(B))
                           => ( ( r2_hidden(E,C)
                                & r2_hidden(F,D) )
                             => r1_xreal_0(A,k4_metric_1(B,E,F)) ) ) )
                   => r1_xreal_0(A,k9_weierstr(B,C,D)) ) ) ) ) ) )).

fof(t14_jordan1k,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] :
              ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
             => ~ ( r4_connsp_1(k15_euclid(A),k3_subset_1(u1_struct_0(k15_euclid(A)),C),B)
                  & ~ r1_jordan2c(A,C,B)
                  & ~ r2_jordan2c(A,C,B) ) ) ) ) )).

fof(t15_jordan1k,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ( r1_xreal_0(np__1,A)
       => k1_jordan2c(A,k1_pre_topc(k15_euclid(A))) = k1_pre_topc(k15_euclid(A)) ) ) )).

fof(t16_jordan1k,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => k1_jordan2c(A,k2_pre_topc(k15_euclid(A))) = k1_pre_topc(k15_euclid(A)) ) )).

fof(t17_jordan1k,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ( r1_xreal_0(np__1,A)
       => k2_jordan2c(A,k1_pre_topc(k15_euclid(A))) = k2_pre_topc(k15_euclid(A)) ) ) )).

fof(t18_jordan1k,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => k2_jordan2c(A,k2_pre_topc(k15_euclid(A))) = k1_pre_topc(k15_euclid(A)) ) )).

fof(t19_jordan1k,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ( v2_connsp_1(B,k15_euclid(A))
            & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(A)))) )
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
             => ~ ( r1_xboole_0(B,C)
                  & ~ r1_tarski(B,k2_jordan2c(A,C))
                  & ~ r1_tarski(B,k1_jordan2c(A,C)) ) ) ) ) )).

fof(t20_jordan1k,axiom,(
    ! [A] :
      ( m1_subset_1(A,u1_struct_0(k15_euclid(np__2)))
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => k1_gobrd14(np__2,k23_euclid(np__0,np__0),k18_euclid(B,np__2,A)) = k4_real_1(k18_complex1(B),k1_gobrd14(np__2,k23_euclid(np__0,np__0),A)) ) ) )).

fof(t21_jordan1k,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)))
             => k1_gobrd14(np__2,k17_euclid(np__2,A,B),k17_euclid(np__2,C,B)) = k1_gobrd14(np__2,A,C) ) ) ) )).

fof(t22_jordan1k,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)))
         => ~ ( A != B
              & r1_xreal_0(k1_gobrd14(np__2,A,B),np__0) ) ) ) )).

fof(t23_jordan1k,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)))
             => k1_gobrd14(np__2,k20_euclid(np__2,A,B),k20_euclid(np__2,C,B)) = k1_gobrd14(np__2,A,C) ) ) ) )).

fof(t24_jordan1k,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)))
         => k1_gobrd14(np__2,A,B) = k1_gobrd14(np__2,k19_euclid(np__2,A),k19_euclid(np__2,B)) ) ) )).

fof(t25_jordan1k,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)))
             => k1_gobrd14(np__2,k20_euclid(np__2,A,B),k20_euclid(np__2,A,C)) = k1_gobrd14(np__2,B,C) ) ) ) )).

fof(t26_jordan1k,axiom,(
    ! [A] :
      ( m1_subset_1(A,u1_struct_0(k15_euclid(np__2)))
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(k15_euclid(np__2)))
         => ! [C] :
              ( m1_subset_1(C,k1_numbers)
             => k1_gobrd14(np__2,k18_euclid(C,np__2,A),k18_euclid(C,np__2,B)) = k4_real_1(k18_complex1(C),k1_gobrd14(np__2,A,B)) ) ) ) )).

fof(t27_jordan1k,axiom,(
    ! [A] :
      ( m1_subset_1(A,u1_struct_0(k15_euclid(np__2)))
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(k15_euclid(np__2)))
         => ! [C] :
              ( m1_subset_1(C,k1_numbers)
             => ( r1_xreal_0(C,np__1)
               => k1_gobrd14(np__2,A,k17_euclid(np__2,k18_euclid(C,np__2,A),k18_euclid(k5_real_1(np__1,C),np__2,B))) = k4_real_1(k5_real_1(np__1,C),k1_gobrd14(np__2,A,B)) ) ) ) ) )).

