SET007 Axioms: SET007+732.ax


%------------------------------------------------------------------------------
% File     : SET007+732 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : On the Decomposition of a Simple Closed Curve into Two Arcs
% 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    : jordan16 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   44 (   0 unit)
%            Number of atoms       :  357 (  40 equality)
%            Maximal formula depth :   30 (  11 average)
%            Number of connectives :  361 (  48 ~  ;   4  |; 140  &)
%                                         (   3 <=>; 166 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   32 (   0 propositional; 1-5 arity)
%            Number of functors    :   40 (   7 constant; 0-6 arity)
%            Number of variables   :  154 (   0 singleton; 152 !;   2 ?)
%            Maximal term depth    :    4 (   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_jordan16,axiom,(
    ! [A] :
      ( m1_subset_1(A,k5_numbers)
     => ? [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
          & v1_realset1(B) ) ) )).

fof(fc1_jordan16,axiom,(
    ! [A,B] :
      ( ( v1_xreal_0(A)
        & v1_xreal_0(B) )
     => ( ~ v1_xboole_0(k1_jordan16(A,B))
        & v1_relat_1(k1_jordan16(A,B))
        & v1_funct_1(k1_jordan16(A,B))
        & v1_funct_2(k1_jordan16(A,B),k1_numbers,k1_numbers)
        & v1_seq_1(k1_jordan16(A,B))
        & v1_partfun1(k1_jordan16(A,B),k1_numbers,k1_numbers)
        & v1_jordan16(k1_jordan16(A,B)) ) ) )).

fof(rc2_jordan16,axiom,(
    ? [A] :
      ( m1_relset_1(A,k1_numbers,k1_numbers)
      & ~ v1_xboole_0(A)
      & v1_relat_1(A)
      & v1_funct_1(A)
      & v1_funct_2(A,k1_numbers,k1_numbers)
      & v1_seq_1(A)
      & v1_partfun1(A,k1_numbers,k1_numbers)
      & v1_jordan16(A) ) )).

fof(t1_jordan16,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_finset_1(A)
        & m1_subset_1(A,k1_zfmisc_1(k1_numbers)) )
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ~ ( ! [C] :
                  ( m1_subset_1(C,k1_numbers)
                 => ~ ( r2_hidden(C,A)
                      & r1_xreal_0(B,C) ) )
              & r1_xreal_0(B,k1_pre_circ(A)) ) ) ) )).

fof(t2_jordan16,axiom,(
    ! [A,B,C,D] :
      ( ( r2_hidden(A,D)
        & r2_hidden(B,D)
        & r2_hidden(C,D) )
     => r1_tarski(k1_enumset1(A,B,C),D) ) )).

fof(t3_jordan16,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => v1_jordan2c(k1_pre_topc(k15_euclid(A)),A) ) )).

fof(t4_jordan16,axiom,(
    ! [A] :
      ( ( v1_topreal2(A)
        & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
     => k9_jordan6(A) != k8_jordan6(A) ) )).

fof(t5_jordan16,axiom,(
    ! [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)))
             => ! [D] :
                  ( m1_subset_1(D,u1_struct_0(k15_euclid(np__2)))
                 => ! [E] :
                      ( m1_subset_1(E,u1_struct_0(k15_euclid(np__2)))
                     => r1_tarski(k5_jordan6(A,B,C,D,E),A) ) ) ) ) ) )).

fof(t6_jordan16,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)))
             => ( r1_tarski(B,C)
               => m1_pre_topc(k3_pre_topc(A,B),k3_pre_topc(A,C)) ) ) ) ) )).

fof(t7_jordan16,axiom,(
    ! [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)))
             => ! [D] :
                  ( m1_subset_1(D,u1_struct_0(k15_euclid(np__2)))
                 => ( ( r1_topreal1(k15_euclid(np__2),B,C,A)
                      & r2_hidden(D,A) )
                   => r2_hidden(D,k3_jordan6(A,B,C,D)) ) ) ) ) ) )).

fof(t8_jordan16,axiom,(
    ! [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)))
             => ! [D] :
                  ( m1_subset_1(D,u1_struct_0(k15_euclid(np__2)))
                 => ( ( r1_topreal1(k15_euclid(np__2),B,C,A)
                      & r2_hidden(D,A) )
                   => r2_hidden(D,k4_jordan6(A,B,C,D)) ) ) ) ) ) )).

