SET007 Axioms: SET007+910.ax


%------------------------------------------------------------------------------
% File     : SET007+910 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : The Maclaurin Expansions
% 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    : taylor_2 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   30 (   0 unit)
%            Number of atoms       :  225 (  32 equality)
%            Maximal formula depth :   23 (  12 average)
%            Number of connectives :  247 (  52 ~  ;   9  |;  82  &)
%                                         (   0 <=>; 104 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   14 (   0 propositional; 1-3 arity)
%            Number of functors    :   35 (   8 constant; 0-4 arity)
%            Number of variables   :  102 (   0 singleton;  91 !;  11 ?)
%            Maximal term depth    :    9 (   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_taylor_2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => k18_complex1(k3_prepower(A,B)) = k3_prepower(k18_complex1(A),B) ) ) )).

fof(d1_taylor_2,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & m2_relset_1(A,k1_numbers,k1_numbers) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(k1_numbers))
         => ! [C] :
              ( v1_xreal_0(C)
             => k1_taylor_2(A,B,C) = k5_taylor_1(A,B,np__0,C) ) ) ) )).

fof(t2_taylor_2,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ( v1_funct_1(B)
            & m2_relset_1(B,k1_numbers,k1_numbers) )
         => ! [C] :
              ( m1_subset_1(C,k1_numbers)
             => ( r1_taylor_1(B,k1_nat_1(A,np__1),k2_rcomp_1(k1_real_1(C),C))
               => ( r1_xreal_0(C,np__0)
                  | ! [D] :
                      ( m1_subset_1(D,k1_numbers)
                     => ~ ( r2_hidden(D,k2_rcomp_1(k1_real_1(C),C))
                          & ! [E] :
                              ( m1_subset_1(E,k1_numbers)
                             => ~ ( ~ r1_xreal_0(E,np__0)
                                  & ~ r1_xreal_0(np__1,E)
                                  & k2_seq_1(k1_numbers,k1_numbers,B,D) = k3_real_1(k2_seq_1(k5_numbers,k1_numbers,k1_series_1(k1_taylor_2(B,k2_rcomp_1(k1_real_1(C),C),D)),A),k6_real_1(k4_real_1(k2_seq_1(k1_numbers,k1_numbers,k1_seqfunc(k1_numbers,k1_numbers,k4_taylor_1(B,k2_rcomp_1(k1_real_1(C),C)),k1_nat_1(A,np__1)),k4_real_1(E,D)),k3_prepower(D,k1_nat_1(A,np__1))),k5_sin_cos(k1_nat_1(A,np__1)))) ) ) ) ) ) ) ) ) ) )).

fof(t3_taylor_2,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ( v1_funct_1(B)
            & m2_relset_1(B,k1_numbers,k1_numbers) )
         => ! [C] :
              ( m1_subset_1(C,k1_numbers)
             => ! [D] :
                  ( m1_subset_1(D,k1_numbers)
                 => ( r1_taylor_1(B,k1_nat_1(A,np__1),k2_rcomp_1(k5_real_1(C,D),k3_real_1(C,D)))
                   => ( r1_xreal_0(D,np__0)
                      | ! [E] :
                          ( m1_subset_1(E,k1_numbers)
                         => ~ ( r2_hidden(E,k2_rcomp_1(k5_real_1(C,D),k3_real_1(C,D)))
                              & ! [F] :
                                  ( m1_subset_1(F,k1_numbers)
                                 => ~ ( ~ r1_xreal_0(F,np__0)
                                      & ~ r1_xreal_0(np__1,F)
                                      & k18_complex1(k5_real_1(k2_seq_1(k1_numbers,k1_numbers,B,E),k2_seq_1(k5_numbers,k1_numbers,k1_series_1(k5_taylor_1(B,k2_rcomp_1(k5_real_1(C,D),k3_real_1(C,D)),C,E)),A))) = k18_complex1(k6_real_1(k4_real_1(k2_seq_1(k1_numbers,k1_numbers,k1_seqfunc(k1_numbers,k1_numbers,k4_taylor_1(B,k2_rcomp_1(k5_real_1(C,D),k3_real_1(C,D))),k1_nat_1(A,np__1)),k3_real_1(C,k4_real_1(F,k5_real_1(E,C)))),k3_prepower(k5_real_1(E,C),k1_nat_1(A,np__1))),k5_sin_cos(k1_nat_1(A,np__1)))) ) ) ) ) ) ) ) ) ) ) )).

