SET007 Axioms: SET007+149.ax


%------------------------------------------------------------------------------
% File     : SET007+149 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Real Function One-Side Differentiability
% 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    : fdiff_3 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   30 (   0 unit)
%            Number of atoms       :  451 (  55 equality)
%            Maximal formula depth :   20 (  13 average)
%            Number of connectives :  463 (  42 ~  ;   6  |; 247  &)
%                                         (   6 <=>; 162 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   18 (   0 propositional; 1-3 arity)
%            Number of functors    :   31 (   3 constant; 0-4 arity)
%            Number of variables   :  136 (   0 singleton; 128 !;   8 ?)
%            Maximal term depth    :    6 (   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_fdiff_3,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & m2_relset_1(A,k1_numbers,k1_numbers) )
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ~ ( ? [C] :
                  ( m1_subset_1(C,k1_numbers)
                  & ~ r1_xreal_0(C,np__0)
                  & r1_tarski(k1_rcomp_1(k5_real_1(B,C),B),k4_relset_1(k1_numbers,k1_numbers,A)) )
              & ! [C] :
                  ( ( v1_funct_1(C)
                    & v1_funct_2(C,k5_numbers,k1_numbers)
                    & v1_fdiff_1(C)
                    & m2_relset_1(C,k5_numbers,k1_numbers) )
                 => ! [D] :
                      ( ( v1_funct_1(D)
                        & v1_funct_2(D,k5_numbers,k1_numbers)
                        & v5_seqm_3(D)
                        & m2_relset_1(D,k5_numbers,k1_numbers) )
                     => ~ ( k1_rfunct_2(D) = k1_tarski(B)
                          & r1_tarski(k1_rfunct_2(k9_seq_1(C,D)),k4_relset_1(k1_numbers,k1_numbers,A))
                          & ! [E] :
                              ( m2_subset_1(E,k1_numbers,k5_numbers)
                             => ~ r1_xreal_0(np__0,k2_seq_1(k5_numbers,k1_numbers,C,E)) ) ) ) ) ) ) ) )).

fof(t2_fdiff_3,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & m2_relset_1(A,k1_numbers,k1_numbers) )
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ~ ( ? [C] :
                  ( m1_subset_1(C,k1_numbers)
                  & ~ r1_xreal_0(C,np__0)
                  & r1_tarski(k1_rcomp_1(B,k3_real_1(B,C)),k4_relset_1(k1_numbers,k1_numbers,A)) )
              & ! [C] :
                  ( ( v1_funct_1(C)
                    & v1_funct_2(C,k5_numbers,k1_numbers)
                    & v1_fdiff_1(C)
                    & m2_relset_1(C,k5_numbers,k1_numbers) )
                 => ! [D] :
                      ( ( v1_funct_1(D)
                        & v1_funct_2(D,k5_numbers,k1_numbers)
                        & v5_seqm_3(D)
                        & m2_relset_1(D,k5_numbers,k1_numbers) )
                     => ~ ( k1_rfunct_2(D) = k1_tarski(B)
                          & r1_tarski(k1_rfunct_2(k9_seq_1(C,D)),k4_relset_1(k1_numbers,k1_numbers,A))
                          & ! [E] :
                              ( m2_subset_1(E,k1_numbers,k5_numbers)
                             => ~ r1_xreal_0(k2_seq_1(k5_numbers,k1_numbers,C,E),np__0) ) ) ) ) ) ) ) )).