fof(t9_jordan16,axiom,(
    ! [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)))
             => ! [D] :
                  ( m1_subset_1(D,u1_struct_0(k15_euclid(np__2)))
                 => ! [E] :
                      ( m1_subset_1(E,u1_struct_0(k15_euclid(np__2)))
                     => ( ( r1_topreal1(k15_euclid(np__2),B,C,A)
                          & r1_jordan5c(A,B,C,D,E) )
                       => ( r2_hidden(D,k5_jordan6(A,B,C,D,E))
                          & r2_hidden(E,k5_jordan6(A,B,C,D,E)) ) ) ) ) ) ) ) )).

fof(t10_jordan16,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)))
             => r1_tarski(k1_jordan7(A,B,C),A) ) ) ) )).

fof(t11_jordan16,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,A)
                  & r2_hidden(C,A)
                  & ~ r1_jordan6(A,B,C)
                  & ~ r1_jordan6(A,C,B) ) ) ) ) )).

fof(t12_jordan16,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & v2_pre_topc(B)
            & l1_pre_topc(B) )
         => ! [C] :
              ( ( ~ v3_struct_0(C)
                & m1_pre_topc(C,B) )
             => ! [D] :
                  ( ( v1_funct_1(D)
                    & v1_funct_2(D,u1_struct_0(A),u1_struct_0(B))
                    & m2_relset_1(D,u1_struct_0(A),u1_struct_0(B)) )
                 => ! [E] :
                      ( ( v1_funct_1(E)
                        & v1_funct_2(E,u1_struct_0(A),u1_struct_0(C))
                        & m2_relset_1(E,u1_struct_0(A),u1_struct_0(C)) )
                     => ( ( D = E
                          & v5_pre_topc(D,A,B) )
                       => v5_pre_topc(E,A,C) ) ) ) ) ) ) )).

fof(t13_jordan16,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & v2_pre_topc(B)
            & l1_pre_topc(B) )
         => ! [C] :
              ( ( ~ v3_struct_0(C)
                & m1_pre_topc(C,A) )
             => ! [D] :
                  ( ( ~ v3_struct_0(D)
                    & m1_pre_topc(D,B) )
                 => ! [E] :
                      ( ( v1_funct_1(E)
                        & v1_funct_2(E,u1_struct_0(A),u1_struct_0(B))
                        & m2_relset_1(E,u1_struct_0(A),u1_struct_0(B)) )
                     => ( v3_tops_2(E,A,B)
                       => ! [F] :
                            ( ( v1_funct_1(F)
                              & v1_funct_2(F,u1_struct_0(C),u1_struct_0(D))
                              & m2_relset_1(F,u1_struct_0(C),u1_struct_0(D)) )
                           => ( ( F = k2_tmap_1(A,B,C,E)
                                & v2_funct_2(F,u1_struct_0(C),u1_struct_0(D)) )
                             => v3_tops_2(F,C,D) ) ) ) ) ) ) ) ) )).

fof(t14_jordan16,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
             => ! [D] :
                  ( m1_subset_1(D,u1_struct_0(k15_euclid(np__2)))
                 => ! [E] :
                      ( m1_subset_1(E,u1_struct_0(k15_euclid(np__2)))
                     => ~ ( r1_topreal1(k15_euclid(np__2),D,E,A)
                          & r1_topreal1(k15_euclid(np__2),D,E,B)
                          & r1_topreal1(k15_euclid(np__2),D,E,C)
                          & k5_subset_1(u1_struct_0(k15_euclid(np__2)),B,C) = k2_struct_0(k15_euclid(np__2),D,E)
                          & r1_tarski(A,k4_subset_1(u1_struct_0(k15_euclid(np__2)),B,C))
                          & A != B
                          & A != C ) ) ) ) ) ) )).

fof(t15_jordan16,axiom,(
    ! [A] :
      ( ( v1_topreal2(A)
        & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
             => ! [D] :
                  ( m1_subset_1(D,u1_struct_0(k15_euclid(np__2)))
                 => ! [E] :
                      ( m1_subset_1(E,u1_struct_0(k15_euclid(np__2)))
                     => ( ( r1_topreal1(k15_euclid(np__2),D,E,B)
                          & r1_topreal1(k15_euclid(np__2),D,E,C)
                          & r1_tarski(B,A)
                          & r1_tarski(C,A) )
                       => ( B = C
                          | ( k4_subset_1(u1_struct_0(k15_euclid(np__2)),B,C) = A
                            & k5_subset_1(u1_struct_0(k15_euclid(np__2)),B,C) = k2_struct_0(k15_euclid(np__2),D,E) ) ) ) ) ) ) ) ) )).