fof(t4_taylor_2,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ( v1_funct_1(B)
            & m2_relset_1(B,k1_numbers,k1_numbers) )
         => ! [C] :
              ( m1_subset_1(C,k1_numbers)
             => ( r1_taylor_1(B,k1_nat_1(A,np__1),k2_rcomp_1(k1_real_1(C),C))
               => ( r1_xreal_0(C,np__0)
                  | ! [D] :
                      ( m1_subset_1(D,k1_numbers)
                     => ~ ( r2_hidden(D,k2_rcomp_1(k1_real_1(C),C))
                          & ! [E] :
                              ( m1_subset_1(E,k1_numbers)
                             => ~ ( ~ r1_xreal_0(E,np__0)
                                  & ~ r1_xreal_0(np__1,E)
                                  & k18_complex1(k5_real_1(k2_seq_1(k1_numbers,k1_numbers,B,D),k2_seq_1(k5_numbers,k1_numbers,k1_series_1(k1_taylor_2(B,k2_rcomp_1(k1_real_1(C),C),D)),A))) = k18_complex1(k6_real_1(k4_real_1(k2_seq_1(k1_numbers,k1_numbers,k1_seqfunc(k1_numbers,k1_numbers,k4_taylor_1(B,k2_rcomp_1(k1_real_1(C),C)),k1_nat_1(A,np__1)),k4_real_1(E,D)),k3_prepower(D,k1_nat_1(A,np__1))),k5_sin_cos(k1_nat_1(A,np__1)))) ) ) ) ) ) ) ) ) ) )).

fof(t5_taylor_2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ( k2_fdiff_1(k26_sin_cos,k2_rcomp_1(k1_real_1(A),A)) = k2_partfun1(k1_numbers,k1_numbers,k26_sin_cos,k2_rcomp_1(k1_real_1(A),A))
        & k1_relat_1(k2_partfun1(k1_numbers,k1_numbers,k26_sin_cos,k2_rcomp_1(k1_real_1(A),A))) = k2_rcomp_1(k1_real_1(A),A) ) ) )).

fof(t6_taylor_2,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => k1_seqfunc(k1_numbers,k1_numbers,k4_taylor_1(k26_sin_cos,k2_rcomp_1(k1_real_1(B),B)),A) = k2_partfun1(k1_numbers,k1_numbers,k26_sin_cos,k2_rcomp_1(k1_real_1(B),B)) ) ) )).

fof(t7_taylor_2,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ! [C] :
              ( m1_subset_1(C,k1_numbers)
             => ( r2_hidden(C,k2_rcomp_1(k1_real_1(B),B))
               => k2_seq_1(k1_numbers,k1_numbers,k1_seqfunc(k1_numbers,k1_numbers,k4_taylor_1(k26_sin_cos,k2_rcomp_1(k1_real_1(B),B)),A),C) = k2_seq_1(k1_numbers,k1_numbers,k26_sin_cos,C) ) ) ) ) )).

fof(t8_taylor_2,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ! [C] :
              ( m1_subset_1(C,k1_numbers)
             => ( ~ r1_xreal_0(B,np__0)
               => k2_seq_1(k5_numbers,k1_numbers,k1_taylor_2(k26_sin_cos,k2_rcomp_1(k1_real_1(B),B),C),A) = k6_real_1(k3_prepower(C,A),k5_sin_cos(A)) ) ) ) ) )).