fof(t3_fdiff_3,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & m2_relset_1(A,k1_numbers,k1_numbers) )
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ( ( ! [C] :
                  ( ( v1_funct_1(C)
                    & v1_funct_2(C,k5_numbers,k1_numbers)
                    & v1_fdiff_1(C)
                    & m2_relset_1(C,k5_numbers,k1_numbers) )
                 => ! [D] :
                      ( ( v1_funct_1(D)
                        & v1_funct_2(D,k5_numbers,k1_numbers)
                        & v5_seqm_3(D)
                        & m2_relset_1(D,k5_numbers,k1_numbers) )
                     => ( ( k1_rfunct_2(D) = k1_tarski(B)
                          & r1_tarski(k1_rfunct_2(k9_seq_1(C,D)),k4_relset_1(k1_numbers,k1_numbers,A))
                          & ! [E] :
                              ( m2_subset_1(E,k1_numbers,k5_numbers)
                             => ~ r1_xreal_0(np__0,k2_seq_1(k5_numbers,k1_numbers,C,E)) ) )
                       => v4_seq_2(k11_seq_1(k18_seq_1(C),k10_seq_1(k2_rfunct_2(A,k9_seq_1(C,D)),k2_rfunct_2(A,D)))) ) ) )
              & r1_tarski(k1_tarski(B),k4_relset_1(k1_numbers,k1_numbers,A)) )
           => ! [C] :
                ( ( v1_funct_1(C)
                  & v1_funct_2(C,k5_numbers,k1_numbers)
                  & v1_fdiff_1(C)
                  & m2_relset_1(C,k5_numbers,k1_numbers) )
               => ! [D] :
                    ( ( v1_funct_1(D)
                      & v1_funct_2(D,k5_numbers,k1_numbers)
                      & v1_fdiff_1(D)
                      & m2_relset_1(D,k5_numbers,k1_numbers) )
                   => ! [E] :
                        ( ( v1_funct_1(E)
                          & v1_funct_2(E,k5_numbers,k1_numbers)
                          & v5_seqm_3(E)
                          & m2_relset_1(E,k5_numbers,k1_numbers) )
                       => ( ( k1_rfunct_2(E) = k1_tarski(B)
                            & r1_tarski(k1_rfunct_2(k9_seq_1(C,E)),k4_relset_1(k1_numbers,k1_numbers,A))
                            & ! [F] :
                                ( m2_subset_1(F,k1_numbers,k5_numbers)
                               => ~ r1_xreal_0(np__0,k2_seq_1(k5_numbers,k1_numbers,C,F)) )
                            & r1_tarski(k1_rfunct_2(k9_seq_1(D,E)),k4_relset_1(k1_numbers,k1_numbers,A))
                            & ! [F] :
                                ( m2_subset_1(F,k1_numbers,k5_numbers)
                               => ~ r1_xreal_0(np__0,k2_seq_1(k5_numbers,k1_numbers,D,F)) ) )
                         => k2_seq_2(k11_seq_1(k18_seq_1(C),k10_seq_1(k2_rfunct_2(A,k9_seq_1(C,E)),k2_rfunct_2(A,E)))) = k2_seq_2(k11_seq_1(k18_seq_1(D),k10_seq_1(k2_rfunct_2(A,k9_seq_1(D,E)),k2_rfunct_2(A,E)))) ) ) ) ) ) ) ) )).

fof(t4_fdiff_3,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & m2_relset_1(A,k1_numbers,k1_numbers) )
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ( ( ! [C] :
                  ( ( v1_funct_1(C)
                    & v1_funct_2(C,k5_numbers,k1_numbers)
                    & v1_fdiff_1(C)
                    & m2_relset_1(C,k5_numbers,k1_numbers) )
                 => ! [D] :
                      ( ( v1_funct_1(D)
                        & v1_funct_2(D,k5_numbers,k1_numbers)
                        & v5_seqm_3(D)
                        & m2_relset_1(D,k5_numbers,k1_numbers) )
                     => ( ( k1_rfunct_2(D) = k1_tarski(B)
                          & r1_tarski(k1_rfunct_2(k9_seq_1(C,D)),k4_relset_1(k1_numbers,k1_numbers,A))
                          & ! [E] :
                              ( m2_subset_1(E,k1_numbers,k5_numbers)
                             => ~ r1_xreal_0(k2_seq_1(k5_numbers,k1_numbers,C,E),np__0) ) )
                       => v4_seq_2(k11_seq_1(k18_seq_1(C),k10_seq_1(k2_rfunct_2(A,k9_seq_1(C,D)),k2_rfunct_2(A,D)))) ) ) )
              & r1_tarski(k1_tarski(B),k4_relset_1(k1_numbers,k1_numbers,A)) )
           => ! [C] :
                ( ( v1_funct_1(C)
                  & v1_funct_2(C,k5_numbers,k1_numbers)
                  & v1_fdiff_1(C)
                  & m2_relset_1(C,k5_numbers,k1_numbers) )
               => ! [D] :
                    ( ( v1_funct_1(D)
                      & v1_funct_2(D,k5_numbers,k1_numbers)
                      & v1_fdiff_1(D)
                      & m2_relset_1(D,k5_numbers,k1_numbers) )
                   => ! [E] :
                        ( ( v1_funct_1(E)
                          & v1_funct_2(E,k5_numbers,k1_numbers)
                          & v5_seqm_3(E)
                          & m2_relset_1(E,k5_numbers,k1_numbers) )
                       => ( ( k1_rfunct_2(E) = k1_tarski(B)
                            & r1_tarski(k1_rfunct_2(k9_seq_1(C,E)),k4_relset_1(k1_numbers,k1_numbers,A))
                            & r1_tarski(k1_rfunct_2(k9_seq_1(D,E)),k4_relset_1(k1_numbers,k1_numbers,A))
                            & ! [F] :
                                ( m2_subset_1(F,k1_numbers,k5_numbers)
                               => ~ r1_xreal_0(k2_seq_1(k5_numbers,k1_numbers,C,F),np__0) )
                            & ! [F] :
                                ( m2_subset_1(F,k1_numbers,k5_numbers)
                               => ~ r1_xreal_0(k2_seq_1(k5_numbers,k1_numbers,D,F),np__0) ) )
                         => k2_seq_2(k11_seq_1(k18_seq_1(C),k10_seq_1(k2_rfunct_2(A,k9_seq_1(C,E)),k2_rfunct_2(A,E)))) = k2_seq_2(k11_seq_1(k18_seq_1(D),k10_seq_1(k2_rfunct_2(A,k9_seq_1(D,E)),k2_rfunct_2(A,E)))) ) ) ) ) ) ) ) )).

