SET007 Axioms: SET007+353.ax


%------------------------------------------------------------------------------
% File     : SET007+353 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : The Jordan's Property for Certain Subsets 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    : jordan1 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   66 (   3 unit)
%            Number of atoms       :  549 (  96 equality)
%            Maximal formula depth :   27 (  12 average)
%            Number of connectives :  533 (  50 ~  ;  20  |; 192  &)
%                                         (  16 <=>; 255 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   21 (   1 propositional; 0-3 arity)
%            Number of functors    :   38 (   7 constant; 0-4 arity)
%            Number of variables   :  287 (   0 singleton; 256 !;  31 ?)
%            Maximal term depth    :    7 (   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(t1_jordan1,axiom,(
    ! [A] :
      ( l1_pre_topc(A)
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => u1_struct_0(k3_pre_topc(A,B)) = B ) ) )).

fof(t2_jordan1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ( ! [B] :
            ( m1_subset_1(B,u1_struct_0(A))
           => ! [C] :
                ( m1_subset_1(C,u1_struct_0(A))
               => ? [D] :
                    ( ~ v3_struct_0(D)
                    & v2_pre_topc(D)
                    & l1_pre_topc(D)
                    & v1_connsp_1(D)
                    & ? [E] :
                        ( v1_funct_1(E)
                        & v1_funct_2(E,u1_struct_0(D),u1_struct_0(A))
                        & m2_relset_1(E,u1_struct_0(D),u1_struct_0(A))
                        & v5_pre_topc(E,D,A)
                        & r2_hidden(B,k2_relat_1(E))
                        & r2_hidden(C,k2_relat_1(E)) ) ) ) )
       => v1_connsp_1(A) ) ) )).

fof(t3_jordan1,axiom,(
    $true )).

fof(t4_jordan1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ( ! [B] :
            ( m1_subset_1(B,u1_struct_0(A))
           => ! [C] :
                ( m1_subset_1(C,u1_struct_0(A))
               => ? [D] :
                    ( v1_funct_1(D)
                    & v1_funct_2(D,u1_struct_0(k22_borsuk_1),u1_struct_0(A))
                    & m2_relset_1(D,u1_struct_0(k22_borsuk_1),u1_struct_0(A))
                    & v5_pre_topc(D,k22_borsuk_1,A)
                    & B = k1_funct_1(D,np__0)
                    & C = k1_funct_1(D,np__1) ) ) )
       => v1_connsp_1(A) ) ) )).

fof(t5_jordan1,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,u1_struct_0(A))
               => ! [D] :
                    ( m1_subset_1(D,u1_struct_0(A))
                   => ~ ( r2_hidden(C,B)
                        & r2_hidden(D,B)
                        & C != D
                        & ! [E] :
                            ( ( v1_funct_1(E)
                              & v1_funct_2(E,u1_struct_0(k22_borsuk_1),u1_struct_0(k3_pre_topc(A,B)))
                              & m2_relset_1(E,u1_struct_0(k22_borsuk_1),u1_struct_0(k3_pre_topc(A,B))) )
                           => ~ ( v5_pre_topc(E,k22_borsuk_1,k3_pre_topc(A,B))
                                & C = k1_funct_1(E,np__0)
                                & D = k1_funct_1(E,np__1) ) ) ) ) )
           => v2_connsp_1(B,A) ) ) ) )).

fof(t6_jordan1,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)))
             => ( ( v2_connsp_1(B,A)
                  & v2_connsp_1(C,A) )
               => ( r1_xboole_0(B,C)
                  | v2_connsp_1(k4_subset_1(u1_struct_0(A),B,C),A) ) ) ) ) ) )).

fof(t7_jordan1,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)))
                 => ( ( v2_connsp_1(B,A)
                      & v2_connsp_1(C,A)
                      & v2_connsp_1(D,A) )
                   => ( r1_xboole_0(B,C)
                      | r1_xboole_0(C,D)
                      | v2_connsp_1(k4_subset_1(u1_struct_0(A),k4_subset_1(u1_struct_0(A),B,C),D),A) ) ) ) ) ) ) )).

