SET007 Axioms: SET007+910.ax
%------------------------------------------------------------------------------
% File : SET007+910 : TPTP v9.0.0. 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 unt; 0 def)
% Number of atoms : 225 ( 32 equ)
% Maximal formula atoms : 14 ( 7 avg)
% Number of connectives : 247 ( 52 ~; 9 |; 82 &)
% ( 0 <=>; 104 =>; 0 <=; 0 <~>)
% Maximal formula depth : 23 ( 12 avg)
% Maximal term depth : 9 ( 2 avg)
% Number of predicates : 14 ( 13 usr; 0 prp; 1-3 aty)
% Number of functors : 35 ( 35 usr; 8 con; 0-4 aty)
% Number of variables : 102 ( 91 !; 11 ?)
% 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) ) ) ).
%------------------------------------------------------------------------------