SET007 Axioms: SET007+135.ax


%------------------------------------------------------------------------------
% File     : SET007+135 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Real Function Differentiability - Part II
% 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_2 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   52 (   0 unit)
%            Number of atoms       :  597 (  50 equality)
%            Maximal formula depth :   18 (  11 average)
%            Number of connectives :  562 (  17 ~  ;  12  |; 301  &)
%                                         (   1 <=>; 231 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   28 (   0 propositional; 1-4 arity)
%            Number of functors    :   48 (   6 constant; 0-6 arity)
%            Number of variables   :  184 (   0 singleton; 163 !;  21 ?)
%            Maximal term depth    :    7 (   1 average)
% SPC      : 

% Comments : The individual reference can be found in [Mat90] by looking for
%            the name provided by [Urb08].
%          : Translated by MPTP from the Mizar Mathematical Library 4.48.930.
%          : These set theory axioms are used in encodings of problems in
%            various domains, including ALG, CAT, GRP, LAT, SET, and TOP.
%------------------------------------------------------------------------------
fof(t1_fdiff_2,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & m2_relset_1(A,k5_numbers,k1_numbers) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,k1_numbers)
            & m2_relset_1(B,k5_numbers,k1_numbers) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,k1_numbers)
                & m2_relset_1(C,k5_numbers,k1_numbers) )
             => ( ( v4_seq_2(A)
                  & v4_seq_2(B)
                  & k2_seq_2(A) = k2_seq_2(B)
                  & ! [D] :
                      ( m2_subset_1(D,k1_numbers,k5_numbers)
                     => ( k2_seq_1(k5_numbers,k1_numbers,C,k2_nat_1(np__2,D)) = k2_seq_1(k5_numbers,k1_numbers,A,D)
                        & k2_seq_1(k5_numbers,k1_numbers,C,k1_nat_1(k2_nat_1(np__2,D),np__1)) = k2_seq_1(k5_numbers,k1_numbers,B,D) ) ) )
               => ( v4_seq_2(C)
                  & k2_seq_2(C) = k2_seq_2(A) ) ) ) ) ) )).

fof(t2_fdiff_2,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & m2_relset_1(A,k5_numbers,k1_numbers) )
     => ( ! [B] :
            ( m2_subset_1(B,k1_numbers,k5_numbers)
           => k2_seq_1(k5_numbers,k1_numbers,A,B) = k2_nat_1(np__2,B) )
       => ( v1_seqm_3(A)
          & v7_seqm_3(A) ) ) ) )).

fof(t3_fdiff_2,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & m2_relset_1(A,k5_numbers,k1_numbers) )
     => ( ! [B] :
            ( m2_subset_1(B,k1_numbers,k5_numbers)
           => k2_seq_1(k5_numbers,k1_numbers,A,B) = k1_nat_1(k2_nat_1(np__2,B),np__1) )
       => ( v1_seqm_3(A)
          & v7_seqm_3(A) ) ) ) )).

fof(t4_fdiff_2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,k1_numbers)
            & v1_fdiff_1(B)
            & m2_relset_1(B,k5_numbers,k1_numbers) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,k1_numbers)
                & v5_seqm_3(C)
                & m2_relset_1(C,k5_numbers,k1_numbers) )
             => ( k1_rfunct_2(C) = k1_tarski(A)
               => ( v4_seq_2(C)
                  & k2_seq_2(C) = A
                  & v4_seq_2(k9_seq_1(B,C))
                  & k2_seq_2(k9_seq_1(B,C)) = A ) ) ) ) ) )).

fof(t5_fdiff_2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,k1_numbers)
            & m2_relset_1(B,k5_numbers,k1_numbers) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,k1_numbers)
                & m2_relset_1(C,k5_numbers,k1_numbers) )
             => ( ( k1_rfunct_2(B) = k1_tarski(A)
                  & k1_rfunct_2(C) = k1_tarski(A) )
               => B = C ) ) ) ) )).

fof(t6_fdiff_2,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & m2_relset_1(A,k5_numbers,k1_numbers) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,k1_numbers)
            & v1_fdiff_1(B)
            & m2_relset_1(B,k5_numbers,k1_numbers) )
         => ( m1_seqm_3(A,B)
           => ( v1_funct_1(A)
              & v1_funct_2(A,k5_numbers,k1_numbers)
              & v1_fdiff_1(A)
              & m2_relset_1(A,k5_numbers,k1_numbers) ) ) ) ) )).

fof(t7_fdiff_2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( ( v1_funct_1(B)
            & m2_relset_1(B,k1_numbers,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(A)
                        & r1_tarski(k1_rfunct_2(k9_seq_1(C,D)),k4_relset_1(k1_numbers,k1_numbers,B))
                        & r1_tarski(k1_tarski(A),k4_relset_1(k1_numbers,k1_numbers,B)) )
                     => v4_seq_2(k11_seq_1(k18_seq_1(C),k10_seq_1(k2_rfunct_2(B,k9_seq_1(C,D)),k2_rfunct_2(B,D)))) ) ) )
           => ! [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(A)
                            & r1_tarski(k1_rfunct_2(k9_seq_1(C,E)),k4_relset_1(k1_numbers,k1_numbers,B))
                            & r1_tarski(k1_rfunct_2(k9_seq_1(D,E)),k4_relset_1(k1_numbers,k1_numbers,B))
                            & r1_tarski(k1_tarski(A),k4_relset_1(k1_numbers,k1_numbers,B)) )
                         => k2_seq_2(k11_seq_1(k18_seq_1(C),k10_seq_1(k2_rfunct_2(B,k9_seq_1(C,E)),k2_rfunct_2(B,E)))) = 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(t8_fdiff_2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( ( v1_funct_1(B)
            & m2_relset_1(B,k1_numbers,k1_numbers) )
         => ~ ( ? [C] :
                  ( m1_rcomp_1(C,A)
                  & r1_tarski(C,k4_relset_1(k1_numbers,k1_numbers,B)) )
              & ! [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(A)
                          & r1_tarski(k1_rfunct_2(k9_seq_1(C,D)),k4_relset_1(k1_numbers,k1_numbers,B))
                          & r1_tarski(k1_tarski(A),k4_relset_1(k1_numbers,k1_numbers,B)) ) ) ) ) ) ) )).

