SET007 Axioms: SET007+811.ax


%------------------------------------------------------------------------------
% File     : SET007+811 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Behaviour of an Arc Crossing a Line
% 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    : jordan20 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   42 (   1 unit)
%            Number of atoms       :  521 (  58 equality)
%            Maximal formula depth :   29 (  16 average)
%            Number of connectives :  552 (  73 ~  ;  18  |; 219  &)
%                                         (   8 <=>; 234 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   30 (   1 propositional; 0-5 arity)
%            Number of functors    :   31 (   7 constant; 0-6 arity)
%            Number of variables   :  221 (   0 singleton; 207 !;  14 ?)
%            Maximal term depth    :    5 (   2 average)
% SPC      : 

% Comments : The individual reference can be found in [Mat90] by looking for
%            the name provided by [Urb08].
%          : Translated by MPTP from the Mizar Mathematical Library 4.48.930.
%          : These set theory axioms are used in encodings of problems in
%            various domains, including ALG, CAT, GRP, LAT, SET, and TOP.
%------------------------------------------------------------------------------
fof(t1_jordan20,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) )
                   => k5_jordan6(A,B,C,D,D) = k1_struct_0(k15_euclid(np__2),D) ) ) ) ) ) )).

fof(t2_jordan20,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)))
             => ! [D] :
                  ( m1_subset_1(D,k1_numbers)
                 => ( ( r2_hidden(C,k3_topreal1(np__2,A,B))
                      & r1_xreal_0(k21_euclid(A),D)
                      & r1_xreal_0(k21_euclid(B),D) )
                   => r1_xreal_0(k21_euclid(C),D) ) ) ) ) ) )).

fof(t3_jordan20,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)))
             => ! [D] :
                  ( m1_subset_1(D,k1_numbers)
                 => ( ( r2_hidden(C,k3_topreal1(np__2,A,B))
                      & r1_xreal_0(D,k21_euclid(A))
                      & r1_xreal_0(D,k21_euclid(B)) )
                   => r1_xreal_0(D,k21_euclid(C)) ) ) ) ) ) )).

fof(t4_jordan20,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)))
             => ! [D] :
                  ( m1_subset_1(D,k1_numbers)
                 => ~ ( r2_hidden(C,k3_topreal1(np__2,A,B))
                      & ~ r1_xreal_0(D,k21_euclid(A))
                      & ~ r1_xreal_0(D,k21_euclid(B))
                      & r1_xreal_0(D,k21_euclid(C)) ) ) ) ) ) )).

fof(t5_jordan20,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)))
             => ! [D] :
                  ( m1_subset_1(D,k1_numbers)
                 => ~ ( r2_hidden(C,k3_topreal1(np__2,A,B))
                      & ~ r1_xreal_0(k21_euclid(A),D)
                      & ~ r1_xreal_0(k21_euclid(B),D)
                      & r1_xreal_0(k21_euclid(C),D) ) ) ) ) ) )).

fof(t6_jordan20,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ( v4_topreal1(B)
            & m2_finseq_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_xreal_0(np__1,A)
                      & r2_hidden(C,k4_topreal1(np__2,B,A))
                      & r2_hidden(D,k4_topreal1(np__2,B,A))
                      & k22_euclid(k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,A)) = k22_euclid(k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,k1_nat_1(A,np__1)))
                      & r1_jordan5c(k5_topreal1(np__2,B),k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,np__1),k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,k3_finseq_1(B)),C,D) )
                   => ( r1_xreal_0(k3_finseq_1(B),A)
                      | r1_xreal_0(k21_euclid(k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,A)),k21_euclid(k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,k1_nat_1(A,np__1))))
                      | r1_xreal_0(k21_euclid(D),k21_euclid(C)) ) ) ) ) ) ) )).

fof(t7_jordan20,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ( v4_topreal1(B)
            & m2_finseq_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_xreal_0(np__1,A)
                      & r2_hidden(C,k4_topreal1(np__2,B,A))
                      & r2_hidden(D,k4_topreal1(np__2,B,A))
                      & k22_euclid(k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,A)) = k22_euclid(k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,k1_nat_1(A,np__1)))
                      & r1_jordan5c(k5_topreal1(np__2,B),k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,np__1),k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,k3_finseq_1(B)),C,D) )
                   => ( r1_xreal_0(k3_finseq_1(B),A)
                      | r1_xreal_0(k21_euclid(k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,k1_nat_1(A,np__1))),k21_euclid(k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,A)))
                      | r1_xreal_0(k21_euclid(C),k21_euclid(D)) ) ) ) ) ) ) )).