fof(t28_jordan1k,axiom,(
    ! [A] :
      ( m1_subset_1(A,u1_struct_0(k15_euclid(np__2)))
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(k15_euclid(np__2)))
         => ! [C] :
              ( m1_subset_1(C,k1_numbers)
             => ( r1_xreal_0(np__0,C)
               => k1_gobrd14(np__2,A,k17_euclid(np__2,k18_euclid(C,np__2,B),k18_euclid(k5_real_1(np__1,C),np__2,A))) = k4_real_1(C,k1_gobrd14(np__2,B,A)) ) ) ) ) )).

fof(t29_jordan1k,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)))
             => ( r2_hidden(A,k3_topreal1(np__2,B,C))
               => k3_real_1(k1_gobrd14(np__2,B,A),k1_gobrd14(np__2,A,C)) = k1_gobrd14(np__2,B,C) ) ) ) ) )).

fof(t30_jordan1k,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)))
             => ~ ( r2_hidden(A,k3_topreal1(np__2,B,C))
                  & A != B
                  & r1_xreal_0(k1_gobrd14(np__2,B,C),k1_gobrd14(np__2,A,C)) ) ) ) ) )).

fof(t32_jordan1k,axiom,(
    ! [A] :
      ( m1_subset_1(A,u1_struct_0(k15_euclid(np__2)))
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ! [C] :
              ( m1_subset_1(C,k1_numbers)
             => ! [D] :
                  ( m1_subset_1(D,k1_numbers)
                 => k8_funct_2(u1_struct_0(k15_euclid(np__2)),u1_struct_0(k15_euclid(np__2)),k2_jgraph_2(B,C,B,D),A) = k17_euclid(np__2,k18_euclid(B,np__2,A),k23_euclid(C,D)) ) ) ) ) )).

fof(t33_jordan1k,axiom,(
    ! [A] :
      ( m1_subset_1(A,u1_struct_0(k15_euclid(np__2)))
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(k15_euclid(np__2)))
         => ! [C] :
              ( m1_subset_1(C,k1_numbers)
             => k8_funct_2(u1_struct_0(k15_euclid(np__2)),u1_struct_0(k15_euclid(np__2)),k2_jgraph_2(C,k21_euclid(A),C,k22_euclid(A)),B) = k17_euclid(np__2,k18_euclid(C,np__2,B),A) ) ) ) )).

fof(t34_jordan1k,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(A,np__0)
                      & ~ r1_xreal_0(B,np__0)
                      & k7_funct_2(u1_struct_0(k15_euclid(np__2)),u1_struct_0(k15_euclid(np__2)),u1_struct_0(k15_euclid(np__2)),k2_jgraph_2(k6_real_1(np__1,A),k1_real_1(k6_real_1(C,A)),k6_real_1(np__1,B),k1_real_1(k6_real_1(D,B))),k2_jgraph_2(A,C,B,D)) != k6_partfun1(k1_euclid(np__2)) ) ) ) ) ) )).

fof(t35_jordan1k,axiom,(
    ! [A] :
      ( m1_subset_1(A,u1_struct_0(k15_euclid(np__2)))
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(k14_euclid(np__2)))
             => ! [D] :
                  ( m1_subset_1(D,u1_struct_0(k14_euclid(np__2)))
                 => ( ( C = k23_euclid(np__0,np__0)
                      & D = A )
                   => ( r1_xreal_0(B,np__0)
                      | k4_pre_topc(k15_euclid(np__2),k15_euclid(np__2),k2_jgraph_2(B,k21_euclid(A),B,k22_euclid(A)),k9_metric_1(k14_euclid(np__2),C,np__1)) = k9_metric_1(k14_euclid(np__2),D,B) ) ) ) ) ) ) )).

fof(t36_jordan1k,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(A,np__0)
                      & ~ r1_xreal_0(C,np__0)
                      & ~ v2_funct_2(k2_jgraph_2(A,B,C,D),u1_struct_0(k15_euclid(np__2)),u1_struct_0(k15_euclid(np__2))) ) ) ) ) ) )).

fof(t37_jordan1k,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(k14_euclid(np__2)))
         => ( v2_connsp_1(k3_subset_1(u1_struct_0(k14_euclid(np__2)),k9_metric_1(k14_euclid(np__2),B,A)),k15_euclid(np__2))
            & m1_subset_1(k3_subset_1(u1_struct_0(k14_euclid(np__2)),k9_metric_1(k14_euclid(np__2),B,A)),k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) ) ) ) )).

