SET007 Axioms: SET007+823.ax


%------------------------------------------------------------------------------
% File     : SET007+823 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : The Taylor 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_1 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   46 (   1 unit)
%            Number of atoms       :  325 (  65 equality)
%            Maximal formula depth :   26 (   9 average)
%            Number of connectives :  296 (  17 ~  ;   6  |; 107  &)
%                                         (   6 <=>; 160 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   26 (   0 propositional; 1-3 arity)
%            Number of functors    :   51 (   7 constant; 0-6 arity)
%            Number of variables   :  135 (   0 singleton; 134 !;   1 ?)
%            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(fc1_taylor_1,axiom,
    ( v1_relat_1(k26_sin_cos)
    & v1_funct_1(k26_sin_cos)
    & v2_funct_1(k26_sin_cos)
    & v1_seq_1(k26_sin_cos) )).

fof(fc2_taylor_1,axiom,
    ( ~ v1_xboole_0(k4_limfunc1(np__0))
    & v1_membered(k4_limfunc1(np__0))
    & v2_membered(k4_limfunc1(np__0)) )).

fof(d1_taylor_1,axiom,(
    ! [A] :
      ( v1_int_1(A)
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k1_numbers,k1_numbers)
            & m2_relset_1(B,k1_numbers,k1_numbers) )
         => ( B = k1_taylor_1(A)
          <=> ! [C] :
                ( v1_xreal_0(C)
               => k2_seq_1(k1_numbers,k1_numbers,B,C) = k6_prepower(C,A) ) ) ) ) )).

fof(t1_taylor_1,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v4_ordinal2(B)
         => ! [C] :
              ( v4_ordinal2(C)
             => k6_prepower(A,k2_xcmplx_0(C,B)) = k3_xcmplx_0(k6_prepower(A,C),k6_prepower(A,B)) ) ) ) )).

fof(t2_taylor_1,axiom,(
    ! [A] :
      ( v4_ordinal2(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ( r1_fdiff_1(k1_taylor_1(A),B)
            & k1_fdiff_1(k1_taylor_1(A),B) = k3_xcmplx_0(A,k6_prepower(B,k6_xcmplx_0(A,np__1))) ) ) ) )).

fof(t3_taylor_1,axiom,(
    ! [A] :
      ( v4_ordinal2(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ! [C] :
              ( ( v1_funct_1(C)
                & m2_relset_1(C,k1_numbers,k1_numbers) )
             => ( r1_fdiff_1(C,B)
               => ( r1_fdiff_1(k1_partfun1(k1_numbers,k1_numbers,k1_numbers,k1_numbers,C,k1_taylor_1(A)),B)
                  & k1_fdiff_1(k1_partfun1(k1_numbers,k1_numbers,k1_numbers,k1_numbers,C,k1_taylor_1(A)),B) = k3_xcmplx_0(k3_xcmplx_0(A,k7_prepower(k2_seq_1(k1_numbers,k1_numbers,C,B),k6_xcmplx_0(A,np__1))),k1_fdiff_1(C,B)) ) ) ) ) ) )).

fof(t4_taylor_1,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => k27_sin_cos(k4_xcmplx_0(A)) = k7_xcmplx_0(np__1,k27_sin_cos(A)) ) )).

fof(t5_taylor_1,axiom,(
    ! [A] :
      ( v1_int_1(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => k12_prepower(k27_sin_cos(B),k7_xcmplx_0(np__1,A)) = k27_sin_cos(k7_xcmplx_0(B,A)) ) ) )).

fof(t6_taylor_1,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_int_1(B)
         => ! [C] :
              ( v1_int_1(C)
             => k12_prepower(k27_sin_cos(A),k7_xcmplx_0(B,C)) = k27_sin_cos(k3_xcmplx_0(k7_xcmplx_0(B,C),A)) ) ) ) )).

fof(t7_taylor_1,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_rat_1(B)
         => k8_prepower(k27_sin_cos(A),B) = k27_sin_cos(k3_xcmplx_0(B,A)) ) ) )).