fof(t9_fdiff_2,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & m2_relset_1(A,k5_numbers,k1_numbers) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & m2_relset_1(B,k1_numbers,k1_numbers) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & m2_relset_1(C,k1_numbers,k1_numbers) )
             => ( r1_tarski(k1_rfunct_2(A),k4_relset_1(k1_numbers,k1_numbers,k1_partfun1(k1_numbers,k1_numbers,k1_numbers,k1_numbers,C,B)))
               => ( r1_tarski(k1_rfunct_2(A),k4_relset_1(k1_numbers,k1_numbers,C))
                  & r1_tarski(k1_rfunct_2(k2_rfunct_2(C,A)),k4_relset_1(k1_numbers,k1_numbers,B)) ) ) ) ) ) )).

fof(t10_fdiff_2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ! [C] :
              ( ( v1_funct_1(C)
                & m2_relset_1(C,k1_numbers,k1_numbers) )
             => ~ ( k2_seq_1(k1_numbers,k1_numbers,C,A) != B
                  & r1_fdiff_1(C,A)
                  & ! [D] :
                      ( m1_rcomp_1(D,A)
                     => ~ ( r1_tarski(D,k4_relset_1(k1_numbers,k1_numbers,C))
                          & ! [E] :
                              ( m1_subset_1(E,k1_numbers)
                             => ~ ( r2_hidden(E,D)
                                  & k2_seq_1(k1_numbers,k1_numbers,C,E) = B ) ) ) ) ) ) ) ) )).

fof(t11_fdiff_2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( ( v1_funct_1(B)
            & m2_relset_1(B,k1_numbers,k1_numbers) )
         => ( r1_fdiff_1(B,A)
          <=> ( ? [C] :
                  ( m1_rcomp_1(C,A)
                  & r1_tarski(C,k4_relset_1(k1_numbers,k1_numbers,B)) )
              & ! [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(A)
                          & r1_tarski(k1_rfunct_2(k9_seq_1(C,D)),k4_relset_1(k1_numbers,k1_numbers,B)) )
                       => v4_seq_2(k11_seq_1(k18_seq_1(C),k10_seq_1(k2_rfunct_2(B,k9_seq_1(C,D)),k2_rfunct_2(B,D)))) ) ) ) ) ) ) ) )).

fof(t12_fdiff_2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ! [C] :
              ( ( v1_funct_1(C)
                & m2_relset_1(C,k1_numbers,k1_numbers) )
             => ( ( ( r1_fdiff_1(C,A)
                    & k1_fdiff_1(C,A) = B )
                 => ( ? [D] :
                        ( m1_rcomp_1(D,A)
                        & r1_tarski(D,k4_relset_1(k1_numbers,k1_numbers,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(A)
                                & r1_tarski(k1_rfunct_2(k9_seq_1(D,E)),k4_relset_1(k1_numbers,k1_numbers,C)) )
                             => ( v4_seq_2(k11_seq_1(k18_seq_1(D),k10_seq_1(k2_rfunct_2(C,k9_seq_1(D,E)),k2_rfunct_2(C,E))))
                                & k2_seq_2(k11_seq_1(k18_seq_1(D),k10_seq_1(k2_rfunct_2(C,k9_seq_1(D,E)),k2_rfunct_2(C,E)))) = 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,C)) )
                           => ( v4_seq_2(k11_seq_1(k18_seq_1(D),k10_seq_1(k2_rfunct_2(C,k9_seq_1(D,E)),k2_rfunct_2(C,E))))
                              & k2_seq_2(k11_seq_1(k18_seq_1(D),k10_seq_1(k2_rfunct_2(C,k9_seq_1(D,E)),k2_rfunct_2(C,E)))) = B ) ) ) )
                 => ( ! [D] :
                        ( m1_rcomp_1(D,A)
                       => ~ r1_tarski(D,k4_relset_1(k1_numbers,k1_numbers,C)) )
                    | ( r1_fdiff_1(C,A)
                      & k1_fdiff_1(C,A) = B ) ) ) ) ) ) ) )).

fof(t13_fdiff_2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( ( v1_funct_1(B)
            & m2_relset_1(B,k1_numbers,k1_numbers) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & m2_relset_1(C,k1_numbers,k1_numbers) )
             => ( ( r1_fdiff_1(B,A)
                  & r1_fdiff_1(C,k2_seq_1(k1_numbers,k1_numbers,B,A)) )
               => ( r1_fdiff_1(k1_partfun1(k1_numbers,k1_numbers,k1_numbers,k1_numbers,B,C),A)
                  & k1_fdiff_1(k1_partfun1(k1_numbers,k1_numbers,k1_numbers,k1_numbers,B,C),A) = k4_real_1(k1_fdiff_1(C,k2_seq_1(k1_numbers,k1_numbers,B,A)),k1_fdiff_1(B,A)) ) ) ) ) ) )).