fof(t16_jordan16,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(k15_euclid(np__2)))
             => ! [D] :
                  ( m1_subset_1(D,u1_struct_0(k15_euclid(np__2)))
                 => ! [E] :
                      ( m1_subset_1(E,u1_struct_0(k15_euclid(np__2)))
                     => ! [F] :
                          ( m1_subset_1(F,u1_struct_0(k15_euclid(np__2)))
                         => ~ ( r1_topreal1(k15_euclid(np__2),C,D,A)
                              & k5_subset_1(u1_struct_0(k15_euclid(np__2)),A,B) = k2_struct_0(k15_euclid(np__2),E,F)
                              & A = B ) ) ) ) ) ) ) )).

fof(t17_jordan16,axiom,(
    ! [A] :
      ( ( v1_topreal2(A)
        & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
             => ! [D] :
                  ( m1_subset_1(D,u1_struct_0(k15_euclid(np__2)))
                 => ! [E] :
                      ( m1_subset_1(E,u1_struct_0(k15_euclid(np__2)))
                     => ( ( r1_topreal1(k15_euclid(np__2),D,E,B)
                          & r1_topreal1(k15_euclid(np__2),D,E,C)
                          & r1_tarski(B,A)
                          & r1_tarski(C,A)
                          & k5_subset_1(u1_struct_0(k15_euclid(np__2)),B,C) = k2_struct_0(k15_euclid(np__2),D,E) )
                       => k4_subset_1(u1_struct_0(k15_euclid(np__2)),B,C) = A ) ) ) ) ) ) )).

fof(t18_jordan16,axiom,(
    ! [A] :
      ( ( v1_topreal2(A)
        & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
             => ! [D] :
                  ( m1_subset_1(D,u1_struct_0(k15_euclid(np__2)))
                 => ! [E] :
                      ( m1_subset_1(E,u1_struct_0(k15_euclid(np__2)))
                     => ( ( r1_tarski(B,A)
                          & r1_tarski(C,A)
                          & r1_topreal1(k15_euclid(np__2),D,E,B)
                          & r1_topreal1(k15_euclid(np__2),D,E,C) )
                       => ( B = C
                          | ! [F] :
                              ( m1_subset_1(F,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
                             => ~ ( r1_topreal1(k15_euclid(np__2),D,E,F)
                                  & r1_tarski(F,A)
                                  & F != B
                                  & F != C ) ) ) ) ) ) ) ) ) )).

fof(t19_jordan16,axiom,(
    ! [A] :
      ( ( v1_topreal2(A)
        & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
         => ~ ( r1_topreal1(k15_euclid(np__2),k30_pscomp_1(A),k34_pscomp_1(A),B)
              & r1_tarski(B,A)
              & B != k9_jordan6(A)
              & B != k8_jordan6(A) ) ) ) )).

fof(t20_jordan16,axiom,(
    ! [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)))
             => ! [D] :
                  ( m1_subset_1(D,u1_struct_0(k15_euclid(np__2)))
                 => ! [E] :
                      ( m1_subset_1(E,u1_struct_0(k15_euclid(np__2)))
                     => ~ ( r1_topreal1(k15_euclid(np__2),B,C,A)
                          & r1_jordan5c(A,B,C,D,E)
                          & ! [F] :
                              ( ( v1_funct_1(F)
                                & v1_funct_2(F,u1_struct_0(k5_topmetr),u1_struct_0(k3_pre_topc(k15_euclid(np__2),A)))
                                & m2_relset_1(F,u1_struct_0(k5_topmetr),u1_struct_0(k3_pre_topc(k15_euclid(np__2),A))) )
                             => ! [G] :
                                  ( m1_subset_1(G,k1_numbers)
                                 => ! [H] :
                                      ( m1_subset_1(H,k1_numbers)
                                     => ~ ( v3_tops_2(F,k5_topmetr,k3_pre_topc(k15_euclid(np__2),A))
                                          & k1_funct_1(F,np__0) = B
                                          & k1_funct_1(F,np__1) = C
                                          & k1_funct_1(F,G) = D
                                          & k1_funct_1(F,H) = E
                                          & r1_xreal_0(np__0,G)
                                          & r1_xreal_0(G,H)
                                          & r1_xreal_0(H,np__1) ) ) ) ) ) ) ) ) ) ) )).