fof(t8_taylor_1,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => k12_prepower(k27_sin_cos(A),B) = k27_sin_cos(k3_xcmplx_0(B,A)) ) ) )).

fof(t9_taylor_1,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ( k12_prepower(k28_sin_cos(np__1),A) = k27_sin_cos(A)
        & k3_power(k28_sin_cos(np__1),A) = k27_sin_cos(A)
        & k3_power(k8_power,A) = k27_sin_cos(A)
        & k12_prepower(k8_power,A) = k27_sin_cos(A) ) ) )).

fof(t10_taylor_1,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ( k12_prepower(k2_seq_1(k1_numbers,k1_numbers,k26_sin_cos,np__1),A) = k2_seq_1(k1_numbers,k1_numbers,k26_sin_cos,A)
        & k3_power(k2_seq_1(k1_numbers,k1_numbers,k26_sin_cos,np__1),A) = k2_seq_1(k1_numbers,k1_numbers,k26_sin_cos,A)
        & k3_power(k8_power,A) = k2_seq_1(k1_numbers,k1_numbers,k26_sin_cos,A)
        & k12_prepower(k8_power,A) = k2_seq_1(k1_numbers,k1_numbers,k26_sin_cos,A) ) ) )).

fof(t11_taylor_1,axiom,(
    r1_xreal_0(np__2,k8_power) )).

fof(t12_taylor_1,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => k5_power(k8_power,k27_sin_cos(A)) = A ) )).

fof(t13_taylor_1,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => k6_power(k8_power,k2_seq_1(k1_numbers,k1_numbers,k26_sin_cos,A)) = A ) )).

fof(t14_taylor_1,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ( ~ r1_xreal_0(A,np__0)
       => k27_sin_cos(k5_power(k8_power,A)) = A ) ) )).

fof(t15_taylor_1,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ( ~ r1_xreal_0(A,np__0)
       => k2_seq_1(k1_numbers,k1_numbers,k26_sin_cos,k5_power(k8_power,A)) = A ) ) )).

fof(t16_taylor_1,axiom,
    ( v2_funct_1(k26_sin_cos)
    & r2_fdiff_1(k26_sin_cos,k1_numbers)
    & r2_fdiff_1(k26_sin_cos,k2_subset_1(k1_numbers))
    & ! [A] :
        ( m1_subset_1(A,k1_numbers)
       => k1_fdiff_1(k26_sin_cos,A) = k2_seq_1(k1_numbers,k1_numbers,k26_sin_cos,A) )
    & ! [A] :
        ( m1_subset_1(A,k1_numbers)
       => ~ r1_xreal_0(k1_fdiff_1(k26_sin_cos,A),np__0) )
    & k1_relat_1(k26_sin_cos) = k1_numbers
    & k1_relat_1(k26_sin_cos) = k2_subset_1(k1_numbers)
    & k2_relat_1(k26_sin_cos) = k4_limfunc1(np__0) )).

fof(t17_taylor_1,axiom,
    ( r2_fdiff_1(k2_partfun2(k1_numbers,k1_numbers,k26_sin_cos),k1_relat_1(k2_partfun2(k1_numbers,k1_numbers,k26_sin_cos)))
    & ! [A] :
        ( v1_xreal_0(A)
       => ( r2_hidden(A,k1_relat_1(k2_partfun2(k1_numbers,k1_numbers,k26_sin_cos)))
         => k1_fdiff_1(k2_partfun2(k1_numbers,k1_numbers,k26_sin_cos),A) = k7_xcmplx_0(np__1,A) ) ) )).

fof(d2_taylor_1,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( ( v1_funct_1(B)
            & m2_relset_1(B,k1_numbers,k1_numbers) )
         => ( B = k2_taylor_1(A)
          <=> ( k1_relat_1(B) = k4_limfunc1(np__0)
              & ! [C] :
                  ( m2_subset_1(C,k1_numbers,k4_limfunc1(np__0))
                 => k2_seq_1(k1_numbers,k1_numbers,B,C) = k5_power(A,C) ) ) ) ) ) )).