fof(d1_fdiff_3,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & m2_relset_1(A,k1_numbers,k1_numbers) )
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ( r1_fdiff_3(A,B)
          <=> ( r2_hidden(B,k4_relset_1(k1_numbers,k1_numbers,A))
              & ! [C] :
                  ( ( v1_funct_1(C)
                    & v1_funct_2(C,k5_numbers,k1_numbers)
                    & m2_relset_1(C,k5_numbers,k1_numbers) )
                 => ( ( r1_tarski(k1_rfunct_2(C),k3_xboole_0(k12_prob_1(B),k4_relset_1(k1_numbers,k1_numbers,A)))
                      & v4_seq_2(C)
                      & k2_seq_2(C) = B )
                   => ( v4_seq_2(k2_rfunct_2(A,C))
                      & k2_seq_1(k1_numbers,k1_numbers,A,B) = k2_seq_2(k2_rfunct_2(A,C)) ) ) ) ) ) ) ) )).

fof(d2_fdiff_3,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & m2_relset_1(A,k1_numbers,k1_numbers) )
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ( r2_fdiff_3(A,B)
          <=> ( r2_hidden(B,k4_relset_1(k1_numbers,k1_numbers,A))
              & ! [C] :
                  ( ( v1_funct_1(C)
                    & v1_funct_2(C,k5_numbers,k1_numbers)
                    & m2_relset_1(C,k5_numbers,k1_numbers) )
                 => ( ( r1_tarski(k1_rfunct_2(C),k3_xboole_0(k4_limfunc1(B),k4_relset_1(k1_numbers,k1_numbers,A)))
                      & v4_seq_2(C)
                      & k2_seq_2(C) = B )
                   => ( v4_seq_2(k2_rfunct_2(A,C))
                      & k2_seq_1(k1_numbers,k1_numbers,A,B) = k2_seq_2(k2_rfunct_2(A,C)) ) ) ) ) ) ) ) )).

fof(d3_fdiff_3,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & m2_relset_1(A,k1_numbers,k1_numbers) )
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ( r3_fdiff_3(A,B)
          <=> ( ? [C] :
                  ( m1_subset_1(C,k1_numbers)
                  & ~ r1_xreal_0(C,np__0)
                  & r1_tarski(k1_rcomp_1(B,k3_real_1(B,C)),k4_relset_1(k1_numbers,k1_numbers,A)) )
              & ! [C] :
                  ( ( v1_funct_1(C)
                    & v1_funct_2(C,k5_numbers,k1_numbers)
                    & v1_fdiff_1(C)
                    & m2_relset_1(C,k5_numbers,k1_numbers) )
                 => ! [D] :
                      ( ( v1_funct_1(D)
                        & v1_funct_2(D,k5_numbers,k1_numbers)
                        & v5_seqm_3(D)
                        & m2_relset_1(D,k5_numbers,k1_numbers) )
                     => ( ( k1_rfunct_2(D) = k1_tarski(B)
                          & r1_tarski(k1_rfunct_2(k9_seq_1(C,D)),k4_relset_1(k1_numbers,k1_numbers,A))
                          & ! [E] :
                              ( m2_subset_1(E,k1_numbers,k5_numbers)
                             => ~ r1_xreal_0(k2_seq_1(k5_numbers,k1_numbers,C,E),np__0) ) )
                       => v4_seq_2(k11_seq_1(k18_seq_1(C),k10_seq_1(k2_rfunct_2(A,k9_seq_1(C,D)),k2_rfunct_2(A,D)))) ) ) ) ) ) ) ) )).