fof(t21_jordan16,axiom,(
    ! [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)))
             => ! [D] :
                  ( m1_subset_1(D,u1_struct_0(k15_euclid(np__2)))
                 => ! [E] :
                      ( m1_subset_1(E,u1_struct_0(k15_euclid(np__2)))
                     => ~ ( r1_topreal1(k15_euclid(np__2),B,C,A)
                          & r1_jordan5c(A,B,C,D,E)
                          & D != E
                          & ! [F] :
                              ( ( v1_funct_1(F)
                                & v1_funct_2(F,u1_struct_0(k5_topmetr),u1_struct_0(k3_pre_topc(k15_euclid(np__2),A)))
                                & m2_relset_1(F,u1_struct_0(k5_topmetr),u1_struct_0(k3_pre_topc(k15_euclid(np__2),A))) )
                             => ! [G] :
                                  ( m1_subset_1(G,k1_numbers)
                                 => ! [H] :
                                      ( m1_subset_1(H,k1_numbers)
                                     => ~ ( v3_tops_2(F,k5_topmetr,k3_pre_topc(k15_euclid(np__2),A))
                                          & k1_funct_1(F,np__0) = B
                                          & k1_funct_1(F,np__1) = C
                                          & k1_funct_1(F,G) = D
                                          & k1_funct_1(F,H) = E
                                          & r1_xreal_0(np__0,G)
                                          & ~ r1_xreal_0(H,G)
                                          & r1_xreal_0(H,np__1) ) ) ) ) ) ) ) ) ) ) )).

fof(t22_jordan16,axiom,(
    ! [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)))
             => ! [D] :
                  ( m1_subset_1(D,u1_struct_0(k15_euclid(np__2)))
                 => ! [E] :
                      ( m1_subset_1(E,u1_struct_0(k15_euclid(np__2)))
                     => ~ ( r1_topreal1(k15_euclid(np__2),B,C,A)
                          & r1_jordan5c(A,B,C,D,E)
                          & v1_xboole_0(k5_jordan6(A,B,C,D,E)) ) ) ) ) ) ) )).

fof(t23_jordan16,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,A)
           => ( r2_hidden(B,k1_jordan7(A,B,k30_pscomp_1(A)))
              & r2_hidden(k30_pscomp_1(A),k1_jordan7(A,B,k30_pscomp_1(A))) ) ) ) ) )).

fof(d1_jordan16,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & m2_relset_1(A,k1_numbers,k1_numbers) )
     => ( v1_jordan16(A)
      <=> r2_fcont_1(A,k1_relat_1(A)) ) ) )).

fof(d2_jordan16,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k1_numbers,k1_numbers)
        & m2_relset_1(A,k1_numbers,k1_numbers) )
     => ( v1_jordan16(A)
      <=> r2_fcont_1(A,k1_numbers) ) ) )).

fof(d3_jordan16,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k1_numbers,k1_numbers)
                & m2_relset_1(C,k1_numbers,k1_numbers) )
             => ( C = k1_jordan16(A,B)
              <=> ! [D] :
                    ( v1_xreal_0(D)
                   => k1_funct_1(C,D) = k2_xcmplx_0(k3_xcmplx_0(A,D),B) ) ) ) ) ) )).

fof(t24_jordan16,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_jordan16(A)
        & m2_relset_1(A,k1_numbers,k1_numbers) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_jordan16(B)
            & m2_relset_1(B,k1_numbers,k1_numbers) )
         => ( v1_funct_1(k1_partfun1(k1_numbers,k1_numbers,k1_numbers,k1_numbers,A,B))
            & v1_jordan16(k1_partfun1(k1_numbers,k1_numbers,k1_numbers,k1_numbers,A,B))
            & m2_relset_1(k1_partfun1(k1_numbers,k1_numbers,k1_numbers,k1_numbers,A,B),k1_numbers,k1_numbers) ) ) ) )).

fof(t25_jordan16,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => k8_funct_2(k1_numbers,k1_numbers,k1_jordan16(A,B),np__0) = B ) ) )).

fof(t26_jordan16,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => k8_funct_2(k1_numbers,k1_numbers,k1_jordan16(A,B),np__1) = k2_xcmplx_0(A,B) ) ) )).

fof(t27_jordan16,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ( A != np__0
           => v2_funct_1(k1_jordan16(A,B)) ) ) ) )).