fof(t14_fdiff_2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( ( v1_funct_1(B)
            & m2_relset_1(B,k1_numbers,k1_numbers) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & m2_relset_1(C,k1_numbers,k1_numbers) )
             => ( ( r1_fdiff_1(C,A)
                  & r1_fdiff_1(B,A) )
               => ( k2_seq_1(k1_numbers,k1_numbers,B,A) = np__0
                  | ( r1_fdiff_1(k2_rfunct_1(k1_numbers,k1_numbers,C,B),A)
                    & k1_fdiff_1(k2_rfunct_1(k1_numbers,k1_numbers,C,B),A) = k6_real_1(k5_real_1(k4_real_1(k1_fdiff_1(C,A),k2_seq_1(k1_numbers,k1_numbers,B,A)),k4_real_1(k1_fdiff_1(B,A),k2_seq_1(k1_numbers,k1_numbers,C,A))),k7_square_1(k2_seq_1(k1_numbers,k1_numbers,B,A))) ) ) ) ) ) ) )).

fof(t15_fdiff_2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( ( v1_funct_1(B)
            & m2_relset_1(B,k1_numbers,k1_numbers) )
         => ( r1_fdiff_1(B,A)
           => ( k2_seq_1(k1_numbers,k1_numbers,B,A) = np__0
              | ( r1_fdiff_1(k4_rfunct_1(k1_numbers,k1_numbers,B),A)
                & k1_fdiff_1(k4_rfunct_1(k1_numbers,k1_numbers,B),A) = k1_real_1(k6_real_1(k1_fdiff_1(B,A),k7_square_1(k2_seq_1(k1_numbers,k1_numbers,B,A)))) ) ) ) ) ) )).

fof(t16_fdiff_2,axiom,(
    ! [A] :
      ( ( v3_rcomp_1(A)
        & m1_subset_1(A,k1_zfmisc_1(k1_numbers)) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & m2_relset_1(B,k1_numbers,k1_numbers) )
         => ( r2_fdiff_1(B,A)
           => ( r2_fdiff_1(k2_partfun1(k1_numbers,k1_numbers,B,A),A)
              & k2_fdiff_1(B,A) = k2_fdiff_1(k2_partfun1(k1_numbers,k1_numbers,B,A),A) ) ) ) ) )).

fof(t17_fdiff_2,axiom,(
    ! [A] :
      ( ( v3_rcomp_1(A)
        & m1_subset_1(A,k1_zfmisc_1(k1_numbers)) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & m2_relset_1(B,k1_numbers,k1_numbers) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & m2_relset_1(C,k1_numbers,k1_numbers) )
             => ( ( r2_fdiff_1(B,A)
                  & r2_fdiff_1(C,A) )
               => ( r2_fdiff_1(k6_seq_1(k1_numbers,k1_numbers,B,C),A)
                  & k2_fdiff_1(k6_seq_1(k1_numbers,k1_numbers,B,C),A) = k6_seq_1(k1_numbers,k1_numbers,k2_fdiff_1(B,A),k2_fdiff_1(C,A)) ) ) ) ) ) )).

fof(t18_fdiff_2,axiom,(
    ! [A] :
      ( ( v3_rcomp_1(A)
        & m1_subset_1(A,k1_zfmisc_1(k1_numbers)) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & m2_relset_1(B,k1_numbers,k1_numbers) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & m2_relset_1(C,k1_numbers,k1_numbers) )
             => ( ( r2_fdiff_1(B,A)
                  & r2_fdiff_1(C,A) )
               => ( r2_fdiff_1(k7_seq_1(k1_numbers,k1_numbers,B,C),A)
                  & k2_fdiff_1(k7_seq_1(k1_numbers,k1_numbers,B,C),A) = k7_seq_1(k1_numbers,k1_numbers,k2_fdiff_1(B,A),k2_fdiff_1(C,A)) ) ) ) ) ) )).

fof(t19_fdiff_2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( ( v3_rcomp_1(B)
            & m1_subset_1(B,k1_zfmisc_1(k1_numbers)) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & m2_relset_1(C,k1_numbers,k1_numbers) )
             => ( r2_fdiff_1(C,B)
               => ( r2_fdiff_1(k13_seq_1(k1_numbers,k1_numbers,C,A),B)
                  & k2_fdiff_1(k13_seq_1(k1_numbers,k1_numbers,C,A),B) = k13_seq_1(k1_numbers,k1_numbers,k2_fdiff_1(C,B),A) ) ) ) ) ) )).

fof(t20_fdiff_2,axiom,(
    ! [A] :
      ( ( v3_rcomp_1(A)
        & m1_subset_1(A,k1_zfmisc_1(k1_numbers)) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & m2_relset_1(B,k1_numbers,k1_numbers) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & m2_relset_1(C,k1_numbers,k1_numbers) )
             => ( ( r2_fdiff_1(B,A)
                  & r2_fdiff_1(C,A) )
               => ( r2_fdiff_1(k8_seq_1(k1_numbers,k1_numbers,B,C),A)
                  & k2_fdiff_1(k8_seq_1(k1_numbers,k1_numbers,B,C),A) = k6_seq_1(k1_numbers,k1_numbers,k8_seq_1(k1_numbers,k1_numbers,k2_fdiff_1(B,A),C),k8_seq_1(k1_numbers,k1_numbers,B,k2_fdiff_1(C,A))) ) ) ) ) ) )).