fof(t18_taylor_1,axiom,
    ( k2_taylor_1(k8_power) = k2_partfun2(k1_numbers,k1_numbers,k26_sin_cos)
    & v2_funct_1(k2_taylor_1(k8_power))
    & k1_relat_1(k2_taylor_1(k8_power)) = k4_limfunc1(np__0)
    & k2_relat_1(k2_taylor_1(k8_power)) = k1_numbers
    & r2_fdiff_1(k2_taylor_1(k8_power),k4_limfunc1(np__0))
    & ! [A] :
        ( m1_subset_1(A,k1_numbers)
       => ( ~ r1_xreal_0(A,np__0)
         => r1_fdiff_1(k2_taylor_1(k8_power),A) ) )
    & ! [A] :
        ( m2_subset_1(A,k1_numbers,k4_limfunc1(np__0))
       => k1_fdiff_1(k2_taylor_1(k8_power),A) = k6_real_1(np__1,A) )
    & ! [A] :
        ( m2_subset_1(A,k1_numbers,k4_limfunc1(np__0))
       => ~ r1_xreal_0(k1_fdiff_1(k2_taylor_1(k8_power),A),np__0) ) )).

fof(t19_taylor_1,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( ( v1_funct_1(B)
            & m2_relset_1(B,k1_numbers,k1_numbers) )
         => ( r1_fdiff_1(B,A)
           => ( r1_fdiff_1(k1_partfun1(k1_numbers,k1_numbers,k1_numbers,k1_numbers,B,k26_sin_cos),A)
              & k1_fdiff_1(k1_partfun1(k1_numbers,k1_numbers,k1_numbers,k1_numbers,B,k26_sin_cos),A) = k4_real_1(k2_seq_1(k1_numbers,k1_numbers,k26_sin_cos,k2_seq_1(k1_numbers,k1_numbers,B,A)),k1_fdiff_1(B,A)) ) ) ) ) )).

fof(t20_taylor_1,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( ( v1_funct_1(B)
            & m2_relset_1(B,k1_numbers,k1_numbers) )
         => ( r1_fdiff_1(B,A)
           => ( r1_xreal_0(k2_seq_1(k1_numbers,k1_numbers,B,A),np__0)
              | ( r1_fdiff_1(k1_partfun1(k1_numbers,k1_numbers,k1_numbers,k1_numbers,B,k2_taylor_1(k8_power)),A)
                & k1_fdiff_1(k1_partfun1(k1_numbers,k1_numbers,k1_numbers,k1_numbers,B,k2_taylor_1(k8_power)),A) = k6_real_1(k1_fdiff_1(B,A),k2_seq_1(k1_numbers,k1_numbers,B,A)) ) ) ) ) ) )).

fof(d3_taylor_1,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( ( v1_funct_1(B)
            & m2_relset_1(B,k1_numbers,k1_numbers) )
         => ( B = k3_taylor_1(A)
          <=> ( k1_relat_1(B) = k4_limfunc1(np__0)
              & ! [C] :
                  ( m2_subset_1(C,k1_numbers,k4_limfunc1(np__0))
                 => k2_seq_1(k1_numbers,k1_numbers,B,C) = k12_prepower(C,A) ) ) ) ) ) )).

fof(t21_taylor_1,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ( ~ r1_xreal_0(A,np__0)
           => ( r1_fdiff_1(k3_taylor_1(B),A)
              & k1_fdiff_1(k3_taylor_1(B),A) = k3_xcmplx_0(B,k12_prepower(A,k6_xcmplx_0(B,np__1))) ) ) ) ) )).