fof(d4_fdiff_3,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & m2_relset_1(A,k1_numbers,k1_numbers) )
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ( r4_fdiff_3(A,B)
          <=> ( ? [C] :
                  ( m1_subset_1(C,k1_numbers)
                  & ~ r1_xreal_0(C,np__0)
                  & r1_tarski(k1_rcomp_1(k5_real_1(B,C),B),k4_relset_1(k1_numbers,k1_numbers,A)) )
              & ! [C] :
                  ( ( v1_funct_1(C)
                    & v1_funct_2(C,k5_numbers,k1_numbers)
                    & v1_fdiff_1(C)
                    & m2_relset_1(C,k5_numbers,k1_numbers) )
                 => ! [D] :
                      ( ( v1_funct_1(D)
                        & v1_funct_2(D,k5_numbers,k1_numbers)
                        & v5_seqm_3(D)
                        & m2_relset_1(D,k5_numbers,k1_numbers) )
                     => ( ( k1_rfunct_2(D) = k1_tarski(B)
                          & r1_tarski(k1_rfunct_2(k9_seq_1(C,D)),k4_relset_1(k1_numbers,k1_numbers,A))
                          & ! [E] :
                              ( m2_subset_1(E,k1_numbers,k5_numbers)
                             => ~ r1_xreal_0(np__0,k2_seq_1(k5_numbers,k1_numbers,C,E)) ) )
                       => v4_seq_2(k11_seq_1(k18_seq_1(C),k10_seq_1(k2_rfunct_2(A,k9_seq_1(C,D)),k2_rfunct_2(A,D)))) ) ) ) ) ) ) ) )).

fof(t5_fdiff_3,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & m2_relset_1(A,k1_numbers,k1_numbers) )
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ( r4_fdiff_3(A,B)
           => r1_fdiff_3(A,B) ) ) ) )).

fof(t6_fdiff_3,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & m2_relset_1(A,k1_numbers,k1_numbers) )
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ! [C] :
              ( m1_subset_1(C,k1_numbers)
             => ~ ( r1_fdiff_3(A,B)
                  & k2_seq_1(k1_numbers,k1_numbers,A,B) != C
                  & ? [D] :
                      ( m1_subset_1(D,k1_numbers)
                      & ~ r1_xreal_0(D,np__0)
                      & r1_tarski(k1_rcomp_1(k5_real_1(B,D),B),k4_relset_1(k1_numbers,k1_numbers,A)) )
                  & ! [D] :
                      ( m1_subset_1(D,k1_numbers)
                     => ~ ( ~ r1_xreal_0(D,np__0)
                          & r1_tarski(k1_rcomp_1(k5_real_1(B,D),B),k4_relset_1(k1_numbers,k1_numbers,A))
                          & ! [E] :
                              ( m1_subset_1(E,k1_numbers)
                             => ~ ( r2_hidden(E,k1_rcomp_1(k5_real_1(B,D),B))
                                  & k2_seq_1(k1_numbers,k1_numbers,A,E) = C ) ) ) ) ) ) ) ) )).

fof(t7_fdiff_3,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & m2_relset_1(A,k1_numbers,k1_numbers) )
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ( r3_fdiff_3(A,B)
           => r2_fdiff_3(A,B) ) ) ) )).

fof(t8_fdiff_3,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & m2_relset_1(A,k1_numbers,k1_numbers) )
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ! [C] :
              ( m1_subset_1(C,k1_numbers)
             => ~ ( r2_fdiff_3(A,B)
                  & k2_seq_1(k1_numbers,k1_numbers,A,B) != C
                  & ? [D] :
                      ( m1_subset_1(D,k1_numbers)
                      & ~ r1_xreal_0(D,np__0)
                      & r1_tarski(k1_rcomp_1(B,k3_real_1(B,D)),k4_relset_1(k1_numbers,k1_numbers,A)) )
                  & ! [D] :
                      ( m1_subset_1(D,k1_numbers)
                     => ~ ( ~ r1_xreal_0(D,np__0)
                          & r1_tarski(k1_rcomp_1(B,k3_real_1(B,D)),k4_relset_1(k1_numbers,k1_numbers,A))
                          & ! [E] :
                              ( m1_subset_1(E,k1_numbers)
                             => ~ ( r2_hidden(E,k1_rcomp_1(B,k3_real_1(B,D)))
                                  & k2_seq_1(k1_numbers,k1_numbers,A,E) = C ) ) ) ) ) ) ) ) )).

fof(d5_fdiff_3,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( ( v1_funct_1(B)
            & m2_relset_1(B,k1_numbers,k1_numbers) )
         => ( r4_fdiff_3(B,A)
           => ! [C] :
                ( m1_subset_1(C,k1_numbers)
               => ( C = k1_fdiff_3(A,B)
                <=> ! [D] :
                      ( ( v1_funct_1(D)
                        & v1_funct_2(D,k5_numbers,k1_numbers)
                        & v1_fdiff_1(D)
                        & m2_relset_1(D,k5_numbers,k1_numbers) )
                     => ! [E] :
                          ( ( v1_funct_1(E)
                            & v1_funct_2(E,k5_numbers,k1_numbers)
                            & v5_seqm_3(E)
                            & m2_relset_1(E,k5_numbers,k1_numbers) )
                         => ( ( k1_rfunct_2(E) = k1_tarski(A)
                              & r1_tarski(k1_rfunct_2(k9_seq_1(D,E)),k4_relset_1(k1_numbers,k1_numbers,B))
                              & ! [F] :
                                  ( m2_subset_1(F,k1_numbers,k5_numbers)
                                 => ~ r1_xreal_0(np__0,k2_seq_1(k5_numbers,k1_numbers,D,F)) ) )
                           => C = k2_seq_2(k11_seq_1(k18_seq_1(D),k10_seq_1(k2_rfunct_2(B,k9_seq_1(D,E)),k2_rfunct_2(B,E)))) ) ) ) ) ) ) ) ) )).