fof(t21_fdiff_2,axiom,(
    ! [A] :
      ( ( v3_rcomp_1(A)
        & m1_subset_1(A,k1_zfmisc_1(k1_numbers)) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & m2_relset_1(B,k1_numbers,k1_numbers) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & m2_relset_1(C,k1_numbers,k1_numbers) )
             => ( ( r2_fdiff_1(B,A)
                  & r2_fdiff_1(C,A)
                  & ! [D] :
                      ( m1_subset_1(D,k1_numbers)
                     => ~ ( r2_hidden(D,A)
                          & k2_seq_1(k1_numbers,k1_numbers,C,D) = np__0 ) ) )
               => ( r2_fdiff_1(k2_rfunct_1(k1_numbers,k1_numbers,B,C),A)
                  & k2_fdiff_1(k2_rfunct_1(k1_numbers,k1_numbers,B,C),A) = k2_rfunct_1(k1_numbers,k1_numbers,k7_seq_1(k1_numbers,k1_numbers,k8_seq_1(k1_numbers,k1_numbers,k2_fdiff_1(B,A),C),k8_seq_1(k1_numbers,k1_numbers,k2_fdiff_1(C,A),B)),k8_seq_1(k1_numbers,k1_numbers,C,C)) ) ) ) ) ) )).

fof(t22_fdiff_2,axiom,(
    ! [A] :
      ( ( v3_rcomp_1(A)
        & m1_subset_1(A,k1_zfmisc_1(k1_numbers)) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & m2_relset_1(B,k1_numbers,k1_numbers) )
         => ( ( r2_fdiff_1(B,A)
              & ! [C] :
                  ( m1_subset_1(C,k1_numbers)
                 => ~ ( r2_hidden(C,A)
                      & k2_seq_1(k1_numbers,k1_numbers,B,C) = np__0 ) ) )
           => ( r2_fdiff_1(k4_rfunct_1(k1_numbers,k1_numbers,B),A)
              & k2_fdiff_1(k4_rfunct_1(k1_numbers,k1_numbers,B),A) = k16_seq_1(k1_numbers,k1_numbers,k2_rfunct_1(k1_numbers,k1_numbers,k2_fdiff_1(B,A),k8_seq_1(k1_numbers,k1_numbers,B,B))) ) ) ) ) )).

fof(t23_fdiff_2,axiom,(
    ! [A] :
      ( ( v3_rcomp_1(A)
        & m1_subset_1(A,k1_zfmisc_1(k1_numbers)) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & m2_relset_1(B,k1_numbers,k1_numbers) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & m2_relset_1(C,k1_numbers,k1_numbers) )
             => ( ( r2_fdiff_1(B,A)
                  & v3_rcomp_1(k10_relset_1(k1_numbers,k1_numbers,B,A))
                  & m1_subset_1(k10_relset_1(k1_numbers,k1_numbers,B,A),k1_zfmisc_1(k1_numbers))
                  & r2_fdiff_1(C,k10_relset_1(k1_numbers,k1_numbers,B,A)) )
               => ( r2_fdiff_1(k1_partfun1(k1_numbers,k1_numbers,k1_numbers,k1_numbers,B,C),A)
                  & k2_fdiff_1(k1_partfun1(k1_numbers,k1_numbers,k1_numbers,k1_numbers,B,C),A) = k8_seq_1(k1_numbers,k1_numbers,k1_partfun1(k1_numbers,k1_numbers,k1_numbers,k1_numbers,B,k2_fdiff_1(C,k10_relset_1(k1_numbers,k1_numbers,B,A))),k2_fdiff_1(B,A)) ) ) ) ) ) )).

fof(t24_fdiff_2,axiom,(
    ! [A] :
      ( ( v3_rcomp_1(A)
        & m1_subset_1(A,k1_zfmisc_1(k1_numbers)) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & m2_relset_1(B,k1_numbers,k1_numbers) )
         => ( ( r1_tarski(A,k4_relset_1(k1_numbers,k1_numbers,B))
              & ! [C] :
                  ( m1_subset_1(C,k1_numbers)
                 => ! [D] :
                      ( m1_subset_1(D,k1_numbers)
                     => ( ( r2_hidden(C,A)
                          & r2_hidden(D,A) )
                       => r1_xreal_0(k18_complex1(k5_real_1(k2_seq_1(k1_numbers,k1_numbers,B,C),k2_seq_1(k1_numbers,k1_numbers,B,D))),k7_square_1(k5_real_1(C,D))) ) ) ) )
           => ( r2_fdiff_1(B,A)
              & ! [C] :
                  ( m1_subset_1(C,k1_numbers)
                 => ( r2_hidden(C,A)
                   => k1_fdiff_1(B,C) = np__0 ) ) ) ) ) ) )).

fof(t25_fdiff_2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ! [C] :
              ( ( v1_funct_1(C)
                & m2_relset_1(C,k1_numbers,k1_numbers) )
             => ( ( ! [D] :
                      ( m1_subset_1(D,k1_numbers)
                     => ! [E] :
                          ( m1_subset_1(E,k1_numbers)
                         => ( ( r2_hidden(D,k2_rcomp_1(A,B))
                              & r2_hidden(E,k2_rcomp_1(A,B)) )
                           => r1_xreal_0(k18_complex1(k5_real_1(k2_seq_1(k1_numbers,k1_numbers,C,D),k2_seq_1(k1_numbers,k1_numbers,C,E))),k7_square_1(k5_real_1(D,E))) ) ) )
                  & r1_tarski(k2_rcomp_1(A,B),k4_relset_1(k1_numbers,k1_numbers,C)) )
               => ( r1_xreal_0(B,A)
                  | ( r2_fdiff_1(C,k2_rcomp_1(A,B))
                    & r1_partfun2(k1_numbers,k1_numbers,C,k2_rcomp_1(A,B)) ) ) ) ) ) ) )).