fof(d1_jordan20,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,k1_numbers)
                     => ( r1_jordan20(A,B,C,D,E)
                      <=> ( r1_topreal1(k15_euclid(np__2),B,C,A)
                          & r2_hidden(D,A)
                          & k21_euclid(D) = E
                          & ? [F] :
                              ( m1_subset_1(F,u1_struct_0(k15_euclid(np__2)))
                              & ~ r1_xreal_0(E,k21_euclid(F))
                              & r1_jordan5c(A,B,C,F,D)
                              & ! [G] :
                                  ( m1_subset_1(G,u1_struct_0(k15_euclid(np__2)))
                                 => ( ( r1_jordan5c(A,B,C,F,G)
                                      & r1_jordan5c(A,B,C,G,D) )
                                   => r1_xreal_0(k21_euclid(G),E) ) ) ) ) ) ) ) ) ) ) )).

fof(d2_jordan20,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,k1_numbers)
                     => ( r2_jordan20(A,B,C,D,E)
                      <=> ( r1_topreal1(k15_euclid(np__2),B,C,A)
                          & r2_hidden(D,A)
                          & k21_euclid(D) = E
                          & ? [F] :
                              ( m1_subset_1(F,u1_struct_0(k15_euclid(np__2)))
                              & ~ r1_xreal_0(k21_euclid(F),E)
                              & r1_jordan5c(A,B,C,F,D)
                              & ! [G] :
                                  ( m1_subset_1(G,u1_struct_0(k15_euclid(np__2)))
                                 => ( ( r1_jordan5c(A,B,C,F,G)
                                      & r1_jordan5c(A,B,C,G,D) )
                                   => r1_xreal_0(E,k21_euclid(G)) ) ) ) ) ) ) ) ) ) ) )).

fof(d3_jordan20,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,k1_numbers)
                     => ( r3_jordan20(A,B,C,D,E)
                      <=> ( r1_topreal1(k15_euclid(np__2),B,C,A)
                          & r2_hidden(D,A)
                          & k21_euclid(D) = E
                          & ? [F] :
                              ( m1_subset_1(F,u1_struct_0(k15_euclid(np__2)))
                              & ~ r1_xreal_0(E,k21_euclid(F))
                              & r1_jordan5c(A,B,C,D,F)
                              & ! [G] :
                                  ( m1_subset_1(G,u1_struct_0(k15_euclid(np__2)))
                                 => ( ( r1_jordan5c(A,B,C,G,F)
                                      & r1_jordan5c(A,B,C,D,G) )
                                   => r1_xreal_0(k21_euclid(G),E) ) ) ) ) ) ) ) ) ) ) )).

fof(d4_jordan20,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,k1_numbers)
                     => ( r4_jordan20(A,B,C,D,E)
                      <=> ( r1_topreal1(k15_euclid(np__2),B,C,A)
                          & r2_hidden(D,A)
                          & k21_euclid(D) = E
                          & ? [F] :
                              ( m1_subset_1(F,u1_struct_0(k15_euclid(np__2)))
                              & ~ r1_xreal_0(k21_euclid(F),E)
                              & r1_jordan5c(A,B,C,D,F)
                              & ! [G] :
                                  ( m1_subset_1(G,u1_struct_0(k15_euclid(np__2)))
                                 => ( ( r1_jordan5c(A,B,C,G,F)
                                      & r1_jordan5c(A,B,C,D,G) )
                                   => r1_xreal_0(E,k21_euclid(G)) ) ) ) ) ) ) ) ) ) ) )).