fof(d1_jordan1k,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] :
              ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
             => ! [D] :
                  ( m1_subset_1(D,k1_numbers)
                 => ( D = k1_jordan1k(A,B,C)
                  <=> ? [E] :
                        ( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(k5_pcomps_1(k14_euclid(A)))))
                        & ? [F] :
                            ( m1_subset_1(F,k1_zfmisc_1(u1_struct_0(k5_pcomps_1(k14_euclid(A)))))
                            & B = E
                            & C = F
                            & D = k9_weierstr(k14_euclid(A),E,F) ) ) ) ) ) ) ) )).

fof(t38_jordan1k,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(A)))) )
         => ! [C] :
              ( ( ~ v1_xboole_0(C)
                & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(A)))) )
             => r1_xreal_0(np__0,k1_jordan1k(A,B,C)) ) ) ) )).

fof(t39_jordan1k,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ( v6_compts_1(B,k15_euclid(A))
            & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(A)))) )
         => ! [C] :
              ( ( v6_compts_1(C,k15_euclid(A))
                & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(A)))) )
             => ( ~ r1_xboole_0(B,C)
               => k1_jordan1k(A,B,C) = np__0 ) ) ) ) )).

fof(t40_jordan1k,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ! [C] :
              ( ( ~ v1_xboole_0(C)
                & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(A)))) )
             => ! [D] :
                  ( ( ~ v1_xboole_0(D)
                    & m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k15_euclid(A)))) )
                 => ( ! [E] :
                        ( m1_subset_1(E,u1_struct_0(k15_euclid(A)))
                       => ! [F] :
                            ( m1_subset_1(F,u1_struct_0(k15_euclid(A)))
                           => ( ( r2_hidden(E,C)
                                & r2_hidden(F,D) )
                             => r1_xreal_0(B,k1_gobrd14(A,E,F)) ) ) )
                   => r1_xreal_0(B,k1_jordan1k(A,C,D)) ) ) ) ) ) )).

fof(t41_jordan1k,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_xboole_0(C)
                & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(A)))) )
             => ! [D] :
                  ( ( ~ v1_xboole_0(D)
                    & m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k15_euclid(A)))) )
                 => ( r1_tarski(D,B)
                   => r1_xreal_0(k1_jordan1k(A,C,B),k1_jordan1k(A,C,D)) ) ) ) ) ) )).

fof(t42_jordan1k,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & v6_compts_1(B,k15_euclid(A))
            & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(A)))) )
         => ! [C] :
              ( ( ~ v1_xboole_0(C)
                & v6_compts_1(C,k15_euclid(A))
                & m1_subset_1(C,k1_zfmisc_1(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)))
                      & r2_hidden(D,B)
                      & r2_hidden(E,C)
                      & k4_jordan1k(A,B,C) = k1_gobrd14(A,D,E) ) ) ) ) ) )).

fof(t43_jordan1k,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)))
             => k4_jordan1k(A,k1_struct_0(k15_euclid(A),B),k1_struct_0(k15_euclid(A),C)) = k1_gobrd14(A,B,C) ) ) ) )).

fof(d2_jordan1k,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))))
             => k5_jordan1k(A,B,C) = k1_jordan1k(A,k1_struct_0(k15_euclid(A),B),C) ) ) ) )).

fof(t44_jordan1k,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(A)))) )
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(k15_euclid(A)))
             => r1_xreal_0(np__0,k5_jordan1k(A,C,B)) ) ) ) )).

fof(t45_jordan1k,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ( v6_compts_1(B,k15_euclid(A))
            & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(A)))) )
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(k15_euclid(A)))
             => ( r2_hidden(C,B)
               => k5_jordan1k(A,C,B) = np__0 ) ) ) ) )).

fof(t46_jordan1k,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & v6_compts_1(B,k15_euclid(A))
            & m1_subset_1(B,k1_zfmisc_1(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(D,B)
                  & k5_jordan1k(A,C,B) = k1_gobrd14(A,C,D) ) ) ) ) )).

fof(t47_jordan1k,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(A)))) )
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
             => ( r1_tarski(B,C)
               => ! [D] :
                    ( m1_subset_1(D,u1_struct_0(k15_euclid(A)))
                   => r1_xreal_0(k5_jordan1k(A,D,C),k5_jordan1k(A,D,B)) ) ) ) ) ) )).