fof(t8_jordan1,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)))
                 => ! [E] :
                      ( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(A)))
                     => ( ( v2_connsp_1(B,A)
                          & v2_connsp_1(C,A)
                          & v2_connsp_1(D,A)
                          & v2_connsp_1(E,A) )
                       => ( r1_xboole_0(B,C)
                          | r1_xboole_0(C,D)
                          | r1_xboole_0(D,E)
                          | v2_connsp_1(k4_subset_1(u1_struct_0(A),k4_subset_1(u1_struct_0(A),k4_subset_1(u1_struct_0(A),B,C),D),E),A) ) ) ) ) ) ) ) )).

fof(d1_jordan1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
         => ( v1_jordan1(B,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(C,B)
                        & r2_hidden(D,B) )
                     => r1_tarski(k3_topreal1(A,C,D),B) ) ) ) ) ) ) )).

fof(t9_jordan1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
         => ( v1_jordan1(B,A)
           => v2_connsp_1(B,k15_euclid(A)) ) ) ) )).

fof(t10_jordan1,axiom,(
    $true )).

fof(t11_jordan1,axiom,(
    $true )).

fof(d2_jordan1,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)
                  & ! [D] :
                      ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k3_pre_topc(k15_euclid(np__2),k3_subset_1(u1_struct_0(k15_euclid(np__2)),A)))))
                     => ! [E] :
                          ( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(k3_pre_topc(k15_euclid(np__2),k3_subset_1(u1_struct_0(k15_euclid(np__2)),A)))))
                         => ( ( D = B
                              & E = C )
                           => ( r3_connsp_1(k3_pre_topc(k15_euclid(np__2),k3_subset_1(u1_struct_0(k15_euclid(np__2)),A)),D)
                              & r3_connsp_1(k3_pre_topc(k15_euclid(np__2),k3_subset_1(u1_struct_0(k15_euclid(np__2)),A)),E) ) ) ) ) ) ) ) ) ) )).

fof(t47_jordan1,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))))
                  & ? [D] :
                      ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k3_pre_topc(k15_euclid(np__2),k3_subset_1(u1_struct_0(k15_euclid(np__2)),A)))))
                      & ? [E] :
                          ( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(k3_pre_topc(k15_euclid(np__2),k3_subset_1(u1_struct_0(k15_euclid(np__2)),A)))))
                          & 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)
                          & D = B
                          & E = C
                          & r3_connsp_1(k3_pre_topc(k15_euclid(np__2),k3_subset_1(u1_struct_0(k15_euclid(np__2)),A)),D)
                          & r3_connsp_1(k3_pre_topc(k15_euclid(np__2),k3_subset_1(u1_struct_0(k15_euclid(np__2)),A)),E)
                          & ! [F] :
                              ( m1_subset_1(F,k1_zfmisc_1(u1_struct_0(k3_pre_topc(k15_euclid(np__2),k3_subset_1(u1_struct_0(k15_euclid(np__2)),A)))))
                             => ~ ( r3_connsp_1(k3_pre_topc(k15_euclid(np__2),k3_subset_1(u1_struct_0(k15_euclid(np__2)),A)),F)
                                  & F != D
                                  & F != E ) ) ) ) ) ) ) ) ) )).

fof(t12_jordan1,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)
                 => a_4_0_jordan1(A,B,C,D) = k3_xboole_0(k3_xboole_0(k3_xboole_0(a_1_0_jordan1(A),a_1_1_jordan1(B)),a_1_2_jordan1(C)),a_1_3_jordan1(D)) ) ) ) ) )).

fof(t13_jordan1,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)
                 => a_4_1_jordan1(A,B,C,D) = k2_xboole_0(k2_xboole_0(k2_xboole_0(a_1_1_jordan1(A),a_1_3_jordan1(C)),a_1_0_jordan1(B)),a_1_2_jordan1(D)) ) ) ) ) )).

fof(t14_jordan1,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)
                 => ! [E] :
                      ( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
                     => ( E = a_4_2_jordan1(A,B,C,D)
                       => v1_jordan1(E,np__2) ) ) ) ) ) ) )).

fof(t15_jordan1,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)
                 => ! [E] :
                      ( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
                     => ( E = a_4_2_jordan1(A,B,C,D)
                       => v2_connsp_1(E,k15_euclid(np__2)) ) ) ) ) ) ) )).

fof(t16_jordan1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
         => ( B = a_1_0_jordan1(A)
           => v1_jordan1(B,np__2) ) ) ) )).

fof(t17_jordan1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
         => ( B = a_1_0_jordan1(A)
           => v2_connsp_1(B,k15_euclid(np__2)) ) ) ) )).