fof(t9_taylor_2,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ! [C] :
              ( m1_subset_1(C,k1_numbers)
             => ! [D] :
                  ( m1_subset_1(D,k1_numbers)
                 => ( r2_hidden(C,k2_rcomp_1(k1_real_1(B),B))
                   => ( r1_xreal_0(D,np__0)
                      | r1_xreal_0(np__1,D)
                      | r1_xreal_0(k18_complex1(k6_real_1(k4_real_1(k2_seq_1(k1_numbers,k1_numbers,k1_seqfunc(k1_numbers,k1_numbers,k4_taylor_1(k26_sin_cos,k2_rcomp_1(k1_real_1(B),B)),k1_nat_1(A,np__1)),k4_real_1(D,C)),k3_prepower(C,k1_nat_1(A,np__1))),k5_sin_cos(k1_nat_1(A,np__1)))),k6_real_1(k4_real_1(k18_complex1(k2_seq_1(k1_numbers,k1_numbers,k26_sin_cos,k4_real_1(D,C))),k3_prepower(k18_complex1(C),k1_nat_1(A,np__1))),k5_sin_cos(k1_nat_1(A,np__1)))) ) ) ) ) ) ) )).

fof(t10_taylor_2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => r1_taylor_1(k26_sin_cos,B,k2_rcomp_1(k1_real_1(A),A)) ) ) )).

fof(t11_taylor_2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ~ ( ~ r1_xreal_0(A,np__0)
          & ! [B] :
              ( m1_subset_1(B,k1_numbers)
             => ! [C] :
                  ( m1_subset_1(C,k1_numbers)
                 => ~ ( r1_xreal_0(np__0,B)
                      & r1_xreal_0(np__0,C)
                      & ! [D] :
                          ( m2_subset_1(D,k1_numbers,k5_numbers)
                         => ! [E] :
                              ( m1_subset_1(E,k1_numbers)
                             => ! [F] :
                                  ( m1_subset_1(F,k1_numbers)
                                 => ( r2_hidden(E,k2_rcomp_1(k1_real_1(A),A))
                                   => ( r1_xreal_0(F,np__0)
                                      | r1_xreal_0(np__1,F)
                                      | r1_xreal_0(k18_complex1(k6_real_1(k4_real_1(k2_seq_1(k1_numbers,k1_numbers,k1_seqfunc(k1_numbers,k1_numbers,k4_taylor_1(k26_sin_cos,k2_rcomp_1(k1_real_1(A),A)),D),k4_real_1(F,E)),k3_prepower(E,D)),k5_sin_cos(D))),k6_real_1(k4_real_1(B,k3_prepower(C,D)),k5_sin_cos(D))) ) ) ) ) ) ) ) ) ) ) )).

fof(t12_taylor_2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ( ( r1_xreal_0(np__0,A)
              & r1_xreal_0(np__0,B) )
           => ! [C] :
                ( m1_subset_1(C,k1_numbers)
               => ~ ( ~ r1_xreal_0(C,np__0)
                    & ! [D] :
                        ( m2_subset_1(D,k1_numbers,k5_numbers)
                       => ? [E] :
                            ( m2_subset_1(E,k1_numbers,k5_numbers)
                            & r1_xreal_0(D,E)
                            & r1_xreal_0(C,k6_real_1(k4_real_1(A,k3_prepower(B,E)),k5_sin_cos(E))) ) ) ) ) ) ) ) )).

fof(t13_taylor_2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ~ ( ~ r1_xreal_0(A,np__0)
              & ~ r1_xreal_0(B,np__0)
              & ! [C] :
                  ( m2_subset_1(C,k1_numbers,k5_numbers)
                 => ? [D] :
                      ( m2_subset_1(D,k1_numbers,k5_numbers)
                      & r1_xreal_0(C,D)
                      & ? [E] :
                          ( m1_subset_1(E,k1_numbers)
                          & ? [F] :
                              ( m1_subset_1(F,k1_numbers)
                              & r2_hidden(E,k2_rcomp_1(k1_real_1(A),A))
                              & ~ r1_xreal_0(F,np__0)
                              & ~ r1_xreal_0(np__1,F)
                              & r1_xreal_0(B,k18_complex1(k6_real_1(k4_real_1(k2_seq_1(k1_numbers,k1_numbers,k1_seqfunc(k1_numbers,k1_numbers,k4_taylor_1(k26_sin_cos,k2_rcomp_1(k1_real_1(A),A)),D),k4_real_1(F,E)),k3_prepower(E,D)),k5_sin_cos(D)))) ) ) ) ) ) ) ) )).