fof(t48_jordan1k,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ! [C] :
              ( ( ~ v1_xboole_0(C)
                & m1_subset_1(C,k1_zfmisc_1(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)))
                       => ( r2_hidden(E,C)
                         => r1_xreal_0(B,k1_gobrd14(A,D,E)) ) )
                   => r1_xreal_0(B,k5_jordan1k(A,D,C)) ) ) ) ) ) )).

fof(t49_jordan1k,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)))
             => k5_jordan1k(A,B,k1_struct_0(k15_euclid(A),C)) = k1_gobrd14(A,B,C) ) ) ) )).

fof(t50_jordan1k,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & m1_subset_1(B,k1_zfmisc_1(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(D,B)
                   => r1_xreal_0(k5_jordan1k(A,C,B),k1_gobrd14(A,C,D)) ) ) ) ) ) )).

fof(t51_jordan1k,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)))) )
     => ! [B] :
          ( ( v3_pre_topc(B,k15_euclid(np__2))
            & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
         => ( r1_tarski(A,B)
           => ! [C] :
                ( m1_subset_1(C,u1_struct_0(k15_euclid(np__2)))
               => ~ ( ~ r2_hidden(C,B)
                    & r1_xreal_0(k5_jordan1k(np__2,C,A),k5_jordan1k(np__2,C,B)) ) ) ) ) ) )).

fof(t52_jordan1k,axiom,(
    ! [A] :
      ( ( v1_topreal2(A)
        & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
     => ! [B] :
          ( ( v1_topreal2(B)
            & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
         => ~ r2_subset_1(k2_jordan2c(np__2,A),k2_jordan2c(np__2,B)) ) ) )).

fof(t53_jordan1k,axiom,(
    ! [A] :
      ( ( v1_topreal2(A)
        & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(k15_euclid(np__2)))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(k15_euclid(np__2)))
             => ~ ( r2_hidden(B,k2_jordan2c(np__2,A))
                  & r2_hidden(C,k1_jordan2c(np__2,A))
                  & r1_xreal_0(k1_gobrd14(np__2,B,C),k5_jordan1k(np__2,B,A)) ) ) ) ) )).

fof(t54_jordan1k,axiom,(
    ! [A] :
      ( ( v1_topreal2(A)
        & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(k15_euclid(np__2)))
         => ( ~ r2_hidden(B,k1_jordan2c(np__2,A))
           => r1_xreal_0(k5_jordan1k(np__2,B,A),k5_jordan1k(np__2,B,k1_jordan2c(np__2,A))) ) ) ) )).

fof(t55_jordan1k,axiom,(
    ! [A] :
      ( ( v1_topreal2(A)
        & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
     => ! [B] :
          ( ( v1_topreal2(B)
            & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
         => ~ ( r1_tarski(A,k1_jordan2c(np__2,B))
              & r1_tarski(B,k1_jordan2c(np__2,A)) ) ) ) )).

fof(t56_jordan1k,axiom,(
    ! [A] :
      ( ( v1_topreal2(A)
        & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
     => ! [B] :
          ( ( v1_topreal2(B)
            & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
         => ( r1_tarski(A,k1_jordan2c(np__2,B))
           => r1_tarski(B,k2_jordan2c(np__2,A)) ) ) ) )).

fof(t57_jordan1k,axiom,(
    ! [A] :
      ( ( v1_topreal2(A)
        & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => r1_tarski(k5_topreal1(np__2,k1_jordan9(A,B)),k2_jordan2c(np__2,A)) ) ) )).

fof(d3_jordan1k,axiom,(
    ! [A] :
      ( ( v1_topreal2(A)
        & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
     => k6_jordan1k(A) = k1_jordan5c(k9_jordan6(A),k6_jordan6(k6_real_1(k3_real_1(k18_pscomp_1(A),k20_pscomp_1(A)),np__2)),k30_pscomp_1(A),k34_pscomp_1(A)) ) )).

fof(d4_jordan1k,axiom,(
    ! [A] :
      ( ( v1_topreal2(A)
        & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
     => k7_jordan1k(A) = k1_jordan5c(k8_jordan6(A),k6_jordan6(k6_real_1(k3_real_1(k18_pscomp_1(A),k20_pscomp_1(A)),np__2)),k30_pscomp_1(A),k34_pscomp_1(A)) ) )).