fof(t18_jordan1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
         => ( B = a_1_1_jordan1(A)
           => v1_jordan1(B,np__2) ) ) ) )).

fof(t19_jordan1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
         => ( B = a_1_1_jordan1(A)
           => v2_connsp_1(B,k15_euclid(np__2)) ) ) ) )).

fof(t20_jordan1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
         => ( B = a_1_2_jordan1(A)
           => v1_jordan1(B,np__2) ) ) ) )).

fof(t21_jordan1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
         => ( B = a_1_2_jordan1(A)
           => v2_connsp_1(B,k15_euclid(np__2)) ) ) ) )).

fof(t22_jordan1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
         => ( B = a_1_3_jordan1(A)
           => v1_jordan1(B,np__2) ) ) ) )).

fof(t23_jordan1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
         => ( B = a_1_3_jordan1(A)
           => v2_connsp_1(B,k15_euclid(np__2)) ) ) ) )).

fof(t24_jordan1,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)
                 => ! [E] :
                      ( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
                     => ( E = a_4_3_jordan1(A,B,C,D)
                       => v2_connsp_1(E,k15_euclid(np__2)) ) ) ) ) ) ) )).

fof(t25_jordan1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
         => ( B = a_1_0_jordan1(A)
           => v3_pre_topc(B,k15_euclid(np__2)) ) ) ) )).

fof(t26_jordan1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
         => ( B = a_1_1_jordan1(A)
           => v3_pre_topc(B,k15_euclid(np__2)) ) ) ) )).

fof(t27_jordan1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
         => ( B = a_1_2_jordan1(A)
           => v3_pre_topc(B,k15_euclid(np__2)) ) ) ) )).

fof(t28_jordan1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
         => ( B = a_1_3_jordan1(A)
           => v3_pre_topc(B,k15_euclid(np__2)) ) ) ) )).

fof(t29_jordan1,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)
                 => ! [E] :
                      ( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
                     => ( E = a_4_2_jordan1(A,B,C,D)
                       => v3_pre_topc(E,k15_euclid(np__2)) ) ) ) ) ) ) )).

fof(t30_jordan1,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)
                 => ! [E] :
                      ( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
                     => ( E = a_4_3_jordan1(A,B,C,D)
                       => v3_pre_topc(E,k15_euclid(np__2)) ) ) ) ) ) ) )).

fof(t31_jordan1,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)
                 => ! [E] :
                      ( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
                     => ! [F] :
                          ( m1_subset_1(F,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
                         => ( ( E = a_4_2_jordan1(A,B,C,D)
                              & F = a_4_3_jordan1(A,B,C,D) )
                           => r1_xboole_0(E,F) ) ) ) ) ) ) ) )).

fof(t32_jordan1,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)
                 => a_4_4_jordan1(A,B,C,D) = a_4_0_jordan1(A,B,C,D) ) ) ) ) )).

fof(t33_jordan1,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)
                 => a_4_5_jordan1(A,B,C,D) = a_4_1_jordan1(A,B,C,D) ) ) ) ) )).

fof(t34_jordan1,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)
                 => m1_subset_1(a_4_4_jordan1(A,B,C,D),k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) ) ) ) ) )).

fof(t35_jordan1,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)
                 => m1_subset_1(a_4_5_jordan1(A,B,C,D),k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) ) ) ) ) )).

fof(t36_jordan1,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)
                 => ! [E] :
                      ( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
                     => ( E = a_4_6_jordan1(A,B,C,D)
                       => v2_connsp_1(E,k15_euclid(np__2)) ) ) ) ) ) ) )).

fof(t37_jordan1,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)
                 => ! [E] :
                      ( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
                     => ( E = a_4_7_jordan1(A,B,C,D)
                       => v2_connsp_1(E,k15_euclid(np__2)) ) ) ) ) ) ) )).

fof(t38_jordan1,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)
                 => ! [E] :
                      ( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
                     => ( E = a_4_6_jordan1(A,B,C,D)
                       => v3_pre_topc(E,k15_euclid(np__2)) ) ) ) ) ) ) )).

fof(t39_jordan1,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)
                 => ! [E] :
                      ( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
                     => ( E = a_4_7_jordan1(A,B,C,D)
                       => v3_pre_topc(E,k15_euclid(np__2)) ) ) ) ) ) ) )).