fof(d5_jordan20,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,k1_numbers)
                     => ( r5_jordan20(A,B,C,D,E)
                      <=> ( r1_topreal1(k15_euclid(np__2),B,C,A)
                          & r2_hidden(D,A)
                          & k21_euclid(D) = E
                          & ? [F] :
                              ( m1_subset_1(F,u1_struct_0(k15_euclid(np__2)))
                              & r1_jordan5c(A,B,C,F,D)
                              & ! [G] :
                                  ( m1_subset_1(G,u1_struct_0(k15_euclid(np__2)))
                                 => ( ( r1_jordan5c(A,B,C,F,G)
                                      & r1_jordan5c(A,B,C,G,D) )
                                   => k21_euclid(G) = E ) )
                              & ! [G] :
                                  ( m1_subset_1(G,u1_struct_0(k15_euclid(np__2)))
                                 => ( r1_jordan5c(A,B,C,G,F)
                                   => ( G = F
                                      | ( ? [H] :
                                            ( m1_subset_1(H,u1_struct_0(k15_euclid(np__2)))
                                            & r1_jordan5c(A,B,C,G,H)
                                            & r1_jordan5c(A,B,C,H,F)
                                            & ~ r1_xreal_0(k21_euclid(H),E) )
                                        & ? [H] :
                                            ( m1_subset_1(H,u1_struct_0(k15_euclid(np__2)))
                                            & r1_jordan5c(A,B,C,G,H)
                                            & r1_jordan5c(A,B,C,H,F)
                                            & ~ r1_xreal_0(E,k21_euclid(H)) ) ) ) ) ) ) ) ) ) ) ) ) ) )).

fof(d6_jordan20,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,k1_numbers)
                     => ( r6_jordan20(A,B,C,D,E)
                      <=> ( r1_topreal1(k15_euclid(np__2),B,C,A)
                          & r2_hidden(D,A)
                          & k21_euclid(D) = E
                          & ? [F] :
                              ( m1_subset_1(F,u1_struct_0(k15_euclid(np__2)))
                              & r1_jordan5c(A,B,C,D,F)
                              & ! [G] :
                                  ( m1_subset_1(G,u1_struct_0(k15_euclid(np__2)))
                                 => ( ( r1_jordan5c(A,B,C,G,F)
                                      & r1_jordan5c(A,B,C,D,G) )
                                   => k21_euclid(G) = E ) )
                              & ! [G] :
                                  ( m1_subset_1(G,u1_struct_0(k15_euclid(np__2)))
                                 => ( r1_jordan5c(A,B,C,F,G)
                                   => ( G = F
                                      | ( ? [H] :
                                            ( m1_subset_1(H,u1_struct_0(k15_euclid(np__2)))
                                            & r1_jordan5c(A,B,C,H,G)
                                            & r1_jordan5c(A,B,C,F,H)
                                            & ~ r1_xreal_0(k21_euclid(H),E) )
                                        & ? [H] :
                                            ( m1_subset_1(H,u1_struct_0(k15_euclid(np__2)))
                                            & r1_jordan5c(A,B,C,H,G)
                                            & r1_jordan5c(A,B,C,F,H)
                                            & ~ r1_xreal_0(E,k21_euclid(H)) ) ) ) ) ) ) ) ) ) ) ) ) ) )).

fof(t8_jordan20,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,k1_numbers)
                     => ~ ( r1_topreal1(k15_euclid(np__2),B,C,A)
                          & r1_xreal_0(k21_euclid(B),E)
                          & r1_xreal_0(E,k21_euclid(C))
                          & ! [F] :
                              ( m1_subset_1(F,u1_struct_0(k15_euclid(np__2)))
                             => ~ ( r2_hidden(F,A)
                                  & k21_euclid(F) = E ) ) ) ) ) ) ) ) )).

fof(t9_jordan20,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(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,k1_numbers)
                     => ~ ( r1_topreal1(k15_euclid(np__2),B,C,A)
                          & ~ r1_xreal_0(E,k21_euclid(B))
                          & ~ r1_xreal_0(k21_euclid(C),E)
                          & r2_hidden(D,A)
                          & k21_euclid(D) = E
                          & ~ r1_jordan20(A,B,C,D,E)
                          & ~ r2_jordan20(A,B,C,D,E)
                          & ~ r5_jordan20(A,B,C,D,E) ) ) ) ) ) ) )).