fof(d6_fdiff_3,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( ( v1_funct_1(B)
            & m2_relset_1(B,k1_numbers,k1_numbers) )
         => ( r3_fdiff_3(B,A)
           => ! [C] :
                ( m1_subset_1(C,k1_numbers)
               => ( C = k2_fdiff_3(A,B)
                <=> ! [D] :
                      ( ( v1_funct_1(D)
                        & v1_funct_2(D,k5_numbers,k1_numbers)
                        & v1_fdiff_1(D)
                        & m2_relset_1(D,k5_numbers,k1_numbers) )
                     => ! [E] :
                          ( ( v1_funct_1(E)
                            & v1_funct_2(E,k5_numbers,k1_numbers)
                            & v5_seqm_3(E)
                            & m2_relset_1(E,k5_numbers,k1_numbers) )
                         => ( ( k1_rfunct_2(E) = k1_tarski(A)
                              & r1_tarski(k1_rfunct_2(k9_seq_1(D,E)),k4_relset_1(k1_numbers,k1_numbers,B))
                              & ! [F] :
                                  ( m2_subset_1(F,k1_numbers,k5_numbers)
                                 => ~ r1_xreal_0(k2_seq_1(k5_numbers,k1_numbers,D,F),np__0) ) )
                           => C = k2_seq_2(k11_seq_1(k18_seq_1(D),k10_seq_1(k2_rfunct_2(B,k9_seq_1(D,E)),k2_rfunct_2(B,E)))) ) ) ) ) ) ) ) ) )).

fof(t9_fdiff_3,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & m2_relset_1(A,k1_numbers,k1_numbers) )
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ! [C] :
              ( m1_subset_1(C,k1_numbers)
             => ( ( ( r4_fdiff_3(A,B)
                    & k1_fdiff_3(B,A) = C )
                 => ( ? [D] :
                        ( m1_subset_1(D,k1_numbers)
                        & ~ r1_xreal_0(D,np__0)
                        & r1_tarski(k1_rcomp_1(k5_real_1(B,D),B),k4_relset_1(k1_numbers,k1_numbers,A)) )
                    & ! [D] :
                        ( ( v1_funct_1(D)
                          & v1_funct_2(D,k5_numbers,k1_numbers)
                          & v1_fdiff_1(D)
                          & m2_relset_1(D,k5_numbers,k1_numbers) )
                       => ! [E] :
                            ( ( v1_funct_1(E)
                              & v1_funct_2(E,k5_numbers,k1_numbers)
                              & v5_seqm_3(E)
                              & m2_relset_1(E,k5_numbers,k1_numbers) )
                           => ( ( k1_rfunct_2(E) = k1_tarski(B)
                                & r1_tarski(k1_rfunct_2(k9_seq_1(D,E)),k4_relset_1(k1_numbers,k1_numbers,A))
                                & ! [F] :
                                    ( m2_subset_1(F,k1_numbers,k5_numbers)
                                   => ~ r1_xreal_0(np__0,k2_seq_1(k5_numbers,k1_numbers,D,F)) ) )
                             => ( v4_seq_2(k11_seq_1(k18_seq_1(D),k10_seq_1(k2_rfunct_2(A,k9_seq_1(D,E)),k2_rfunct_2(A,E))))
                                & k2_seq_2(k11_seq_1(k18_seq_1(D),k10_seq_1(k2_rfunct_2(A,k9_seq_1(D,E)),k2_rfunct_2(A,E)))) = C ) ) ) ) ) )
                & ( ! [D] :
                      ( ( v1_funct_1(D)
                        & v1_funct_2(D,k5_numbers,k1_numbers)
                        & v1_fdiff_1(D)
                        & m2_relset_1(D,k5_numbers,k1_numbers) )
                     => ! [E] :
                          ( ( v1_funct_1(E)
                            & v1_funct_2(E,k5_numbers,k1_numbers)
                            & v5_seqm_3(E)
                            & m2_relset_1(E,k5_numbers,k1_numbers) )
                         => ( ( k1_rfunct_2(E) = k1_tarski(B)
                              & r1_tarski(k1_rfunct_2(k9_seq_1(D,E)),k4_relset_1(k1_numbers,k1_numbers,A))
                              & ! [F] :
                                  ( m2_subset_1(F,k1_numbers,k5_numbers)
                                 => ~ r1_xreal_0(np__0,k2_seq_1(k5_numbers,k1_numbers,D,F)) ) )
                           => ( v4_seq_2(k11_seq_1(k18_seq_1(D),k10_seq_1(k2_rfunct_2(A,k9_seq_1(D,E)),k2_rfunct_2(A,E))))
                              & k2_seq_2(k11_seq_1(k18_seq_1(D),k10_seq_1(k2_rfunct_2(A,k9_seq_1(D,E)),k2_rfunct_2(A,E)))) = C ) ) ) )
                 => ( ! [D] :
                        ( m1_subset_1(D,k1_numbers)
                       => ~ ( ~ r1_xreal_0(D,np__0)
                            & r1_tarski(k1_rcomp_1(k5_real_1(B,D),B),k4_relset_1(k1_numbers,k1_numbers,A)) ) )
                    | ( r4_fdiff_3(A,B)
                      & k1_fdiff_3(B,A) = C ) ) ) ) ) ) ) )).