fof(t22_taylor_1,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ! [C] :
              ( ( v1_funct_1(C)
                & m2_relset_1(C,k1_numbers,k1_numbers) )
             => ( r1_fdiff_1(C,A)
               => ( r1_xreal_0(k2_seq_1(k1_numbers,k1_numbers,C,A),np__0)
                  | ( r1_fdiff_1(k1_partfun1(k1_numbers,k1_numbers,k1_numbers,k1_numbers,C,k3_taylor_1(B)),A)
                    & k1_fdiff_1(k1_partfun1(k1_numbers,k1_numbers,k1_numbers,k1_numbers,C,k3_taylor_1(B)),A) = k3_xcmplx_0(k3_xcmplx_0(B,k12_prepower(k2_seq_1(k1_numbers,k1_numbers,C,A),k6_xcmplx_0(B,np__1))),k1_fdiff_1(C,A)) ) ) ) ) ) ) )).

fof(d4_taylor_1,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] :
              ( m1_seqfunc(C,k1_numbers,k1_numbers)
             => ( C = k4_taylor_1(A,B)
              <=> ( k1_seqfunc(k1_numbers,k1_numbers,C,np__0) = k2_partfun1(k1_numbers,k1_numbers,A,B)
                  & ! [D] :
                      ( v4_ordinal2(D)
                     => k1_seqfunc(k1_numbers,k1_numbers,C,k2_xcmplx_0(D,np__1)) = k2_fdiff_1(k1_seqfunc(k1_numbers,k1_numbers,C,D),B) ) ) ) ) ) ) )).

fof(d5_taylor_1,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & m2_relset_1(A,k1_numbers,k1_numbers) )
     => ! [B] :
          ( v4_ordinal2(B)
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(k1_numbers))
             => ( r1_taylor_1(A,B,C)
              <=> ! [D] :
                    ( m2_subset_1(D,k1_numbers,k5_numbers)
                   => ( r1_xreal_0(D,k6_xcmplx_0(B,np__1))
                     => r2_fdiff_1(k1_seqfunc(k1_numbers,k1_numbers,k4_taylor_1(A,C),D),C) ) ) ) ) ) ) )).

fof(t23_taylor_1,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] :
              ( v4_ordinal2(C)
             => ( r1_taylor_1(A,C,B)
               => ! [D] :
                    ( v4_ordinal2(D)
                   => ( r1_xreal_0(D,C)
                     => r1_taylor_1(A,D,B) ) ) ) ) ) ) )).

fof(d6_taylor_1,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)
             => ! [D] :
                  ( v1_xreal_0(D)
                 => ! [E] :
                      ( ( v1_funct_1(E)
                        & v1_funct_2(E,k5_numbers,k1_numbers)
                        & m2_relset_1(E,k5_numbers,k1_numbers) )
                     => ( E = k5_taylor_1(A,B,C,D)
                      <=> ! [F] :
                            ( v4_ordinal2(F)
                           => k2_seq_1(k5_numbers,k1_numbers,E,F) = k7_xcmplx_0(k3_xcmplx_0(k2_seq_1(k1_numbers,k1_numbers,k1_seqfunc(k1_numbers,k1_numbers,k4_taylor_1(A,B),F),C),k2_newton(k6_xcmplx_0(D,C),F)),k6_newton(F)) ) ) ) ) ) ) ) )).

fof(t24_taylor_1,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] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => ( r1_taylor_1(A,C,B)
               => ! [D] :
                    ( m1_subset_1(D,k1_numbers)
                   => ! [E] :
                        ( m1_subset_1(E,k1_numbers)
                       => ( r1_tarski(k2_rcomp_1(D,E),B)
                         => ( r1_xreal_0(E,D)
                            | k2_partfun1(k1_numbers,k1_numbers,k1_seqfunc(k1_numbers,k1_numbers,k4_taylor_1(A,B),C),k2_rcomp_1(D,E)) = k1_seqfunc(k1_numbers,k1_numbers,k4_taylor_1(A,k2_rcomp_1(D,E)),C) ) ) ) ) ) ) ) ) )).