fof(t26_fdiff_2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( ( v1_funct_1(B)
            & m2_relset_1(B,k1_numbers,k1_numbers) )
         => ( ( r1_tarski(k12_prob_1(A),k4_relset_1(k1_numbers,k1_numbers,B))
              & ! [C] :
                  ( m1_subset_1(C,k1_numbers)
                 => ! [D] :
                      ( m1_subset_1(D,k1_numbers)
                     => ( ( r2_hidden(C,k12_prob_1(A))
                          & r2_hidden(D,k12_prob_1(A)) )
                       => r1_xreal_0(k18_complex1(k5_real_1(k2_seq_1(k1_numbers,k1_numbers,B,C),k2_seq_1(k1_numbers,k1_numbers,B,D))),k7_square_1(k5_real_1(C,D))) ) ) ) )
           => ( r2_fdiff_1(B,k12_prob_1(A))
              & r1_partfun2(k1_numbers,k1_numbers,B,k12_prob_1(A)) ) ) ) ) )).

fof(t27_fdiff_2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( ( v1_funct_1(B)
            & m2_relset_1(B,k1_numbers,k1_numbers) )
         => ( ( r1_tarski(k4_limfunc1(A),k4_relset_1(k1_numbers,k1_numbers,B))
              & ! [C] :
                  ( m1_subset_1(C,k1_numbers)
                 => ! [D] :
                      ( m1_subset_1(D,k1_numbers)
                     => ( ( r2_hidden(C,k4_limfunc1(A))
                          & r2_hidden(D,k4_limfunc1(A)) )
                       => r1_xreal_0(k18_complex1(k5_real_1(k2_seq_1(k1_numbers,k1_numbers,B,C),k2_seq_1(k1_numbers,k1_numbers,B,D))),k7_square_1(k5_real_1(C,D))) ) ) ) )
           => ( r2_fdiff_1(B,k4_limfunc1(A))
              & r1_partfun2(k1_numbers,k1_numbers,B,k4_limfunc1(A)) ) ) ) ) )).

fof(t28_fdiff_2,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & m2_relset_1(A,k1_numbers,k1_numbers) )
     => ( ( v1_partfun1(A,k1_numbers,k1_numbers)
          & ! [B] :
              ( m1_subset_1(B,k1_numbers)
             => ! [C] :
                  ( m1_subset_1(C,k1_numbers)
                 => r1_xreal_0(k18_complex1(k5_real_1(k2_seq_1(k1_numbers,k1_numbers,A,B),k2_seq_1(k1_numbers,k1_numbers,A,C))),k7_square_1(k5_real_1(B,C))) ) ) )
       => ( r2_fdiff_1(A,k2_subset_1(k1_numbers))
          & r1_partfun2(k1_numbers,k1_numbers,A,k2_subset_1(k1_numbers)) ) ) ) )).

fof(t29_fdiff_2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( ( v1_funct_1(B)
            & m2_relset_1(B,k1_numbers,k1_numbers) )
         => ( ( r2_fdiff_1(B,k12_prob_1(A))
              & ! [C] :
                  ( m1_subset_1(C,k1_numbers)
                 => ~ ( r2_hidden(C,k12_prob_1(A))
                      & r1_xreal_0(k1_fdiff_1(B,C),np__0) ) ) )
           => ( r1_rfunct_2(B,k12_prob_1(A))
              & v2_funct_1(k2_partfun1(k1_numbers,k1_numbers,B,k12_prob_1(A))) ) ) ) ) )).

fof(t30_fdiff_2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( ( v1_funct_1(B)
            & m2_relset_1(B,k1_numbers,k1_numbers) )
         => ( ( r2_fdiff_1(B,k12_prob_1(A))
              & ! [C] :
                  ( m1_subset_1(C,k1_numbers)
                 => ~ ( r2_hidden(C,k12_prob_1(A))
                      & r1_xreal_0(np__0,k1_fdiff_1(B,C)) ) ) )
           => ( r2_rfunct_2(B,k12_prob_1(A))
              & v2_funct_1(k2_partfun1(k1_numbers,k1_numbers,B,k12_prob_1(A))) ) ) ) ) )).

fof(t31_fdiff_2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( ( v1_funct_1(B)
            & m2_relset_1(B,k1_numbers,k1_numbers) )
         => ( ( r2_fdiff_1(B,k12_prob_1(A))
              & ! [C] :
                  ( m1_subset_1(C,k1_numbers)
                 => ( r2_hidden(C,k12_prob_1(A))
                   => r1_xreal_0(np__0,k1_fdiff_1(B,C)) ) ) )
           => r3_rfunct_2(B,k12_prob_1(A)) ) ) ) )).

fof(t32_fdiff_2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( ( v1_funct_1(B)
            & m2_relset_1(B,k1_numbers,k1_numbers) )
         => ( ( r2_fdiff_1(B,k12_prob_1(A))
              & ! [C] :
                  ( m1_subset_1(C,k1_numbers)
                 => ( r2_hidden(C,k12_prob_1(A))
                   => r1_xreal_0(k1_fdiff_1(B,C),np__0) ) ) )
           => r4_rfunct_2(B,k12_prob_1(A)) ) ) ) )).

fof(t33_fdiff_2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( ( v1_funct_1(B)
            & m2_relset_1(B,k1_numbers,k1_numbers) )
         => ( ( r2_fdiff_1(B,k4_limfunc1(A))
              & ! [C] :
                  ( m1_subset_1(C,k1_numbers)
                 => ~ ( r2_hidden(C,k4_limfunc1(A))
                      & r1_xreal_0(k1_fdiff_1(B,C),np__0) ) ) )
           => ( r1_rfunct_2(B,k4_limfunc1(A))
              & v2_funct_1(k2_partfun1(k1_numbers,k1_numbers,B,k4_limfunc1(A))) ) ) ) ) )).

fof(t34_fdiff_2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( ( v1_funct_1(B)
            & m2_relset_1(B,k1_numbers,k1_numbers) )
         => ( ( r2_fdiff_1(B,k4_limfunc1(A))
              & ! [C] :
                  ( m1_subset_1(C,k1_numbers)
                 => ~ ( r2_hidden(C,k4_limfunc1(A))
                      & r1_xreal_0(np__0,k1_fdiff_1(B,C)) ) ) )
           => ( r2_rfunct_2(B,k4_limfunc1(A))
              & v2_funct_1(k2_partfun1(k1_numbers,k1_numbers,B,k4_limfunc1(A))) ) ) ) ) )).