fof(t10_jordan20,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(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,k1_numbers)
                     => ~ ( r1_topreal1(k15_euclid(np__2),B,C,A)
                          & ~ r1_xreal_0(E,k21_euclid(B))
                          & ~ r1_xreal_0(k21_euclid(C),E)
                          & r2_hidden(D,A)
                          & k21_euclid(D) = E
                          & ~ r3_jordan20(A,B,C,D,E)
                          & ~ r4_jordan20(A,B,C,D,E)
                          & ~ r6_jordan20(A,B,C,D,E) ) ) ) ) ) ) )).

fof(t11_jordan20,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k5_topmetr)))
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ( A = k1_rcomp_2(np__0,B)
           => v3_pre_topc(A,k5_topmetr) ) ) ) )).

fof(t12_jordan20,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k5_topmetr)))
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ( A = k2_rcomp_2(B,np__1)
           => v3_pre_topc(A,k5_topmetr) ) ) ) )).

fof(t17_jordan20,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(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,u1_struct_0(A))
                 => ! [E] :
                      ( m1_subset_1(E,u1_struct_0(A))
                     => ( ( k5_subset_1(u1_struct_0(A),B,C) = k1_xboole_0
                          & k4_subset_1(u1_struct_0(A),B,C) = u1_struct_0(A)
                          & r2_hidden(D,B)
                          & r2_hidden(E,C)
                          & v3_pre_topc(B,A)
                          & v3_pre_topc(C,A) )
                       => ! [F] :
                            ( ( v1_funct_1(F)
                              & v1_funct_2(F,u1_struct_0(k5_topmetr),u1_struct_0(A))
                              & m2_relset_1(F,u1_struct_0(k5_topmetr),u1_struct_0(A)) )
                           => ~ ( v5_pre_topc(F,k5_topmetr,A)
                                & k1_funct_1(F,np__0) = D
                                & k1_funct_1(F,np__1) = E ) ) ) ) ) ) ) ) )).

fof(t18_jordan20,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(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(k3_pre_topc(k15_euclid(np__2),A))))
         => ! [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),C,D,A)
                          & r2_hidden(E,A)
                          & B = k6_subset_1(u1_struct_0(k15_euclid(np__2)),A,k1_struct_0(k15_euclid(np__2),E)) )
                       => ( E = C
                          | E = D
                          | ( ~ v2_connsp_1(B,k3_pre_topc(k15_euclid(np__2),A))
                            & ! [F] :
                                ( ( v1_funct_1(F)
                                  & v1_funct_2(F,u1_struct_0(k5_topmetr),u1_struct_0(k3_pre_topc(k3_pre_topc(k15_euclid(np__2),A),B)))
                                  & m2_relset_1(F,u1_struct_0(k5_topmetr),u1_struct_0(k3_pre_topc(k3_pre_topc(k15_euclid(np__2),A),B))) )
                               => ~ ( v5_pre_topc(F,k5_topmetr,k3_pre_topc(k3_pre_topc(k15_euclid(np__2),A),B))
                                    & k1_funct_1(F,np__0) = C
                                    & k1_funct_1(F,np__1) = D ) ) ) ) ) ) ) ) ) ) )).

fof(t19_jordan20,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(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)
                          & r2_hidden(D,A)
                          & r2_hidden(E,A)
                          & ~ r1_jordan5c(A,B,C,D,E)
                          & ~ r1_jordan5c(A,B,C,E,D) ) ) ) ) ) ) )).

fof(t20_jordan20,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(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) )
                   => ( B = D
                      | r1_topreal1(k15_euclid(np__2),B,D,k5_jordan6(A,B,C,B,D)) ) ) ) ) ) ) )).

fof(t21_jordan20,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(k15_euclid(A)))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(k15_euclid(A)))
             => ! [D] :
                  ( ( ~ v1_xboole_0(D)
                    & m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k15_euclid(A)))) )
                 => ! [E] :
                      ( ( ~ v1_xboole_0(E)
                        & m1_subset_1(E,k1_zfmisc_1(u1_struct_0(k15_euclid(A)))) )
                     => ( ( r1_topreal1(k15_euclid(A),B,C,D)
                          & r1_topreal1(k15_euclid(A),B,C,E)
                          & r1_tarski(E,D) )
                       => E = D ) ) ) ) ) ) )).