fof(t40_jordan1,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)
                 => ! [E] :
                      ( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
                     => ! [F] :
                          ( m1_subset_1(F,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
                         => ( ( E = a_4_6_jordan1(A,B,C,D)
                              & F = a_4_7_jordan1(A,B,C,D) )
                           => r1_xboole_0(E,F) ) ) ) ) ) ) ) )).

fof(t41_jordan1,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)
                 => ! [E] :
                      ( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
                     => ! [F] :
                          ( m1_subset_1(F,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
                         => ! [G] :
                              ( m1_subset_1(G,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
                             => ( ( E = a_4_8_jordan1(A,B,C,D)
                                  & F = a_4_6_jordan1(A,B,C,D)
                                  & G = a_4_7_jordan1(A,B,C,D) )
                               => ( r1_xreal_0(C,A)
                                  | r1_xreal_0(D,B)
                                  | ( k3_subset_1(u1_struct_0(k15_euclid(np__2)),E) = k4_subset_1(u1_struct_0(k15_euclid(np__2)),F,G)
                                    & k3_subset_1(u1_struct_0(k15_euclid(np__2)),E) != k1_xboole_0
                                    & r1_xboole_0(F,G)
                                    & ! [H] :
                                        ( m1_subset_1(H,k1_zfmisc_1(u1_struct_0(k3_pre_topc(k15_euclid(np__2),k3_subset_1(u1_struct_0(k15_euclid(np__2)),E)))))
                                       => ! [I] :
                                            ( m1_subset_1(I,k1_zfmisc_1(u1_struct_0(k3_pre_topc(k15_euclid(np__2),k3_subset_1(u1_struct_0(k15_euclid(np__2)),E)))))
                                           => ( ( H = F
                                                & I = G )
                                             => ( r3_connsp_1(k3_pre_topc(k15_euclid(np__2),k3_subset_1(u1_struct_0(k15_euclid(np__2)),E)),H)
                                                & r3_connsp_1(k3_pre_topc(k15_euclid(np__2),k3_subset_1(u1_struct_0(k15_euclid(np__2)),E)),I) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )).

fof(t42_jordan1,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)
                 => ! [E] :
                      ( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
                     => ! [F] :
                          ( m1_subset_1(F,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
                         => ! [G] :
                              ( m1_subset_1(G,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
                             => ( ( E = a_4_8_jordan1(A,B,C,D)
                                  & F = a_4_6_jordan1(A,B,C,D)
                                  & G = a_4_7_jordan1(A,B,C,D) )
                               => ( r1_xreal_0(C,A)
                                  | r1_xreal_0(D,B)
                                  | ( E = k6_subset_1(u1_struct_0(k15_euclid(np__2)),k6_pre_topc(k15_euclid(np__2),F),F)
                                    & E = k6_subset_1(u1_struct_0(k15_euclid(np__2)),k6_pre_topc(k15_euclid(np__2),G),G) ) ) ) ) ) ) ) ) ) ) )).

fof(t43_jordan1,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)
                 => ! [E] :
                      ( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
                     => ! [F] :
                          ( m1_subset_1(F,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
                         => ( ( E = a_4_9_jordan1(A,B,C,D)
                              & F = a_4_4_jordan1(A,B,C,D) )
                           => r1_tarski(F,k2_pre_topc(k3_pre_topc(k15_euclid(np__2),k3_subset_1(u1_struct_0(k15_euclid(np__2)),E)))) ) ) ) ) ) ) ) )).

fof(t44_jordan1,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)
                 => ! [E] :
                      ( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
                     => ! [F] :
                          ( m1_subset_1(F,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
                         => ( ( E = a_4_9_jordan1(A,B,C,D)
                              & F = a_4_4_jordan1(A,B,C,D) )
                           => m1_subset_1(F,k1_zfmisc_1(u1_struct_0(k3_pre_topc(k15_euclid(np__2),k3_subset_1(u1_struct_0(k15_euclid(np__2)),E))))) ) ) ) ) ) ) ) )).

fof(t45_jordan1,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)
                 => ! [E] :
                      ( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
                     => ! [F] :
                          ( m1_subset_1(F,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
                         => ( ( E = a_4_9_jordan1(A,B,C,D)
                              & F = a_4_5_jordan1(A,B,C,D) )
                           => ( r1_xreal_0(B,A)
                              | r1_xreal_0(D,C)
                              | r1_tarski(F,k2_pre_topc(k3_pre_topc(k15_euclid(np__2),k3_subset_1(u1_struct_0(k15_euclid(np__2)),E)))) ) ) ) ) ) ) ) ) )).