fof(t28_jordan16,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ! [C] :
              ( v1_xreal_0(C)
             => ! [D] :
                  ( v1_xreal_0(D)
                 => ~ ( ~ r1_xreal_0(A,np__0)
                      & ~ r1_xreal_0(D,C)
                      & r1_xreal_0(k1_funct_1(k1_jordan16(A,B),D),k1_funct_1(k1_jordan16(A,B),C)) ) ) ) ) ) )).

fof(t29_jordan16,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ! [C] :
              ( v1_xreal_0(C)
             => ! [D] :
                  ( v1_xreal_0(D)
                 => ~ ( ~ r1_xreal_0(np__0,A)
                      & ~ r1_xreal_0(D,C)
                      & r1_xreal_0(k1_funct_1(k1_jordan16(A,B),C),k1_funct_1(k1_jordan16(A,B),D)) ) ) ) ) ) )).

fof(t30_jordan16,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ! [C] :
              ( v1_xreal_0(C)
             => ! [D] :
                  ( v1_xreal_0(D)
                 => ( ( r1_xreal_0(np__0,A)
                      & r1_xreal_0(C,D) )
                   => r1_xreal_0(k1_funct_1(k1_jordan16(A,B),C),k1_funct_1(k1_jordan16(A,B),D)) ) ) ) ) ) )).

fof(t31_jordan16,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ! [C] :
              ( v1_xreal_0(C)
             => ! [D] :
                  ( v1_xreal_0(D)
                 => ( ( r1_xreal_0(A,np__0)
                      & r1_xreal_0(C,D) )
                   => r1_xreal_0(k1_funct_1(k1_jordan16(A,B),D),k1_funct_1(k1_jordan16(A,B),C)) ) ) ) ) ) )).

fof(t32_jordan16,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ( A != np__0
           => k1_pscomp_1(k1_numbers,k1_numbers,k1_jordan16(A,B)) = k1_numbers ) ) ) )).

fof(t33_jordan16,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ( A != np__0
           => k2_funct_1(k1_jordan16(A,B)) = k1_jordan16(k5_xcmplx_0(A),k4_xcmplx_0(k7_xcmplx_0(B,A))) ) ) ) )).

fof(t34_jordan16,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ( ~ r1_xreal_0(A,np__0)
           => k2_funct_2(k1_numbers,k1_numbers,k1_jordan16(A,B),k1_rcomp_1(np__0,np__1)) = k1_rcomp_1(B,k2_xcmplx_0(A,B)) ) ) ) )).

fof(t35_jordan16,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,u1_struct_0(k3_topmetr),u1_struct_0(k3_topmetr))
        & m2_relset_1(A,u1_struct_0(k3_topmetr),u1_struct_0(k3_topmetr)) )
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ! [C] :
              ( m1_subset_1(C,k1_numbers)
             => ( A = k1_jordan16(B,C)
               => ( B = np__0
                  | v3_tops_2(A,k3_topmetr,k3_topmetr) ) ) ) ) ) )).

fof(t36_jordan16,axiom,(
    ! [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)))
             => ! [D] :
                  ( m1_subset_1(D,u1_struct_0(k15_euclid(np__2)))
                 => ! [E] :
                      ( m1_subset_1(E,u1_struct_0(k15_euclid(np__2)))
                     => ( ( r1_topreal1(k15_euclid(np__2),B,C,A)
                          & r1_jordan5c(A,B,C,D,E) )
                       => ( D = E
                          | r1_topreal1(k15_euclid(np__2),D,E,k5_jordan6(A,B,C,D,E)) ) ) ) ) ) ) ) )).

fof(t37_jordan16,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)))
             => ! [D] :
                  ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
                 => ~ ( r1_tarski(D,A)
                      & r1_topreal1(k15_euclid(np__2),B,C,D)
                      & r2_hidden(k30_pscomp_1(A),D)
                      & r2_hidden(k34_pscomp_1(A),D)
                      & ~ r1_tarski(k8_jordan6(A),D)
                      & ~ r1_tarski(k9_jordan6(A),D) ) ) ) ) ) )).

fof(dt_k1_jordan16,axiom,(
    ! [A,B] :
      ( ( v1_xreal_0(A)
        & v1_xreal_0(B) )
     => ( v1_funct_1(k1_jordan16(A,B))
        & v1_funct_2(k1_jordan16(A,B),k1_numbers,k1_numbers)
        & m2_relset_1(k1_jordan16(A,B),k1_numbers,k1_numbers) ) ) )).
%------------------------------------------------------------------------------