fof(t10_fdiff_3,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & m2_relset_1(A,k1_numbers,k1_numbers) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & m2_relset_1(B,k1_numbers,k1_numbers) )
         => ! [C] :
              ( m1_subset_1(C,k1_numbers)
             => ( ( r4_fdiff_3(A,C)
                  & r4_fdiff_3(B,C) )
               => ( r4_fdiff_3(k6_seq_1(k1_numbers,k1_numbers,A,B),C)
                  & k1_fdiff_3(C,k6_seq_1(k1_numbers,k1_numbers,A,B)) = k3_real_1(k1_fdiff_3(C,A),k1_fdiff_3(C,B)) ) ) ) ) ) )).

fof(t11_fdiff_3,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & m2_relset_1(A,k1_numbers,k1_numbers) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & m2_relset_1(B,k1_numbers,k1_numbers) )
         => ! [C] :
              ( m1_subset_1(C,k1_numbers)
             => ( ( r4_fdiff_3(A,C)
                  & r4_fdiff_3(B,C) )
               => ( r4_fdiff_3(k7_seq_1(k1_numbers,k1_numbers,A,B),C)
                  & k1_fdiff_3(C,k7_seq_1(k1_numbers,k1_numbers,A,B)) = k5_real_1(k1_fdiff_3(C,A),k1_fdiff_3(C,B)) ) ) ) ) ) )).

fof(t12_fdiff_3,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & m2_relset_1(A,k1_numbers,k1_numbers) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & m2_relset_1(B,k1_numbers,k1_numbers) )
         => ! [C] :
              ( m1_subset_1(C,k1_numbers)
             => ( ( r4_fdiff_3(A,C)
                  & r4_fdiff_3(B,C) )
               => ( r4_fdiff_3(k8_seq_1(k1_numbers,k1_numbers,A,B),C)
                  & k1_fdiff_3(C,k8_seq_1(k1_numbers,k1_numbers,A,B)) = k3_real_1(k4_real_1(k1_fdiff_3(C,A),k2_seq_1(k1_numbers,k1_numbers,B,C)),k4_real_1(k1_fdiff_3(C,B),k2_seq_1(k1_numbers,k1_numbers,A,C))) ) ) ) ) ) )).

fof(t13_fdiff_3,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & m2_relset_1(A,k1_numbers,k1_numbers) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & m2_relset_1(B,k1_numbers,k1_numbers) )
         => ! [C] :
              ( m1_subset_1(C,k1_numbers)
             => ( ( r4_fdiff_3(A,C)
                  & r4_fdiff_3(B,C) )
               => ( k2_seq_1(k1_numbers,k1_numbers,B,C) = np__0
                  | ( r4_fdiff_3(k2_rfunct_1(k1_numbers,k1_numbers,A,B),C)
                    & k1_fdiff_3(C,k2_rfunct_1(k1_numbers,k1_numbers,A,B)) = k6_real_1(k5_real_1(k4_real_1(k1_fdiff_3(C,A),k2_seq_1(k1_numbers,k1_numbers,B,C)),k4_real_1(k1_fdiff_3(C,B),k2_seq_1(k1_numbers,k1_numbers,A,C))),k7_square_1(k2_seq_1(k1_numbers,k1_numbers,B,C))) ) ) ) ) ) ) )).

fof(t14_fdiff_3,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & m2_relset_1(A,k1_numbers,k1_numbers) )
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ( r4_fdiff_3(A,B)
           => ( k2_seq_1(k1_numbers,k1_numbers,A,B) = np__0
              | ( r4_fdiff_3(k4_rfunct_1(k1_numbers,k1_numbers,A),B)
                & k1_fdiff_3(B,k4_rfunct_1(k1_numbers,k1_numbers,A)) = k1_real_1(k6_real_1(k1_fdiff_3(B,A),k7_square_1(k2_seq_1(k1_numbers,k1_numbers,A,B)))) ) ) ) ) ) )).