fof(t14_taylor_2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ~ ( ~ r1_xreal_0(A,np__0)
              & ~ r1_xreal_0(B,np__0)
              & ! [C] :
                  ( m2_subset_1(C,k1_numbers,k5_numbers)
                 => ? [D] :
                      ( m2_subset_1(D,k1_numbers,k5_numbers)
                      & r1_xreal_0(C,D)
                      & ? [E] :
                          ( v1_xreal_0(E)
                          & r2_hidden(E,k2_rcomp_1(k1_real_1(A),A))
                          & r1_xreal_0(B,k18_complex1(k5_real_1(k2_seq_1(k1_numbers,k1_numbers,k26_sin_cos,E),k2_seq_1(k5_numbers,k1_numbers,k1_series_1(k1_taylor_2(k26_sin_cos,k2_rcomp_1(k1_real_1(A),A),E)),D)))) ) ) ) ) ) ) )).

fof(t15_taylor_2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => v2_series_1(k7_sin_cos(A)) ) )).

fof(t16_taylor_2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ( ~ r1_xreal_0(A,np__0)
           => ( k1_taylor_2(k26_sin_cos,k2_rcomp_1(k1_real_1(A),A),B) = k7_sin_cos(B)
              & v2_series_1(k1_taylor_2(k26_sin_cos,k2_rcomp_1(k1_real_1(A),A),B))
              & k2_seq_1(k1_numbers,k1_numbers,k26_sin_cos,B) = k2_series_1(k1_taylor_2(k26_sin_cos,k2_rcomp_1(k1_real_1(A),A),B)) ) ) ) ) )).

fof(t17_taylor_2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ( k2_fdiff_1(k18_sin_cos,k2_rcomp_1(k1_real_1(A),A)) = k2_partfun1(k1_numbers,k1_numbers,k21_sin_cos,k2_rcomp_1(k1_real_1(A),A))
        & k2_fdiff_1(k21_sin_cos,k2_rcomp_1(k1_real_1(A),A)) = k2_partfun1(k1_numbers,k1_numbers,k16_seq_1(k1_numbers,k1_numbers,k18_sin_cos),k2_rcomp_1(k1_real_1(A),A))
        & k1_relat_1(k2_partfun1(k1_numbers,k1_numbers,k18_sin_cos,k2_rcomp_1(k1_real_1(A),A))) = k2_rcomp_1(k1_real_1(A),A)
        & k1_relat_1(k2_partfun1(k1_numbers,k1_numbers,k21_sin_cos,k2_rcomp_1(k1_real_1(A),A))) = k2_rcomp_1(k1_real_1(A),A) ) ) )).

fof(t18_taylor_2,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & m2_relset_1(A,k1_numbers,k1_numbers) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(k1_numbers))
         => ( r2_fdiff_1(A,B)
           => k2_fdiff_1(k16_seq_1(k1_numbers,k1_numbers,A),B) = k16_seq_1(k1_numbers,k1_numbers,k2_fdiff_1(A,B)) ) ) ) )).

