SET007 Axioms: SET007+823.ax
%------------------------------------------------------------------------------
% File : SET007+823 : TPTP v9.0.0. 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 unt; 0 def)
% Number of atoms : 325 ( 65 equ)
% Maximal formula atoms : 25 ( 7 avg)
% Number of connectives : 296 ( 17 ~; 6 |; 107 &)
% ( 6 <=>; 160 =>; 0 <=; 0 <~>)
% Maximal formula depth : 26 ( 9 avg)
% Maximal term depth : 9 ( 1 avg)
% Number of predicates : 26 ( 25 usr; 0 prp; 1-3 aty)
% Number of functors : 51 ( 51 usr; 7 con; 0-6 aty)
% Number of variables : 135 ( 134 !; 1 ?)
% 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) ) ) ).
%------------------------------------------------------------------------------