fof(t22_jordan20,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(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) )
                   => ( C = D
                      | r1_topreal1(k15_euclid(np__2),D,C,k5_jordan6(A,B,C,D,C)) ) ) ) ) ) ) )).

fof(t23_jordan20,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(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)))
                     => ! [F] :
                          ( m1_subset_1(F,u1_struct_0(k15_euclid(np__2)))
                         => ( ( r1_topreal1(k15_euclid(np__2),B,C,A)
                              & r1_jordan5c(A,B,C,D,E)
                              & r1_jordan5c(A,B,C,E,F) )
                           => k4_subset_1(u1_struct_0(k15_euclid(np__2)),k5_jordan6(A,B,C,D,E),k5_jordan6(A,B,C,E,F)) = k5_jordan6(A,B,C,D,F) ) ) ) ) ) ) ) )).

fof(t24_jordan20,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(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)))
                     => ! [F] :
                          ( m1_subset_1(F,u1_struct_0(k15_euclid(np__2)))
                         => ( ( r1_topreal1(k15_euclid(np__2),B,C,A)
                              & r1_jordan5c(A,B,C,D,E)
                              & r1_jordan5c(A,B,C,E,F) )
                           => k5_subset_1(u1_struct_0(k15_euclid(np__2)),k5_jordan6(A,B,C,D,E),k5_jordan6(A,B,C,E,F)) = k1_struct_0(k15_euclid(np__2),E) ) ) ) ) ) ) ) )).

fof(t25_jordan20,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(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_topreal1(k15_euclid(np__2),B,C,A)
               => k5_jordan6(A,B,C,B,C) = A ) ) ) ) )).

fof(t26_jordan20,axiom,(
    $true )).

fof(t27_jordan20,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] :
                  ( m1_subset_1(D,u1_struct_0(A))
                 => ! [E] :
                      ( m1_borsuk_2(E,A,B,C)
                     => ! [F] :
                          ( m1_borsuk_2(F,A,C,D)
                         => ( ( v5_pre_topc(E,k5_topmetr,A)
                              & v5_pre_topc(F,k5_topmetr,A)
                              & k1_funct_1(E,np__0) = B
                              & k1_funct_1(E,np__1) = C
                              & k1_funct_1(F,np__0) = C
                              & k1_funct_1(F,np__1) = D )
                           => ( v5_pre_topc(k1_borsuk_2(A,B,C,D,E,F),k5_topmetr,A)
                              & k1_funct_1(k1_borsuk_2(A,B,C,D,E,F),np__0) = B
                              & k1_funct_1(k1_borsuk_2(A,B,C,D,E,F),np__1) = D ) ) ) ) ) ) ) ) )).

fof(t28_jordan20,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(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)))) )
         => ! [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)
                              & r1_topreal1(k15_euclid(np__2),E,F,B)
                              & r1_jordan5c(A,C,D,E,F)
                              & r1_tarski(B,A) )
                           => B = k5_jordan6(A,C,D,E,F) ) ) ) ) ) ) ) )).

fof(t29_jordan20,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(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)))
                     => ! [F] :
                          ( m1_subset_1(F,u1_struct_0(k15_euclid(np__2)))
                         => ! [G] :
                              ( m1_subset_1(G,k1_numbers)
                             => ( ( r1_jordan20(A,B,C,D,G)
                                  & k21_euclid(E) = G
                                  & r1_tarski(k3_topreal1(np__2,D,E),A)
                                  & r2_hidden(F,k3_topreal1(np__2,D,E)) )
                               => ( r1_xreal_0(G,k21_euclid(B))
                                  | r1_xreal_0(k21_euclid(C),G)
                                  | r1_jordan20(A,B,C,F,G) ) ) ) ) ) ) ) ) ) )).