fof(t19_taylor_2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ( k1_seqfunc(k1_numbers,k1_numbers,k4_taylor_1(k18_sin_cos,k2_rcomp_1(k1_real_1(A),A)),k2_nat_1(np__2,B)) = k13_seq_1(k1_numbers,k1_numbers,k2_partfun1(k1_numbers,k1_numbers,k18_sin_cos,k2_rcomp_1(k1_real_1(A),A)),k3_prepower(k1_real_1(np__1),B))
            & k1_seqfunc(k1_numbers,k1_numbers,k4_taylor_1(k18_sin_cos,k2_rcomp_1(k1_real_1(A),A)),k1_nat_1(k2_nat_1(np__2,B),np__1)) = k13_seq_1(k1_numbers,k1_numbers,k2_partfun1(k1_numbers,k1_numbers,k21_sin_cos,k2_rcomp_1(k1_real_1(A),A)),k3_prepower(k1_real_1(np__1),B))
            & k1_seqfunc(k1_numbers,k1_numbers,k4_taylor_1(k21_sin_cos,k2_rcomp_1(k1_real_1(A),A)),k2_nat_1(np__2,B)) = k13_seq_1(k1_numbers,k1_numbers,k2_partfun1(k1_numbers,k1_numbers,k21_sin_cos,k2_rcomp_1(k1_real_1(A),A)),k3_prepower(k1_real_1(np__1),B))
            & k1_seqfunc(k1_numbers,k1_numbers,k4_taylor_1(k21_sin_cos,k2_rcomp_1(k1_real_1(A),A)),k1_nat_1(k2_nat_1(np__2,B),np__1)) = k13_seq_1(k1_numbers,k1_numbers,k2_partfun1(k1_numbers,k1_numbers,k18_sin_cos,k2_rcomp_1(k1_real_1(A),A)),k3_prepower(k1_real_1(np__1),k1_nat_1(B,np__1))) ) ) ) )).

fof(t20_taylor_2,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ! [C] :
              ( m1_subset_1(C,k1_numbers)
             => ( ~ r1_xreal_0(B,np__0)
               => ( k2_seq_1(k5_numbers,k1_numbers,k1_taylor_2(k18_sin_cos,k2_rcomp_1(k1_real_1(B),B),C),k2_nat_1(np__2,A)) = np__0
                  & k2_seq_1(k5_numbers,k1_numbers,k1_taylor_2(k18_sin_cos,k2_rcomp_1(k1_real_1(B),B),C),k1_nat_1(k2_nat_1(np__2,A),np__1)) = k6_real_1(k4_real_1(k3_prepower(k1_real_1(np__1),A),k3_prepower(C,k1_nat_1(k2_nat_1(np__2,A),np__1))),k5_sin_cos(k1_nat_1(k2_nat_1(np__2,A),np__1)))
                  & k2_seq_1(k5_numbers,k1_numbers,k1_taylor_2(k21_sin_cos,k2_rcomp_1(k1_real_1(B),B),C),k2_nat_1(np__2,A)) = k6_real_1(k4_real_1(k3_prepower(k1_real_1(np__1),A),k3_prepower(C,k2_nat_1(np__2,A))),k5_sin_cos(k2_nat_1(np__2,A)))
                  & k2_seq_1(k5_numbers,k1_numbers,k1_taylor_2(k21_sin_cos,k2_rcomp_1(k1_real_1(B),B),C),k1_nat_1(k2_nat_1(np__2,A),np__1)) = np__0 ) ) ) ) ) )).

fof(t21_taylor_2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ( r1_taylor_1(k18_sin_cos,B,k2_rcomp_1(k1_real_1(A),A))
            & r1_taylor_1(k21_sin_cos,B,k2_rcomp_1(k1_real_1(A),A)) ) ) ) )).