fof(t46_jordan1,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)
                 => ! [E] :
                      ( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
                     => ! [F] :
                          ( m1_subset_1(F,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
                         => ( ( E = a_4_9_jordan1(A,B,C,D)
                              & F = a_4_5_jordan1(A,B,C,D) )
                           => ( r1_xreal_0(B,A)
                              | r1_xreal_0(D,C)
                              | m1_subset_1(F,k1_zfmisc_1(u1_struct_0(k3_pre_topc(k15_euclid(np__2),k3_subset_1(u1_struct_0(k15_euclid(np__2)),E))))) ) ) ) ) ) ) ) ) )).

fof(t48_jordan1,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)
                 => ! [E] :
                      ( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
                     => ( E = a_4_9_jordan1(A,B,C,D)
                       => ( r1_xreal_0(B,A)
                          | r1_xreal_0(D,C)
                          | v2_jordan1(E) ) ) ) ) ) ) ) )).

fof(t49_jordan1,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)
                 => ! [E] :
                      ( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
                     => ! [F] :
                          ( m1_subset_1(F,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
                         => ( ( E = a_4_8_jordan1(A,B,C,D)
                              & F = a_4_7_jordan1(A,B,C,D) )
                           => ( r1_xreal_0(C,A)
                              | r1_xreal_0(D,B)
                              | k6_pre_topc(k15_euclid(np__2),F) = k4_subset_1(u1_struct_0(k15_euclid(np__2)),E,F) ) ) ) ) ) ) ) ) )).

fof(t50_jordan1,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)
                 => ! [E] :
                      ( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
                     => ! [F] :
                          ( m1_subset_1(F,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
                         => ( ( E = a_4_8_jordan1(A,B,C,D)
                              & F = a_4_6_jordan1(A,B,C,D) )
                           => ( r1_xreal_0(C,A)
                              | r1_xreal_0(D,B)
                              | k6_pre_topc(k15_euclid(np__2),F) = k4_subset_1(u1_struct_0(k15_euclid(np__2)),E,F) ) ) ) ) ) ) ) ) )).

fof(fraenkel_a_4_0_jordan1,axiom,(
    ! [A,B,C,D,E] :
      ( ( m1_subset_1(B,k1_numbers)
        & m1_subset_1(C,k1_numbers)
        & m1_subset_1(D,k1_numbers)
        & m1_subset_1(E,k1_numbers) )
     => ( r2_hidden(A,a_4_0_jordan1(B,C,D,E))
      <=> ? [F,G] :
            ( m1_subset_1(F,k1_numbers)
            & m1_subset_1(G,k1_numbers)
            & A = k23_euclid(F,G)
            & ~ r1_xreal_0(F,B)
            & ~ r1_xreal_0(C,F)
            & ~ r1_xreal_0(G,D)
            & ~ r1_xreal_0(E,G) ) ) ) )).

fof(fraenkel_a_1_0_jordan1,axiom,(
    ! [A,B] :
      ( m1_subset_1(B,k1_numbers)
     => ( r2_hidden(A,a_1_0_jordan1(B))
      <=> ? [C,D] :
            ( m1_subset_1(C,k1_numbers)
            & m1_subset_1(D,k1_numbers)
            & A = k23_euclid(C,D)
            & ~ r1_xreal_0(C,B) ) ) ) )).

fof(fraenkel_a_1_1_jordan1,axiom,(
    ! [A,B] :
      ( m1_subset_1(B,k1_numbers)
     => ( r2_hidden(A,a_1_1_jordan1(B))
      <=> ? [C,D] :
            ( m1_subset_1(C,k1_numbers)
            & m1_subset_1(D,k1_numbers)
            & A = k23_euclid(C,D)
            & ~ r1_xreal_0(B,C) ) ) ) )).

fof(fraenkel_a_1_2_jordan1,axiom,(
    ! [A,B] :
      ( m1_subset_1(B,k1_numbers)
     => ( r2_hidden(A,a_1_2_jordan1(B))
      <=> ? [C,D] :
            ( m1_subset_1(C,k1_numbers)
            & m1_subset_1(D,k1_numbers)
            & A = k23_euclid(C,D)
            & ~ r1_xreal_0(D,B) ) ) ) )).