fof(t15_fdiff_3,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & m2_relset_1(A,k1_numbers,k1_numbers) )
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ! [C] :
              ( m1_subset_1(C,k1_numbers)
             => ( ( ( r3_fdiff_3(A,B)
                    & k2_fdiff_3(B,A) = C )
                 => ( ? [D] :
                        ( m1_subset_1(D,k1_numbers)
                        & ~ r1_xreal_0(D,np__0)
                        & r1_tarski(k1_rcomp_1(B,k3_real_1(B,D)),k4_relset_1(k1_numbers,k1_numbers,A)) )
                    & ! [D] :
                        ( ( v1_funct_1(D)
                          & v1_funct_2(D,k5_numbers,k1_numbers)
                          & v1_fdiff_1(D)
                          & m2_relset_1(D,k5_numbers,k1_numbers) )
                       => ! [E] :
                            ( ( v1_funct_1(E)
                              & v1_funct_2(E,k5_numbers,k1_numbers)
                              & v5_seqm_3(E)
                              & m2_relset_1(E,k5_numbers,k1_numbers) )
                           => ( ( k1_rfunct_2(E) = k1_tarski(B)
                                & r1_tarski(k1_rfunct_2(k9_seq_1(D,E)),k4_relset_1(k1_numbers,k1_numbers,A))
                                & ! [F] :
                                    ( m2_subset_1(F,k1_numbers,k5_numbers)
                                   => ~ r1_xreal_0(k2_seq_1(k5_numbers,k1_numbers,D,F),np__0) ) )
                             => ( v4_seq_2(k11_seq_1(k18_seq_1(D),k10_seq_1(k2_rfunct_2(A,k9_seq_1(D,E)),k2_rfunct_2(A,E))))
                                & k2_seq_2(k11_seq_1(k18_seq_1(D),k10_seq_1(k2_rfunct_2(A,k9_seq_1(D,E)),k2_rfunct_2(A,E)))) = C ) ) ) ) ) )
                & ( ! [D] :
                      ( ( v1_funct_1(D)
                        & v1_funct_2(D,k5_numbers,k1_numbers)
                        & v1_fdiff_1(D)
                        & m2_relset_1(D,k5_numbers,k1_numbers) )
                     => ! [E] :
                          ( ( v1_funct_1(E)
                            & v1_funct_2(E,k5_numbers,k1_numbers)
                            & v5_seqm_3(E)
                            & m2_relset_1(E,k5_numbers,k1_numbers) )
                         => ( ( k1_rfunct_2(E) = k1_tarski(B)
                              & r1_tarski(k1_rfunct_2(k9_seq_1(D,E)),k4_relset_1(k1_numbers,k1_numbers,A))
                              & ! [F] :
                                  ( m2_subset_1(F,k1_numbers,k5_numbers)
                                 => ~ r1_xreal_0(k2_seq_1(k5_numbers,k1_numbers,D,F),np__0) ) )
                           => ( v4_seq_2(k11_seq_1(k18_seq_1(D),k10_seq_1(k2_rfunct_2(A,k9_seq_1(D,E)),k2_rfunct_2(A,E))))
                              & k2_seq_2(k11_seq_1(k18_seq_1(D),k10_seq_1(k2_rfunct_2(A,k9_seq_1(D,E)),k2_rfunct_2(A,E)))) = C ) ) ) )
                 => ( ! [D] :
                        ( m1_subset_1(D,k1_numbers)
                       => ~ ( ~ r1_xreal_0(D,np__0)
                            & r1_tarski(k1_rcomp_1(B,k3_real_1(B,D)),k4_relset_1(k1_numbers,k1_numbers,A)) ) )
                    | ( r3_fdiff_3(A,B)
                      & k2_fdiff_3(B,A) = C ) ) ) ) ) ) ) )).

fof(t16_fdiff_3,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & m2_relset_1(A,k1_numbers,k1_numbers) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & m2_relset_1(B,k1_numbers,k1_numbers) )
         => ! [C] :
              ( m1_subset_1(C,k1_numbers)
             => ( ( r3_fdiff_3(A,C)
                  & r3_fdiff_3(B,C) )
               => ( r3_fdiff_3(k6_seq_1(k1_numbers,k1_numbers,A,B),C)
                  & k2_fdiff_3(C,k6_seq_1(k1_numbers,k1_numbers,A,B)) = k3_real_1(k2_fdiff_3(C,A),k2_fdiff_3(C,B)) ) ) ) ) ) )).