fof(t22_taylor_2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ~ ( ~ r1_xreal_0(A,np__0)
          & ! [B] :
              ( m1_subset_1(B,k1_numbers)
             => ! [C] :
                  ( m1_subset_1(C,k1_numbers)
                 => ~ ( r1_xreal_0(np__0,B)
                      & r1_xreal_0(np__0,C)
                      & ! [D] :
                          ( m2_subset_1(D,k1_numbers,k5_numbers)
                         => ! [E] :
                              ( m1_subset_1(E,k1_numbers)
                             => ! [F] :
                                  ( m1_subset_1(F,k1_numbers)
                                 => ( r2_hidden(E,k2_rcomp_1(k1_real_1(A),A))
                                   => ( r1_xreal_0(F,np__0)
                                      | r1_xreal_0(np__1,F)
                                      | ( r1_xreal_0(k18_complex1(k6_real_1(k4_real_1(k2_seq_1(k1_numbers,k1_numbers,k1_seqfunc(k1_numbers,k1_numbers,k4_taylor_1(k18_sin_cos,k2_rcomp_1(k1_real_1(A),A)),D),k4_real_1(F,E)),k3_prepower(E,D)),k5_sin_cos(D))),k6_real_1(k4_real_1(B,k3_prepower(C,D)),k5_sin_cos(D)))
                                        & r1_xreal_0(k18_complex1(k6_real_1(k4_real_1(k2_seq_1(k1_numbers,k1_numbers,k1_seqfunc(k1_numbers,k1_numbers,k4_taylor_1(k21_sin_cos,k2_rcomp_1(k1_real_1(A),A)),D),k4_real_1(F,E)),k3_prepower(E,D)),k5_sin_cos(D))),k6_real_1(k4_real_1(B,k3_prepower(C,D)),k5_sin_cos(D))) ) ) ) ) ) ) ) ) ) ) ) )).

fof(t23_taylor_2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ~ ( ~ r1_xreal_0(A,np__0)
              & ~ r1_xreal_0(B,np__0)
              & ! [C] :
                  ( m2_subset_1(C,k1_numbers,k5_numbers)
                 => ? [D] :
                      ( m2_subset_1(D,k1_numbers,k5_numbers)
                      & r1_xreal_0(C,D)
                      & ? [E] :
                          ( m1_subset_1(E,k1_numbers)
                          & ? [F] :
                              ( m1_subset_1(F,k1_numbers)
                              & r2_hidden(E,k2_rcomp_1(k1_real_1(A),A))
                              & ~ r1_xreal_0(F,np__0)
                              & ~ r1_xreal_0(np__1,F)
                              & ~ ( ~ r1_xreal_0(B,k18_complex1(k6_real_1(k4_real_1(k2_seq_1(k1_numbers,k1_numbers,k1_seqfunc(k1_numbers,k1_numbers,k4_taylor_1(k18_sin_cos,k2_rcomp_1(k1_real_1(A),A)),D),k4_real_1(F,E)),k3_prepower(E,D)),k5_sin_cos(D))))
                                  & ~ r1_xreal_0(B,k18_complex1(k6_real_1(k4_real_1(k2_seq_1(k1_numbers,k1_numbers,k1_seqfunc(k1_numbers,k1_numbers,k4_taylor_1(k21_sin_cos,k2_rcomp_1(k1_real_1(A),A)),D),k4_real_1(F,E)),k3_prepower(E,D)),k5_sin_cos(D)))) ) ) ) ) ) ) ) ) )).

fof(t24_taylor_2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ~ ( ~ r1_xreal_0(A,np__0)
              & ~ r1_xreal_0(B,np__0)
              & ! [C] :
                  ( m2_subset_1(C,k1_numbers,k5_numbers)
                 => ? [D] :
                      ( m2_subset_1(D,k1_numbers,k5_numbers)
                      & r1_xreal_0(C,D)
                      & ? [E] :
                          ( v1_xreal_0(E)
                          & r2_hidden(E,k2_rcomp_1(k1_real_1(A),A))
                          & ~ ( ~ r1_xreal_0(B,k18_complex1(k5_real_1(k2_seq_1(k1_numbers,k1_numbers,k18_sin_cos,E),k2_seq_1(k5_numbers,k1_numbers,k1_series_1(k1_taylor_2(k18_sin_cos,k2_rcomp_1(k1_real_1(A),A),E)),D))))
                              & ~ r1_xreal_0(B,k18_complex1(k5_real_1(k2_seq_1(k1_numbers,k1_numbers,k21_sin_cos,E),k2_seq_1(k5_numbers,k1_numbers,k1_series_1(k1_taylor_2(k21_sin_cos,k2_rcomp_1(k1_real_1(A),A),E)),D)))) ) ) ) ) ) ) ) )).