fof(t58_jordan1k,axiom,(
    ! [A] :
      ( ( v1_topreal2(A)
        & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
     => ~ r1_xboole_0(k9_jordan6(A),k6_jordan6(k6_real_1(k3_real_1(k18_pscomp_1(A),k20_pscomp_1(A)),np__2))) ) )).

fof(t59_jordan1k,axiom,(
    ! [A] :
      ( ( v1_topreal2(A)
        & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
     => ~ r1_xboole_0(k8_jordan6(A),k6_jordan6(k6_real_1(k3_real_1(k18_pscomp_1(A),k20_pscomp_1(A)),np__2))) ) )).

fof(t60_jordan1k,axiom,(
    ! [A] :
      ( ( v1_topreal2(A)
        & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
     => k21_euclid(k6_jordan1k(A)) = k6_real_1(k3_real_1(k18_pscomp_1(A),k20_pscomp_1(A)),np__2) ) )).

fof(t61_jordan1k,axiom,(
    ! [A] :
      ( ( v1_topreal2(A)
        & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
     => k21_euclid(k7_jordan1k(A)) = k6_real_1(k3_real_1(k18_pscomp_1(A),k20_pscomp_1(A)),np__2) ) )).

fof(t62_jordan1k,axiom,(
    ! [A] :
      ( ( v1_topreal2(A)
        & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
     => r2_hidden(k6_jordan1k(A),k9_jordan6(A)) ) )).

fof(t63_jordan1k,axiom,(
    ! [A] :
      ( ( v1_topreal2(A)
        & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
     => r2_hidden(k7_jordan1k(A),k8_jordan6(A)) ) )).

fof(dt_k1_jordan1k,axiom,(
    ! [A,B,C] :
      ( ( m1_subset_1(A,k5_numbers)
        & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
        & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(A)))) )
     => m1_subset_1(k1_jordan1k(A,B,C),k1_numbers) ) )).

fof(dt_k2_jordan1k,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(A)
        & v6_metric_1(A)
        & v7_metric_1(A)
        & v8_metric_1(A)
        & v9_metric_1(A)
        & l1_metric_1(A)
        & ~ v1_xboole_0(B)
        & v6_compts_1(B,k5_pcomps_1(A))
        & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k5_pcomps_1(A))))
        & ~ v1_xboole_0(C)
        & v6_compts_1(C,k5_pcomps_1(A))
        & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k5_pcomps_1(A)))) )
     => m1_subset_1(k2_jordan1k(A,B,C),k1_numbers) ) )).

fof(commutativity_k2_jordan1k,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(A)
        & v6_metric_1(A)
        & v7_metric_1(A)
        & v8_metric_1(A)
        & v9_metric_1(A)
        & l1_metric_1(A)
        & ~ v1_xboole_0(B)
        & v6_compts_1(B,k5_pcomps_1(A))
        & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k5_pcomps_1(A))))
        & ~ v1_xboole_0(C)
        & v6_compts_1(C,k5_pcomps_1(A))
        & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k5_pcomps_1(A)))) )
     => k2_jordan1k(A,B,C) = k2_jordan1k(A,C,B) ) )).

fof(redefinition_k2_jordan1k,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(A)
        & v6_metric_1(A)
        & v7_metric_1(A)
        & v8_metric_1(A)
        & v9_metric_1(A)
        & l1_metric_1(A)
        & ~ v1_xboole_0(B)
        & v6_compts_1(B,k5_pcomps_1(A))
        & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k5_pcomps_1(A))))
        & ~ v1_xboole_0(C)
        & v6_compts_1(C,k5_pcomps_1(A))
        & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k5_pcomps_1(A)))) )
     => k2_jordan1k(A,B,C) = k9_weierstr(A,B,C) ) )).

fof(dt_k3_jordan1k,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(A)
        & v6_metric_1(A)
        & v7_metric_1(A)
        & v8_metric_1(A)
        & v9_metric_1(A)
        & l1_metric_1(A)
        & ~ v1_xboole_0(B)
        & v6_compts_1(B,k5_pcomps_1(A))
        & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k5_pcomps_1(A))))
        & ~ v1_xboole_0(C)
        & v6_compts_1(C,k5_pcomps_1(A))
        & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k5_pcomps_1(A)))) )
     => m1_subset_1(k3_jordan1k(A,B,C),k1_numbers) ) )).