fof(t30_jordan20,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(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)))
                     => ! [F] :
                          ( m1_subset_1(F,u1_struct_0(k15_euclid(np__2)))
                         => ! [G] :
                              ( m1_subset_1(G,k1_numbers)
                             => ( ( r2_jordan20(A,B,C,D,G)
                                  & k21_euclid(E) = G
                                  & r1_tarski(k3_topreal1(np__2,D,E),A)
                                  & r2_hidden(F,k3_topreal1(np__2,D,E)) )
                               => ( r1_xreal_0(G,k21_euclid(B))
                                  | r1_xreal_0(k21_euclid(C),G)
                                  | r2_jordan20(A,B,C,F,G) ) ) ) ) ) ) ) ) ) )).

fof(t31_jordan20,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(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)))
                     => ! [F] :
                          ( m1_subset_1(F,u1_struct_0(k15_euclid(np__2)))
                         => ! [G] :
                              ( m1_subset_1(G,k1_numbers)
                             => ( ( r3_jordan20(A,B,C,D,G)
                                  & k21_euclid(E) = G
                                  & r1_tarski(k3_topreal1(np__2,D,E),A)
                                  & r2_hidden(F,k3_topreal1(np__2,D,E)) )
                               => ( r1_xreal_0(G,k21_euclid(B))
                                  | r1_xreal_0(k21_euclid(C),G)
                                  | r3_jordan20(A,B,C,F,G) ) ) ) ) ) ) ) ) ) )).

fof(t32_jordan20,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(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)))
                     => ! [F] :
                          ( m1_subset_1(F,u1_struct_0(k15_euclid(np__2)))
                         => ! [G] :
                              ( m1_subset_1(G,k1_numbers)
                             => ( ( r4_jordan20(A,B,C,D,G)
                                  & k21_euclid(E) = G
                                  & r1_tarski(k3_topreal1(np__2,D,E),A)
                                  & r2_hidden(F,k3_topreal1(np__2,D,E)) )
                               => ( r1_xreal_0(G,k21_euclid(B))
                                  | r1_xreal_0(k21_euclid(C),G)
                                  | r4_jordan20(A,B,C,F,G) ) ) ) ) ) ) ) ) ) )).

fof(t33_jordan20,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(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,k1_numbers)
                     => ~ ( r1_topreal4(A,B,C)
                          & ~ r1_xreal_0(E,k21_euclid(B))
                          & ~ r1_xreal_0(k21_euclid(C),E)
                          & r2_hidden(D,A)
                          & k21_euclid(D) = E
                          & ~ r1_jordan20(A,B,C,D,E)
                          & ~ r2_jordan20(A,B,C,D,E) ) ) ) ) ) ) )).

fof(t34_jordan20,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(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,k1_numbers)
                     => ~ ( r1_topreal4(A,B,C)
                          & ~ r1_xreal_0(E,k21_euclid(B))
                          & ~ r1_xreal_0(k21_euclid(C),E)
                          & r2_hidden(D,A)
                          & k21_euclid(D) = E
                          & ~ r3_jordan20(A,B,C,D,E)
                          & ~ r4_jordan20(A,B,C,D,E) ) ) ) ) ) ) )).

fof(t13_jordan20,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(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(k3_pre_topc(k15_euclid(np__2),A))))
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k5_topmetr)))
             => ! [D] :
                  ( ( v1_funct_1(D)
                    & v1_funct_2(D,u1_struct_0(k5_topmetr),u1_struct_0(k3_pre_topc(k15_euclid(np__2),A)))
                    & m2_relset_1(D,u1_struct_0(k5_topmetr),u1_struct_0(k3_pre_topc(k15_euclid(np__2),A))) )
                 => ! [E] :
                      ( m1_subset_1(E,k1_numbers)
                     => ( ( r1_xreal_0(E,np__1)
                          & B = a_3_0_jordan20(A,D,E)
                          & C = k1_rcomp_2(np__0,E) )
                       => k4_pre_topc(k5_topmetr,k3_pre_topc(k15_euclid(np__2),A),D,C) = B ) ) ) ) ) ) )).