fof(t25_taylor_2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => ( ~ r1_xreal_0(A,np__0)
               => ( k2_seq_1(k5_numbers,k1_numbers,k1_series_1(k1_taylor_2(k18_sin_cos,k2_rcomp_1(k1_real_1(A),A),B)),k1_nat_1(k2_nat_1(np__2,C),np__1)) = k2_seq_1(k5_numbers,k1_numbers,k1_series_1(k24_sin_cos(B)),C)
                  & k2_seq_1(k5_numbers,k1_numbers,k1_series_1(k1_taylor_2(k21_sin_cos,k2_rcomp_1(k1_real_1(A),A),B)),k1_nat_1(k2_nat_1(np__2,C),np__1)) = k2_seq_1(k5_numbers,k1_numbers,k1_series_1(k25_sin_cos(B)),C) ) ) ) ) ) )).

fof(t26_taylor_2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => ~ ( ~ r1_xreal_0(A,np__0)
                  & ~ r1_xreal_0(C,np__0)
                  & ~ ( k2_seq_1(k5_numbers,k1_numbers,k1_series_1(k1_taylor_2(k18_sin_cos,k2_rcomp_1(k1_real_1(A),A),B)),k2_nat_1(np__2,C)) = k2_seq_1(k5_numbers,k1_numbers,k1_series_1(k24_sin_cos(B)),k5_real_1(C,np__1))
                      & k2_seq_1(k5_numbers,k1_numbers,k1_series_1(k1_taylor_2(k21_sin_cos,k2_rcomp_1(k1_real_1(A),A),B)),k2_nat_1(np__2,C)) = k2_seq_1(k5_numbers,k1_numbers,k1_series_1(k25_sin_cos(B)),C) ) ) ) ) ) )).

fof(t27_taylor_2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => ( ~ r1_xreal_0(A,np__0)
               => k2_seq_1(k5_numbers,k1_numbers,k1_series_1(k1_taylor_2(k21_sin_cos,k2_rcomp_1(k1_real_1(A),A),B)),k2_nat_1(np__2,C)) = k2_seq_1(k5_numbers,k1_numbers,k1_series_1(k25_sin_cos(B)),C) ) ) ) ) )).

fof(t28_taylor_2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ( ~ r1_xreal_0(A,np__0)
           => ( v4_seq_2(k1_series_1(k1_taylor_2(k18_sin_cos,k2_rcomp_1(k1_real_1(A),A),B)))
              & k2_seq_1(k1_numbers,k1_numbers,k18_sin_cos,B) = k2_series_1(k1_taylor_2(k18_sin_cos,k2_rcomp_1(k1_real_1(A),A),B))
              & v4_seq_2(k1_series_1(k1_taylor_2(k21_sin_cos,k2_rcomp_1(k1_real_1(A),A),B)))
              & k2_seq_1(k1_numbers,k1_numbers,k21_sin_cos,B) = k2_series_1(k1_taylor_2(k21_sin_cos,k2_rcomp_1(k1_real_1(A),A),B)) ) ) ) ) )).

fof(dt_k1_taylor_2,axiom,(
    ! [A,B,C] :
      ( ( v1_funct_1(A)
        & m1_relset_1(A,k1_numbers,k1_numbers)
        & m1_subset_1(B,k1_zfmisc_1(k1_numbers))
        & v1_xreal_0(C) )
     => ( v1_funct_1(k1_taylor_2(A,B,C))
        & v1_funct_2(k1_taylor_2(A,B,C),k5_numbers,k1_numbers)
        & m2_relset_1(k1_taylor_2(A,B,C),k5_numbers,k1_numbers) ) ) )).
%------------------------------------------------------------------------------