fof(commutativity_k3_jordan1k,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(A)
        & v6_metric_1(A)
        & v7_metric_1(A)
        & v8_metric_1(A)
        & v9_metric_1(A)
        & l1_metric_1(A)
        & ~ v1_xboole_0(B)
        & v6_compts_1(B,k5_pcomps_1(A))
        & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k5_pcomps_1(A))))
        & ~ v1_xboole_0(C)
        & v6_compts_1(C,k5_pcomps_1(A))
        & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k5_pcomps_1(A)))) )
     => k3_jordan1k(A,B,C) = k3_jordan1k(A,C,B) ) )).

fof(redefinition_k3_jordan1k,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(A)
        & v6_metric_1(A)
        & v7_metric_1(A)
        & v8_metric_1(A)
        & v9_metric_1(A)
        & l1_metric_1(A)
        & ~ v1_xboole_0(B)
        & v6_compts_1(B,k5_pcomps_1(A))
        & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k5_pcomps_1(A))))
        & ~ v1_xboole_0(C)
        & v6_compts_1(C,k5_pcomps_1(A))
        & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k5_pcomps_1(A)))) )
     => k3_jordan1k(A,B,C) = k12_weierstr(A,B,C) ) )).

fof(dt_k4_jordan1k,axiom,(
    ! [A,B,C] :
      ( ( m1_subset_1(A,k5_numbers)
        & ~ v1_xboole_0(B)
        & v6_compts_1(B,k15_euclid(A))
        & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
        & ~ v1_xboole_0(C)
        & v6_compts_1(C,k15_euclid(A))
        & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(A)))) )
     => m1_subset_1(k4_jordan1k(A,B,C),k1_numbers) ) )).

fof(commutativity_k4_jordan1k,axiom,(
    ! [A,B,C] :
      ( ( m1_subset_1(A,k5_numbers)
        & ~ v1_xboole_0(B)
        & v6_compts_1(B,k15_euclid(A))
        & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
        & ~ v1_xboole_0(C)
        & v6_compts_1(C,k15_euclid(A))
        & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(A)))) )
     => k4_jordan1k(A,B,C) = k4_jordan1k(A,C,B) ) )).

fof(redefinition_k4_jordan1k,axiom,(
    ! [A,B,C] :
      ( ( m1_subset_1(A,k5_numbers)
        & ~ v1_xboole_0(B)
        & v6_compts_1(B,k15_euclid(A))
        & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
        & ~ v1_xboole_0(C)
        & v6_compts_1(C,k15_euclid(A))
        & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(A)))) )
     => k4_jordan1k(A,B,C) = k1_jordan1k(A,B,C) ) )).

fof(dt_k5_jordan1k,axiom,(
    ! [A,B,C] :
      ( ( m1_subset_1(A,k5_numbers)
        & m1_subset_1(B,u1_struct_0(k15_euclid(A)))
        & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(A)))) )
     => m1_subset_1(k5_jordan1k(A,B,C),k1_numbers) ) )).

fof(dt_k6_jordan1k,axiom,(
    ! [A] :
      ( ( v1_topreal2(A)
        & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
     => m1_subset_1(k6_jordan1k(A),u1_struct_0(k15_euclid(np__2))) ) )).

fof(dt_k7_jordan1k,axiom,(
    ! [A] :
      ( ( v1_topreal2(A)
        & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
     => m1_subset_1(k7_jordan1k(A),u1_struct_0(k15_euclid(np__2))) ) )).

fof(t31_jordan1k,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(k14_euclid(np__2)))
         => ( B = k23_euclid(np__0,np__0)
           => k9_metric_1(k14_euclid(np__2),B,A) = a_1_0_jordan1k(A) ) ) ) )).

fof(fraenkel_a_1_0_jordan1k,axiom,(
    ! [A,B] :
      ( m1_subset_1(B,k1_numbers)
     => ( r2_hidden(A,a_1_0_jordan1k(B))
      <=> ? [C] :
            ( m1_subset_1(C,u1_struct_0(k15_euclid(np__2)))
            & A = C
            & ~ r1_xreal_0(B,k5_toprns_1(np__2,C)) ) ) ) )).
%------------------------------------------------------------------------------