fof(t35_fdiff_2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( ( v1_funct_1(B)
            & m2_relset_1(B,k1_numbers,k1_numbers) )
         => ( ( r2_fdiff_1(B,k4_limfunc1(A))
              & ! [C] :
                  ( m1_subset_1(C,k1_numbers)
                 => ( r2_hidden(C,k4_limfunc1(A))
                   => r1_xreal_0(np__0,k1_fdiff_1(B,C)) ) ) )
           => r3_rfunct_2(B,k4_limfunc1(A)) ) ) ) )).

fof(t36_fdiff_2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( ( v1_funct_1(B)
            & m2_relset_1(B,k1_numbers,k1_numbers) )
         => ( ( r2_fdiff_1(B,k4_limfunc1(A))
              & ! [C] :
                  ( m1_subset_1(C,k1_numbers)
                 => ( r2_hidden(C,k4_limfunc1(A))
                   => r1_xreal_0(k1_fdiff_1(B,C),np__0) ) ) )
           => r4_rfunct_2(B,k4_limfunc1(A)) ) ) ) )).

fof(t37_fdiff_2,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & m2_relset_1(A,k1_numbers,k1_numbers) )
     => ( ( r2_fdiff_1(A,k2_subset_1(k1_numbers))
          & ! [B] :
              ( m1_subset_1(B,k1_numbers)
             => ~ r1_xreal_0(k1_fdiff_1(A,B),np__0) ) )
       => ( r1_rfunct_2(A,k2_subset_1(k1_numbers))
          & v2_funct_1(A) ) ) ) )).

fof(t38_fdiff_2,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & m2_relset_1(A,k1_numbers,k1_numbers) )
     => ( ( r2_fdiff_1(A,k2_subset_1(k1_numbers))
          & ! [B] :
              ( m1_subset_1(B,k1_numbers)
             => ~ r1_xreal_0(np__0,k1_fdiff_1(A,B)) ) )
       => ( r2_rfunct_2(A,k2_subset_1(k1_numbers))
          & v2_funct_1(A) ) ) ) )).

fof(t39_fdiff_2,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & m2_relset_1(A,k1_numbers,k1_numbers) )
     => ( ( r2_fdiff_1(A,k2_subset_1(k1_numbers))
          & ! [B] :
              ( m1_subset_1(B,k1_numbers)
             => r1_xreal_0(np__0,k1_fdiff_1(A,B)) ) )
       => r3_rfunct_2(A,k2_subset_1(k1_numbers)) ) ) )).

fof(t40_fdiff_2,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & m2_relset_1(A,k1_numbers,k1_numbers) )
     => ( ( r2_fdiff_1(A,k2_subset_1(k1_numbers))
          & ! [B] :
              ( m1_subset_1(B,k1_numbers)
             => r1_xreal_0(k1_fdiff_1(A,B),np__0) ) )
       => r4_rfunct_2(A,k2_subset_1(k1_numbers)) ) ) )).

fof(t41_fdiff_2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ! [C] :
              ( ( v1_funct_1(C)
                & m2_relset_1(C,k1_numbers,k1_numbers) )
             => ( r2_fdiff_1(C,k2_rcomp_1(A,B))
               => ( ( ? [D] :
                        ( m1_subset_1(D,k1_numbers)
                        & r2_hidden(D,k2_rcomp_1(A,B))
                        & r1_xreal_0(k1_fdiff_1(C,D),np__0) )
                    & ? [D] :
                        ( m1_subset_1(D,k1_numbers)
                        & r2_hidden(D,k2_rcomp_1(A,B))
                        & r1_xreal_0(np__0,k1_fdiff_1(C,D)) ) )
                  | v3_rcomp_1(k5_relset_1(k1_numbers,k1_numbers,k2_partfun1(k1_numbers,k1_numbers,C,k2_rcomp_1(A,B)))) ) ) ) ) ) )).

fof(t42_fdiff_2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( ( v1_funct_1(B)
            & m2_relset_1(B,k1_numbers,k1_numbers) )
         => ( r2_fdiff_1(B,k12_prob_1(A))
           => ( ( ? [C] :
                    ( m1_subset_1(C,k1_numbers)
                    & r2_hidden(C,k12_prob_1(A))
                    & r1_xreal_0(k1_fdiff_1(B,C),np__0) )
                & ? [C] :
                    ( m1_subset_1(C,k1_numbers)
                    & r2_hidden(C,k12_prob_1(A))
                    & r1_xreal_0(np__0,k1_fdiff_1(B,C)) ) )
              | v3_rcomp_1(k5_relset_1(k1_numbers,k1_numbers,k2_partfun1(k1_numbers,k1_numbers,B,k12_prob_1(A)))) ) ) ) ) )).

fof(t43_fdiff_2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( ( v1_funct_1(B)
            & m2_relset_1(B,k1_numbers,k1_numbers) )
         => ( r2_fdiff_1(B,k4_limfunc1(A))
           => ( ( ? [C] :
                    ( m1_subset_1(C,k1_numbers)
                    & r2_hidden(C,k4_limfunc1(A))
                    & r1_xreal_0(k1_fdiff_1(B,C),np__0) )
                & ? [C] :
                    ( m1_subset_1(C,k1_numbers)
                    & r2_hidden(C,k4_limfunc1(A))
                    & r1_xreal_0(np__0,k1_fdiff_1(B,C)) ) )
              | v3_rcomp_1(k5_relset_1(k1_numbers,k1_numbers,k2_partfun1(k1_numbers,k1_numbers,B,k4_limfunc1(A)))) ) ) ) ) )).