fof(t14_jordan20,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(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(k3_pre_topc(k15_euclid(np__2),A))))
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k5_topmetr)))
             => ! [D] :
                  ( ( v1_funct_1(D)
                    & v1_funct_2(D,u1_struct_0(k5_topmetr),u1_struct_0(k3_pre_topc(k15_euclid(np__2),A)))
                    & m2_relset_1(D,u1_struct_0(k5_topmetr),u1_struct_0(k3_pre_topc(k15_euclid(np__2),A))) )
                 => ! [E] :
                      ( m1_subset_1(E,k1_numbers)
                     => ( ( r1_xreal_0(np__0,E)
                          & B = a_3_1_jordan20(A,D,E)
                          & C = k2_rcomp_2(E,np__1) )
                       => k4_pre_topc(k5_topmetr,k3_pre_topc(k15_euclid(np__2),A),D,C) = B ) ) ) ) ) ) )).

fof(t15_jordan20,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(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(k3_pre_topc(k15_euclid(np__2),A))))
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,u1_struct_0(k5_topmetr),u1_struct_0(k3_pre_topc(k15_euclid(np__2),A)))
                & m2_relset_1(C,u1_struct_0(k5_topmetr),u1_struct_0(k3_pre_topc(k15_euclid(np__2),A))) )
             => ! [D] :
                  ( m1_subset_1(D,k1_numbers)
                 => ( ( r1_xreal_0(D,np__1)
                      & v3_tops_2(C,k5_topmetr,k3_pre_topc(k15_euclid(np__2),A))
                      & B = a_3_0_jordan20(A,C,D) )
                   => v3_pre_topc(B,k3_pre_topc(k15_euclid(np__2),A)) ) ) ) ) ) )).

fof(t16_jordan20,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(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(k3_pre_topc(k15_euclid(np__2),A))))
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,u1_struct_0(k5_topmetr),u1_struct_0(k3_pre_topc(k15_euclid(np__2),A)))
                & m2_relset_1(C,u1_struct_0(k5_topmetr),u1_struct_0(k3_pre_topc(k15_euclid(np__2),A))) )
             => ! [D] :
                  ( m1_subset_1(D,k1_numbers)
                 => ( ( r1_xreal_0(np__0,D)
                      & v3_tops_2(C,k5_topmetr,k3_pre_topc(k15_euclid(np__2),A))
                      & B = a_3_1_jordan20(A,C,D) )
                   => v3_pre_topc(B,k3_pre_topc(k15_euclid(np__2),A)) ) ) ) ) ) )).

fof(fraenkel_a_3_0_jordan20,axiom,(
    ! [A,B,C,D] :
      ( ( ~ v1_xboole_0(B)
        & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
        & v1_funct_1(C)
        & v1_funct_2(C,u1_struct_0(k5_topmetr),u1_struct_0(k3_pre_topc(k15_euclid(np__2),B)))
        & m2_relset_1(C,u1_struct_0(k5_topmetr),u1_struct_0(k3_pre_topc(k15_euclid(np__2),B)))
        & m1_subset_1(D,k1_numbers) )
     => ( r2_hidden(A,a_3_0_jordan20(B,C,D))
      <=> ? [E] :
            ( m1_subset_1(E,u1_struct_0(k15_euclid(np__2)))
            & A = E
            & ? [F] :
                ( m1_subset_1(F,k1_numbers)
                & r1_xreal_0(np__0,F)
                & ~ r1_xreal_0(D,F)
                & E = k1_funct_1(C,F) ) ) ) ) )).

fof(fraenkel_a_3_1_jordan20,axiom,(
    ! [A,B,C,D] :
      ( ( ~ v1_xboole_0(B)
        & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
        & v1_funct_1(C)
        & v1_funct_2(C,u1_struct_0(k5_topmetr),u1_struct_0(k3_pre_topc(k15_euclid(np__2),B)))
        & m2_relset_1(C,u1_struct_0(k5_topmetr),u1_struct_0(k3_pre_topc(k15_euclid(np__2),B)))
        & m1_subset_1(D,k1_numbers) )
     => ( r2_hidden(A,a_3_1_jordan20(B,C,D))
      <=> ? [E] :
            ( m1_subset_1(E,u1_struct_0(k15_euclid(np__2)))
            & A = E
            & ? [F] :
                ( m1_subset_1(F,k1_numbers)
                & ~ r1_xreal_0(F,D)
                & r1_xreal_0(F,np__1)
                & E = k1_funct_1(C,F) ) ) ) ) )).
%------------------------------------------------------------------------------