fof(t17_fdiff_3,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & m2_relset_1(A,k1_numbers,k1_numbers) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & m2_relset_1(B,k1_numbers,k1_numbers) )
         => ! [C] :
              ( m1_subset_1(C,k1_numbers)
             => ( ( r3_fdiff_3(A,C)
                  & r3_fdiff_3(B,C) )
               => ( r3_fdiff_3(k7_seq_1(k1_numbers,k1_numbers,A,B),C)
                  & k2_fdiff_3(C,k7_seq_1(k1_numbers,k1_numbers,A,B)) = k5_real_1(k2_fdiff_3(C,A),k2_fdiff_3(C,B)) ) ) ) ) ) )).

fof(t18_fdiff_3,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & m2_relset_1(A,k1_numbers,k1_numbers) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & m2_relset_1(B,k1_numbers,k1_numbers) )
         => ! [C] :
              ( m1_subset_1(C,k1_numbers)
             => ( ( r3_fdiff_3(A,C)
                  & r3_fdiff_3(B,C) )
               => ( r3_fdiff_3(k8_seq_1(k1_numbers,k1_numbers,A,B),C)
                  & k2_fdiff_3(C,k8_seq_1(k1_numbers,k1_numbers,A,B)) = k3_real_1(k4_real_1(k2_fdiff_3(C,A),k2_seq_1(k1_numbers,k1_numbers,B,C)),k4_real_1(k2_fdiff_3(C,B),k2_seq_1(k1_numbers,k1_numbers,A,C))) ) ) ) ) ) )).

fof(t19_fdiff_3,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & m2_relset_1(A,k1_numbers,k1_numbers) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & m2_relset_1(B,k1_numbers,k1_numbers) )
         => ! [C] :
              ( m1_subset_1(C,k1_numbers)
             => ( ( r3_fdiff_3(A,C)
                  & r3_fdiff_3(B,C) )
               => ( k2_seq_1(k1_numbers,k1_numbers,B,C) = np__0
                  | ( r3_fdiff_3(k2_rfunct_1(k1_numbers,k1_numbers,A,B),C)
                    & k2_fdiff_3(C,k2_rfunct_1(k1_numbers,k1_numbers,A,B)) = k6_real_1(k5_real_1(k4_real_1(k2_fdiff_3(C,A),k2_seq_1(k1_numbers,k1_numbers,B,C)),k4_real_1(k2_fdiff_3(C,B),k2_seq_1(k1_numbers,k1_numbers,A,C))),k7_square_1(k2_seq_1(k1_numbers,k1_numbers,B,C))) ) ) ) ) ) ) )).

fof(t20_fdiff_3,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & m2_relset_1(A,k1_numbers,k1_numbers) )
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ( r3_fdiff_3(A,B)
           => ( k2_seq_1(k1_numbers,k1_numbers,A,B) = np__0
              | ( r3_fdiff_3(k4_rfunct_1(k1_numbers,k1_numbers,A),B)
                & k2_fdiff_3(B,k4_rfunct_1(k1_numbers,k1_numbers,A)) = k1_real_1(k6_real_1(k2_fdiff_3(B,A),k7_square_1(k2_seq_1(k1_numbers,k1_numbers,A,B)))) ) ) ) ) ) )).

fof(t21_fdiff_3,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & m2_relset_1(A,k1_numbers,k1_numbers) )
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ( ( r3_fdiff_3(A,B)
              & r4_fdiff_3(A,B)
              & k2_fdiff_3(B,A) = k1_fdiff_3(B,A) )
           => ( r1_fdiff_1(A,B)
              & k1_fdiff_1(A,B) = k2_fdiff_3(B,A)
              & k1_fdiff_1(A,B) = k1_fdiff_3(B,A) ) ) ) ) )).

fof(t22_fdiff_3,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & m2_relset_1(A,k1_numbers,k1_numbers) )
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ( r1_fdiff_1(A,B)
           => ( r3_fdiff_3(A,B)
              & r4_fdiff_3(A,B)
              & k1_fdiff_1(A,B) = k2_fdiff_3(B,A)
              & k1_fdiff_1(A,B) = k1_fdiff_3(B,A) ) ) ) ) )).

fof(dt_k1_fdiff_3,axiom,(
    ! [A,B] :
      ( ( m1_subset_1(A,k1_numbers)
        & v1_funct_1(B)
        & m1_relset_1(B,k1_numbers,k1_numbers) )
     => m1_subset_1(k1_fdiff_3(A,B),k1_numbers) ) )).

fof(dt_k2_fdiff_3,axiom,(
    ! [A,B] :
      ( ( m1_subset_1(A,k1_numbers)
        & v1_funct_1(B)
        & m1_relset_1(B,k1_numbers,k1_numbers) )
     => m1_subset_1(k2_fdiff_3(A,B),k1_numbers) ) )).
%------------------------------------------------------------------------------