fof(t44_fdiff_2,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & m2_relset_1(A,k1_numbers,k1_numbers) )
     => ( r2_fdiff_1(A,k2_subset_1(k1_numbers))
       => ( ( ? [B] :
                ( m1_subset_1(B,k1_numbers)
                & r1_xreal_0(k1_fdiff_1(A,B),np__0) )
            & ? [B] :
                ( m1_subset_1(B,k1_numbers)
                & r1_xreal_0(np__0,k1_fdiff_1(A,B)) ) )
          | v3_rcomp_1(k5_relset_1(k1_numbers,k1_numbers,A)) ) ) ) )).

fof(t45_fdiff_2,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v2_funct_1(A)
        & m2_relset_1(A,k1_numbers,k1_numbers) )
     => ( r2_fdiff_1(A,k2_subset_1(k1_numbers))
       => ( ( ? [B] :
                ( m1_subset_1(B,k1_numbers)
                & r1_xreal_0(k1_fdiff_1(A,B),np__0) )
            & ? [B] :
                ( m1_subset_1(B,k1_numbers)
                & r1_xreal_0(np__0,k1_fdiff_1(A,B)) ) )
          | ( v2_funct_1(A)
            & r2_fdiff_1(k2_partfun2(k1_numbers,k1_numbers,A),k4_relset_1(k1_numbers,k1_numbers,k2_partfun2(k1_numbers,k1_numbers,A)))
            & ! [B] :
                ( m1_subset_1(B,k1_numbers)
               => ( r2_hidden(B,k4_relset_1(k1_numbers,k1_numbers,k2_partfun2(k1_numbers,k1_numbers,A)))
                 => k1_fdiff_1(k2_partfun2(k1_numbers,k1_numbers,A),B) = k6_real_1(np__1,k1_fdiff_1(A,k2_seq_1(k1_numbers,k1_numbers,k2_partfun2(k1_numbers,k1_numbers,A),B))) ) ) ) ) ) ) )).

fof(t46_fdiff_2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( ( v1_funct_1(B)
            & v2_funct_1(B)
            & m2_relset_1(B,k1_numbers,k1_numbers) )
         => ( r2_fdiff_1(B,k12_prob_1(A))
           => ( ( ? [C] :
                    ( m1_subset_1(C,k1_numbers)
                    & r2_hidden(C,k12_prob_1(A))
                    & r1_xreal_0(k1_fdiff_1(B,C),np__0) )
                & ? [C] :
                    ( m1_subset_1(C,k1_numbers)
                    & r2_hidden(C,k12_prob_1(A))
                    & r1_xreal_0(np__0,k1_fdiff_1(B,C)) ) )
              | ( v2_funct_1(k2_partfun1(k1_numbers,k1_numbers,B,k12_prob_1(A)))
                & r2_fdiff_1(k2_partfun2(k1_numbers,k1_numbers,k2_partfun1(k1_numbers,k1_numbers,B,k12_prob_1(A))),k4_relset_1(k1_numbers,k1_numbers,k2_partfun2(k1_numbers,k1_numbers,k2_partfun1(k1_numbers,k1_numbers,B,k12_prob_1(A)))))
                & ! [C] :
                    ( m1_subset_1(C,k1_numbers)
                   => ( r2_hidden(C,k4_relset_1(k1_numbers,k1_numbers,k2_partfun2(k1_numbers,k1_numbers,k2_partfun1(k1_numbers,k1_numbers,B,k12_prob_1(A)))))
                     => k1_fdiff_1(k2_partfun2(k1_numbers,k1_numbers,k2_partfun1(k1_numbers,k1_numbers,B,k12_prob_1(A))),C) = k6_real_1(np__1,k1_fdiff_1(B,k2_seq_1(k1_numbers,k1_numbers,k2_partfun2(k1_numbers,k1_numbers,k2_partfun1(k1_numbers,k1_numbers,B,k12_prob_1(A))),C))) ) ) ) ) ) ) ) )).

fof(t47_fdiff_2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( ( v1_funct_1(B)
            & v2_funct_1(B)
            & m2_relset_1(B,k1_numbers,k1_numbers) )
         => ( r2_fdiff_1(B,k4_limfunc1(A))
           => ( ( ? [C] :
                    ( m1_subset_1(C,k1_numbers)
                    & r2_hidden(C,k4_limfunc1(A))
                    & r1_xreal_0(k1_fdiff_1(B,C),np__0) )
                & ? [C] :
                    ( m1_subset_1(C,k1_numbers)
                    & r2_hidden(C,k4_limfunc1(A))
                    & r1_xreal_0(np__0,k1_fdiff_1(B,C)) ) )
              | ( v2_funct_1(k2_partfun1(k1_numbers,k1_numbers,B,k4_limfunc1(A)))
                & r2_fdiff_1(k2_partfun2(k1_numbers,k1_numbers,k2_partfun1(k1_numbers,k1_numbers,B,k4_limfunc1(A))),k4_relset_1(k1_numbers,k1_numbers,k2_partfun2(k1_numbers,k1_numbers,k2_partfun1(k1_numbers,k1_numbers,B,k4_limfunc1(A)))))
                & ! [C] :
                    ( m1_subset_1(C,k1_numbers)
                   => ( r2_hidden(C,k4_relset_1(k1_numbers,k1_numbers,k2_partfun2(k1_numbers,k1_numbers,k2_partfun1(k1_numbers,k1_numbers,B,k4_limfunc1(A)))))
                     => k1_fdiff_1(k2_partfun2(k1_numbers,k1_numbers,k2_partfun1(k1_numbers,k1_numbers,B,k4_limfunc1(A))),C) = k6_real_1(np__1,k1_fdiff_1(B,k2_seq_1(k1_numbers,k1_numbers,k2_partfun2(k1_numbers,k1_numbers,k2_partfun1(k1_numbers,k1_numbers,B,k4_limfunc1(A))),C))) ) ) ) ) ) ) ) )).