fof(fraenkel_a_1_3_jordan1,axiom,(
    ! [A,B] :
      ( m1_subset_1(B,k1_numbers)
     => ( r2_hidden(A,a_1_3_jordan1(B))
      <=> ? [C,D] :
            ( m1_subset_1(C,k1_numbers)
            & m1_subset_1(D,k1_numbers)
            & A = k23_euclid(C,D)
            & ~ r1_xreal_0(B,D) ) ) ) )).

fof(fraenkel_a_4_1_jordan1,axiom,(
    ! [A,B,C,D,E] :
      ( ( m1_subset_1(B,k1_numbers)
        & m1_subset_1(C,k1_numbers)
        & m1_subset_1(D,k1_numbers)
        & m1_subset_1(E,k1_numbers) )
     => ( r2_hidden(A,a_4_1_jordan1(B,C,D,E))
      <=> ? [F,G] :
            ( m1_subset_1(F,k1_numbers)
            & m1_subset_1(G,k1_numbers)
            & A = k23_euclid(F,G)
            & ~ ( r1_xreal_0(B,F)
                & r1_xreal_0(F,C)
                & r1_xreal_0(D,G)
                & r1_xreal_0(G,E) ) ) ) ) )).

fof(fraenkel_a_4_2_jordan1,axiom,(
    ! [A,B,C,D,E] :
      ( ( m1_subset_1(B,k1_numbers)
        & m1_subset_1(C,k1_numbers)
        & m1_subset_1(D,k1_numbers)
        & m1_subset_1(E,k1_numbers) )
     => ( r2_hidden(A,a_4_2_jordan1(B,C,D,E))
      <=> ? [F,G] :
            ( m1_subset_1(F,k1_numbers)
            & m1_subset_1(G,k1_numbers)
            & A = k23_euclid(F,G)
            & ~ r1_xreal_0(F,B)
            & ~ r1_xreal_0(D,F)
            & ~ r1_xreal_0(G,C)
            & ~ r1_xreal_0(E,G) ) ) ) )).

fof(fraenkel_a_4_3_jordan1,axiom,(
    ! [A,B,C,D,E] :
      ( ( m1_subset_1(B,k1_numbers)
        & m1_subset_1(C,k1_numbers)
        & m1_subset_1(D,k1_numbers)
        & m1_subset_1(E,k1_numbers) )
     => ( r2_hidden(A,a_4_3_jordan1(B,C,D,E))
      <=> ? [F,G] :
            ( m1_subset_1(F,k1_numbers)
            & m1_subset_1(G,k1_numbers)
            & A = k23_euclid(F,G)
            & ~ ( r1_xreal_0(B,F)
                & r1_xreal_0(F,D)
                & r1_xreal_0(C,G)
                & r1_xreal_0(G,E) ) ) ) ) )).

fof(fraenkel_a_4_4_jordan1,axiom,(
    ! [A,B,C,D,E] :
      ( ( m1_subset_1(B,k1_numbers)
        & m1_subset_1(C,k1_numbers)
        & m1_subset_1(D,k1_numbers)
        & m1_subset_1(E,k1_numbers) )
     => ( r2_hidden(A,a_4_4_jordan1(B,C,D,E))
      <=> ? [F] :
            ( m1_subset_1(F,u1_struct_0(k15_euclid(np__2)))
            & A = F
            & ~ r1_xreal_0(k21_euclid(F),B)
            & ~ r1_xreal_0(C,k21_euclid(F))
            & ~ r1_xreal_0(k22_euclid(F),D)
            & ~ r1_xreal_0(E,k22_euclid(F)) ) ) ) )).

fof(fraenkel_a_4_5_jordan1,axiom,(
    ! [A,B,C,D,E] :
      ( ( m1_subset_1(B,k1_numbers)
        & m1_subset_1(C,k1_numbers)
        & m1_subset_1(D,k1_numbers)
        & m1_subset_1(E,k1_numbers) )
     => ( r2_hidden(A,a_4_5_jordan1(B,C,D,E))
      <=> ? [F] :
            ( m1_subset_1(F,u1_struct_0(k15_euclid(np__2)))
            & A = F
            & ~ ( r1_xreal_0(B,k21_euclid(F))
                & r1_xreal_0(k21_euclid(F),C)
                & r1_xreal_0(D,k22_euclid(F))
                & r1_xreal_0(k22_euclid(F),E) ) ) ) ) )).