fof(t25_taylor_1,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_zfmisc_1(k1_numbers))
             => ( r1_taylor_1(B,A,C)
               => ! [D] :
                    ( m1_subset_1(D,k1_numbers)
                   => ! [E] :
                        ( m1_subset_1(E,k1_numbers)
                       => ( ( r1_tarski(k1_rcomp_1(D,E),C)
                            & r2_fcont_1(k1_seqfunc(k1_numbers,k1_numbers,k4_taylor_1(B,C),A),k1_rcomp_1(D,E))
                            & r1_taylor_1(B,k1_nat_1(A,np__1),k2_rcomp_1(D,E)) )
                         => ( r1_xreal_0(E,D)
                            | ! [F] :
                                ( m1_subset_1(F,k1_numbers)
                               => ! [G] :
                                    ( ( v1_funct_1(G)
                                      & m2_relset_1(G,k1_numbers,k1_numbers) )
                                   => ( ( k1_relat_1(G) = k1_numbers
                                        & ! [H] :
                                            ( m1_subset_1(H,k1_numbers)
                                           => k2_seq_1(k1_numbers,k1_numbers,G,H) = k5_real_1(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,C,H,E)),A)),k6_real_1(k4_real_1(F,k3_prepower(k5_real_1(E,H),k1_nat_1(A,np__1))),k5_sin_cos(k1_nat_1(A,np__1)))) )
                                        & k5_real_1(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,C,D,E)),A)),k6_real_1(k4_real_1(F,k3_prepower(k5_real_1(E,D),k1_nat_1(A,np__1))),k5_sin_cos(k1_nat_1(A,np__1)))) = np__0 )
                                     => ( r2_fdiff_1(G,k2_rcomp_1(D,E))
                                        & k2_seq_1(k1_numbers,k1_numbers,G,D) = np__0
                                        & k2_seq_1(k1_numbers,k1_numbers,G,E) = np__0
                                        & r2_fcont_1(G,k1_rcomp_1(D,E))
                                        & ! [H] :
                                            ( m1_subset_1(H,k1_numbers)
                                           => ( r2_hidden(H,k2_rcomp_1(D,E))
                                             => k1_fdiff_1(G,H) = k3_real_1(k1_real_1(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(D,E)),k1_nat_1(A,np__1)),H),k3_prepower(k5_real_1(E,H),A)),k5_sin_cos(A))),k6_real_1(k4_real_1(F,k3_prepower(k5_real_1(E,H),A)),k5_sin_cos(A))) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )).

fof(t26_taylor_1,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_zfmisc_1(k1_numbers))
             => ! [D] :
                  ( m1_subset_1(D,k1_numbers)
                 => ! [E] :
                      ( m1_subset_1(E,k1_numbers)
                     => ? [F] :
                          ( v1_funct_1(F)
                          & v1_funct_2(F,k1_numbers,k1_numbers)
                          & m2_relset_1(F,k1_numbers,k1_numbers)
                          & ! [G] :
                              ( m1_subset_1(G,k1_numbers)
                             => k2_seq_1(k1_numbers,k1_numbers,F,G) = k5_real_1(k5_real_1(k2_seq_1(k1_numbers,k1_numbers,B,D),k2_seq_1(k5_numbers,k1_numbers,k1_series_1(k5_taylor_1(B,C,G,D)),A)),k6_real_1(k4_real_1(E,k3_prepower(k5_real_1(D,G),k1_nat_1(A,np__1))),k5_sin_cos(k1_nat_1(A,np__1)))) ) ) ) ) ) ) ) )).

fof(t27_taylor_1,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_zfmisc_1(k1_numbers))
             => ( r1_taylor_1(B,A,C)
               => ! [D] :
                    ( m1_subset_1(D,k1_numbers)
                   => ! [E] :
                        ( m1_subset_1(E,k1_numbers)
                       => ~ ( ~ r1_xreal_0(E,D)
                            & r1_tarski(k1_rcomp_1(D,E),C)
                            & r2_fcont_1(k1_seqfunc(k1_numbers,k1_numbers,k4_taylor_1(B,C),A),k1_rcomp_1(D,E))
                            & r1_taylor_1(B,k1_nat_1(A,np__1),k2_rcomp_1(D,E))
                            & ! [F] :
                                ( m1_subset_1(F,k1_numbers)
                               => ~ ( r2_hidden(F,k2_rcomp_1(D,E))
                                    & k2_seq_1(k1_numbers,k1_numbers,B,E) = k3_real_1(k2_seq_1(k5_numbers,k1_numbers,k1_series_1(k5_taylor_1(B,C,D,E)),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(D,E)),k1_nat_1(A,np__1)),F),k3_prepower(k5_real_1(E,D),k1_nat_1(A,np__1))),k5_sin_cos(k1_nat_1(A,np__1)))) ) ) ) ) ) ) ) ) ) )).