fof(t48_fdiff_2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ! [C] :
              ( ( v1_funct_1(C)
                & v2_funct_1(C)
                & m2_relset_1(C,k1_numbers,k1_numbers) )
             => ( r2_fdiff_1(C,k2_rcomp_1(A,B))
               => ( ( ? [D] :
                        ( m1_subset_1(D,k1_numbers)
                        & r2_hidden(D,k2_rcomp_1(A,B))
                        & r1_xreal_0(k1_fdiff_1(C,D),np__0) )
                    & ? [D] :
                        ( m1_subset_1(D,k1_numbers)
                        & r2_hidden(D,k2_rcomp_1(A,B))
                        & r1_xreal_0(np__0,k1_fdiff_1(C,D)) ) )
                  | ( v2_funct_1(k2_partfun1(k1_numbers,k1_numbers,C,k2_rcomp_1(A,B)))
                    & r2_fdiff_1(k2_partfun2(k1_numbers,k1_numbers,k2_partfun1(k1_numbers,k1_numbers,C,k2_rcomp_1(A,B))),k4_relset_1(k1_numbers,k1_numbers,k2_partfun2(k1_numbers,k1_numbers,k2_partfun1(k1_numbers,k1_numbers,C,k2_rcomp_1(A,B)))))
                    & ! [D] :
                        ( m1_subset_1(D,k1_numbers)
                       => ( r2_hidden(D,k4_relset_1(k1_numbers,k1_numbers,k2_partfun2(k1_numbers,k1_numbers,k2_partfun1(k1_numbers,k1_numbers,C,k2_rcomp_1(A,B)))))
                         => k1_fdiff_1(k2_partfun2(k1_numbers,k1_numbers,k2_partfun1(k1_numbers,k1_numbers,C,k2_rcomp_1(A,B))),D) = k6_real_1(np__1,k1_fdiff_1(C,k2_seq_1(k1_numbers,k1_numbers,k2_partfun2(k1_numbers,k1_numbers,k2_partfun1(k1_numbers,k1_numbers,C,k2_rcomp_1(A,B))),D))) ) ) ) ) ) ) ) ) )).

fof(t49_fdiff_2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( ( v1_funct_1(B)
            & m2_relset_1(B,k1_numbers,k1_numbers) )
         => ( r1_fdiff_1(B,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(A)
                        & r1_tarski(k1_rfunct_2(k9_seq_1(C,D)),k4_relset_1(k1_numbers,k1_numbers,B))
                        & r1_tarski(k1_rfunct_2(k9_seq_1(k1_fdiff_2(C),D)),k4_relset_1(k1_numbers,k1_numbers,B)) )
                     => ( v4_seq_2(k11_seq_1(k18_seq_1(k14_seq_1(C,np__2)),k10_seq_1(k2_rfunct_2(B,k9_seq_1(D,C)),k2_rfunct_2(B,k10_seq_1(D,C)))))
                        & k2_seq_2(k11_seq_1(k18_seq_1(k14_seq_1(C,np__2)),k10_seq_1(k2_rfunct_2(B,k9_seq_1(D,C)),k2_rfunct_2(B,k10_seq_1(D,C))))) = k1_fdiff_1(B,A) ) ) ) ) ) ) ) )).

fof(s1_fdiff_2,axiom,
    ( ! [A] :
        ( m2_subset_1(A,k1_numbers,k5_numbers)
       => ? [B] :
            ( m2_subset_1(B,k1_numbers,k5_numbers)
            & r1_xreal_0(A,B)
            & p1_s1_fdiff_2(k2_seq_1(k5_numbers,k1_numbers,f1_s1_fdiff_2,B)) ) )
   => ? [A] :
        ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & v1_seqm_3(A)
        & v7_seqm_3(A)
        & m2_relset_1(A,k5_numbers,k1_numbers)
        & ! [B] :
            ( m2_subset_1(B,k1_numbers,k5_numbers)
           => p1_s1_fdiff_2(k2_seq_1(k5_numbers,k1_numbers,k3_seqm_3(A,f1_s1_fdiff_2),B)) )
        & ! [B] :
            ( m2_subset_1(B,k1_numbers,k5_numbers)
           => ~ ( ! [C] :
                    ( m1_subset_1(C,k1_numbers)
                   => ( C = k2_seq_1(k5_numbers,k1_numbers,f1_s1_fdiff_2,B)
                     => p1_s1_fdiff_2(C) ) )
                & ! [C] :
                    ( m2_subset_1(C,k1_numbers,k5_numbers)
                   => B != k2_seqm_3(A,C) ) ) ) ) )).

fof(dt_k1_fdiff_2,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & v1_fdiff_1(A)
        & m1_relset_1(A,k5_numbers,k1_numbers) )
     => ( v1_funct_1(k1_fdiff_2(A))
        & v1_funct_2(k1_fdiff_2(A),k5_numbers,k1_numbers)
        & v1_fdiff_1(k1_fdiff_2(A))
        & m2_relset_1(k1_fdiff_2(A),k5_numbers,k1_numbers) ) ) )).

fof(redefinition_k1_fdiff_2,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & v1_fdiff_1(A)
        & m1_relset_1(A,k5_numbers,k1_numbers) )
     => k1_fdiff_2(A) = k15_seq_1(A) ) )).
%------------------------------------------------------------------------------