fof(fraenkel_a_4_6_jordan1,axiom,(
    ! [A,B,C,D,E] :
      ( ( m1_subset_1(B,k1_numbers)
        & m1_subset_1(C,k1_numbers)
        & m1_subset_1(D,k1_numbers)
        & m1_subset_1(E,k1_numbers) )
     => ( r2_hidden(A,a_4_6_jordan1(B,C,D,E))
      <=> ? [F] :
            ( m1_subset_1(F,u1_struct_0(k15_euclid(np__2)))
            & A = F
            & ~ r1_xreal_0(k21_euclid(F),B)
            & ~ r1_xreal_0(D,k21_euclid(F))
            & ~ r1_xreal_0(k22_euclid(F),C)
            & ~ r1_xreal_0(E,k22_euclid(F)) ) ) ) )).

fof(fraenkel_a_4_7_jordan1,axiom,(
    ! [A,B,C,D,E] :
      ( ( m1_subset_1(B,k1_numbers)
        & m1_subset_1(C,k1_numbers)
        & m1_subset_1(D,k1_numbers)
        & m1_subset_1(E,k1_numbers) )
     => ( r2_hidden(A,a_4_7_jordan1(B,C,D,E))
      <=> ? [F] :
            ( m1_subset_1(F,u1_struct_0(k15_euclid(np__2)))
            & A = F
            & ~ ( r1_xreal_0(B,k21_euclid(F))
                & r1_xreal_0(k21_euclid(F),D)
                & r1_xreal_0(C,k22_euclid(F))
                & r1_xreal_0(k22_euclid(F),E) ) ) ) ) )).

fof(fraenkel_a_4_8_jordan1,axiom,(
    ! [A,B,C,D,E] :
      ( ( m1_subset_1(B,k1_numbers)
        & m1_subset_1(C,k1_numbers)
        & m1_subset_1(D,k1_numbers)
        & m1_subset_1(E,k1_numbers) )
     => ( r2_hidden(A,a_4_8_jordan1(B,C,D,E))
      <=> ? [F] :
            ( m1_subset_1(F,u1_struct_0(k15_euclid(np__2)))
            & A = F
            & ~ ( ~ ( k21_euclid(F) = B
                    & r1_xreal_0(k22_euclid(F),E)
                    & r1_xreal_0(C,k22_euclid(F)) )
                & ~ ( r1_xreal_0(k21_euclid(F),D)
                    & r1_xreal_0(B,k21_euclid(F))
                    & k22_euclid(F) = E )
                & ~ ( r1_xreal_0(k21_euclid(F),D)
                    & r1_xreal_0(B,k21_euclid(F))
                    & k22_euclid(F) = C )
                & ~ ( k21_euclid(F) = D
                    & r1_xreal_0(k22_euclid(F),E)
                    & r1_xreal_0(C,k22_euclid(F)) ) ) ) ) ) )).

fof(fraenkel_a_4_9_jordan1,axiom,(
    ! [A,B,C,D,E] :
      ( ( m1_subset_1(B,k1_numbers)
        & m1_subset_1(C,k1_numbers)
        & m1_subset_1(D,k1_numbers)
        & m1_subset_1(E,k1_numbers) )
     => ( r2_hidden(A,a_4_9_jordan1(B,C,D,E))
      <=> ? [F] :
            ( m1_subset_1(F,u1_struct_0(k15_euclid(np__2)))
            & A = F
            & ~ ( ~ ( k21_euclid(F) = B
                    & r1_xreal_0(k22_euclid(F),E)
                    & r1_xreal_0(D,k22_euclid(F)) )
                & ~ ( r1_xreal_0(k21_euclid(F),C)
                    & r1_xreal_0(B,k21_euclid(F))
                    & k22_euclid(F) = E )
                & ~ ( r1_xreal_0(k21_euclid(F),C)
                    & r1_xreal_0(B,k21_euclid(F))
                    & k22_euclid(F) = D )
                & ~ ( k21_euclid(F) = C
                    & r1_xreal_0(k22_euclid(F),E)
                    & r1_xreal_0(D,k22_euclid(F)) ) ) ) ) ) )).
%------------------------------------------------------------------------------