fof(t28_taylor_1,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_zfmisc_1(k1_numbers))
             => ( r1_taylor_1(B,A,C)
               => ! [D] :
                    ( m1_subset_1(D,k1_numbers)
                   => ! [E] :
                        ( m1_subset_1(E,k1_numbers)
                       => ( ( r1_tarski(k1_rcomp_1(D,E),C)
                            & r2_fcont_1(k1_seqfunc(k1_numbers,k1_numbers,k4_taylor_1(B,C),A),k1_rcomp_1(D,E))
                            & r1_taylor_1(B,k1_nat_1(A,np__1),k2_rcomp_1(D,E)) )
                         => ( r1_xreal_0(E,D)
                            | ! [F] :
                                ( m1_subset_1(F,k1_numbers)
                               => ! [G] :
                                    ( ( v1_funct_1(G)
                                      & m2_relset_1(G,k1_numbers,k1_numbers) )
                                   => ( ( k1_relat_1(G) = k1_numbers
                                        & ! [H] :
                                            ( m1_subset_1(H,k1_numbers)
                                           => k2_seq_1(k1_numbers,k1_numbers,G,H) = k5_real_1(k5_real_1(k2_seq_1(k1_numbers,k1_numbers,B,D),k2_seq_1(k5_numbers,k1_numbers,k1_series_1(k5_taylor_1(B,C,H,D)),A)),k6_real_1(k4_real_1(F,k3_prepower(k5_real_1(D,H),k1_nat_1(A,np__1))),k5_sin_cos(k1_nat_1(A,np__1)))) )
                                        & k5_real_1(k5_real_1(k2_seq_1(k1_numbers,k1_numbers,B,D),k2_seq_1(k5_numbers,k1_numbers,k1_series_1(k5_taylor_1(B,C,E,D)),A)),k6_real_1(k4_real_1(F,k3_prepower(k5_real_1(D,E),k1_nat_1(A,np__1))),k5_sin_cos(k1_nat_1(A,np__1)))) = np__0 )
                                     => ( r2_fdiff_1(G,k2_rcomp_1(D,E))
                                        & k2_seq_1(k1_numbers,k1_numbers,G,E) = np__0
                                        & k2_seq_1(k1_numbers,k1_numbers,G,D) = np__0
                                        & r2_fcont_1(G,k1_rcomp_1(D,E))
                                        & ! [H] :
                                            ( m1_subset_1(H,k1_numbers)
                                           => ( r2_hidden(H,k2_rcomp_1(D,E))
                                             => k1_fdiff_1(G,H) = k3_real_1(k1_real_1(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(D,E)),k1_nat_1(A,np__1)),H),k3_prepower(k5_real_1(D,H),A)),k5_sin_cos(A))),k6_real_1(k4_real_1(F,k3_prepower(k5_real_1(D,H),A)),k5_sin_cos(A))) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )).

fof(t29_taylor_1,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_zfmisc_1(k1_numbers))
             => ( r1_taylor_1(B,A,C)
               => ! [D] :
                    ( m1_subset_1(D,k1_numbers)
                   => ! [E] :
                        ( m1_subset_1(E,k1_numbers)
                       => ~ ( ~ r1_xreal_0(E,D)
                            & r1_tarski(k1_rcomp_1(D,E),C)
                            & r2_fcont_1(k1_seqfunc(k1_numbers,k1_numbers,k4_taylor_1(B,C),A),k1_rcomp_1(D,E))
                            & r1_taylor_1(B,k1_nat_1(A,np__1),k2_rcomp_1(D,E))
                            & ! [F] :
                                ( m1_subset_1(F,k1_numbers)
                               => ~ ( r2_hidden(F,k2_rcomp_1(D,E))
                                    & k2_seq_1(k1_numbers,k1_numbers,B,D) = k3_real_1(k2_seq_1(k5_numbers,k1_numbers,k1_series_1(k5_taylor_1(B,C,E,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(D,E)),k1_nat_1(A,np__1)),F),k3_prepower(k5_real_1(D,E),k1_nat_1(A,np__1))),k5_sin_cos(k1_nat_1(A,np__1)))) ) ) ) ) ) ) ) ) ) )).

fof(t30_taylor_1,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] :
              ( ( v3_rcomp_1(C)
                & m1_subset_1(C,k1_zfmisc_1(k1_numbers)) )
             => ( r1_tarski(C,B)
               => ! [D] :
                    ( m2_subset_1(D,k1_numbers,k5_numbers)
                   => ( r1_taylor_1(A,D,B)
                     => k2_partfun1(k1_numbers,k1_numbers,k1_seqfunc(k1_numbers,k1_numbers,k4_taylor_1(A,B),D),C) = k1_seqfunc(k1_numbers,k1_numbers,k4_taylor_1(A,C),D) ) ) ) ) ) ) )).

fof(t31_taylor_1,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] :
              ( ( v3_rcomp_1(C)
                & m1_subset_1(C,k1_zfmisc_1(k1_numbers)) )
             => ( r1_tarski(C,B)
               => ! [D] :
                    ( v4_ordinal2(D)
                   => ( r1_taylor_1(A,k2_xcmplx_0(D,np__1),B)
                     => r1_taylor_1(A,k2_xcmplx_0(D,np__1),C) ) ) ) ) ) ) )).

fof(t32_taylor_1,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] :
              ( m1_subset_1(C,k1_numbers)
             => ( r2_hidden(C,B)
               => ! [D] :
                    ( m2_subset_1(D,k1_numbers,k5_numbers)
                   => k2_seq_1(k1_numbers,k1_numbers,A,C) = k2_seq_1(k5_numbers,k1_numbers,k1_series_1(k5_taylor_1(A,B,C,C)),D) ) ) ) ) ) )).

fof(t33_taylor_1,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)
                                      & k2_seq_1(k1_numbers,k1_numbers,B,E) = k3_real_1(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),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(dt_k1_taylor_1,axiom,(
    ! [A] :
      ( v1_int_1(A)
     => ( v1_funct_1(k1_taylor_1(A))
        & v1_funct_2(k1_taylor_1(A),k1_numbers,k1_numbers)
        & m2_relset_1(k1_taylor_1(A),k1_numbers,k1_numbers) ) ) )).

fof(dt_k2_taylor_1,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ( v1_funct_1(k2_taylor_1(A))
        & m2_relset_1(k2_taylor_1(A),k1_numbers,k1_numbers) ) ) )).

fof(dt_k3_taylor_1,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ( v1_funct_1(k3_taylor_1(A))
        & m2_relset_1(k3_taylor_1(A),k1_numbers,k1_numbers) ) ) )).

fof(dt_k4_taylor_1,axiom,(
    ! [A,B] :
      ( ( v1_funct_1(A)
        & m1_relset_1(A,k1_numbers,k1_numbers)
        & m1_subset_1(B,k1_zfmisc_1(k1_numbers)) )
     => m1_seqfunc(k4_taylor_1(A,B),k1_numbers,k1_numbers) ) )).

fof(dt_k5_taylor_1,axiom,(
    ! [A,B,C,D] :
      ( ( 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_xreal_0(D) )
     => ( v1_funct_1(k5_taylor_1(A,B,C,D))
        & v1_funct_2(k5_taylor_1(A,B,C,D),k5_numbers,k1_numbers)
        & m2_relset_1(k5_taylor_1(A,B,C,D),k5_numbers,k1_numbers) ) ) )).
%------------------------------------------------------------------------------