SET007 Axioms: SET007+923.ax
%------------------------------------------------------------------------------
% File : SET007+923 : TPTP v9.0.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Some Differentiable Formulas of Special Functions
% 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_5 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 25 ( 0 unt; 0 def)
% Number of atoms : 386 ( 77 equ)
% Maximal formula atoms : 22 ( 15 avg)
% Number of connectives : 384 ( 23 ~; 0 |; 157 &)
% ( 0 <=>; 204 =>; 0 <=; 0 <~>)
% Maximal formula depth : 19 ( 14 avg)
% Maximal term depth : 7 ( 1 avg)
% Number of predicates : 10 ( 9 usr; 0 prp; 1-3 aty)
% Number of functors : 33 ( 33 usr; 9 con; 0-6 aty)
% Number of variables : 128 ( 128 !; 0 ?)
% 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_5,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) )
=> ! [D] :
( ( v1_funct_1(D)
& m2_relset_1(D,k1_numbers,k1_numbers) )
=> ( ( r1_tarski(B,k1_relat_1(k2_rfunct_1(k1_numbers,k1_numbers,C,D)))
& ! [E] :
( m1_subset_1(E,k1_numbers)
=> ( r2_hidden(E,B)
=> ( k2_seq_1(k1_numbers,k1_numbers,C,E) = k3_real_1(A,E)
& k2_seq_1(k1_numbers,k1_numbers,D,E) = k5_real_1(A,E)
& k2_seq_1(k1_numbers,k1_numbers,D,E) != np__0 ) ) ) )
=> ( r2_fdiff_1(k2_rfunct_1(k1_numbers,k1_numbers,C,D),B)
& ! [E] :
( m1_subset_1(E,k1_numbers)
=> ( r2_hidden(E,B)
=> k2_seq_1(k1_numbers,k1_numbers,k2_fdiff_1(k2_rfunct_1(k1_numbers,k1_numbers,C,D),B),E) = k6_real_1(k4_real_1(np__2,A),k7_square_1(k5_real_1(A,E))) ) ) ) ) ) ) ) ) ).
fof(t2_fdiff_5,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) )
=> ! [D] :
( ( v1_funct_1(D)
& m2_relset_1(D,k1_numbers,k1_numbers) )
=> ( ( r1_tarski(B,k1_relat_1(k2_rfunct_1(k1_numbers,k1_numbers,C,D)))
& ! [E] :
( m1_subset_1(E,k1_numbers)
=> ( r2_hidden(E,B)
=> ( k2_seq_1(k1_numbers,k1_numbers,C,E) = k5_real_1(E,A)
& k2_seq_1(k1_numbers,k1_numbers,D,E) = k3_real_1(E,A)
& k2_seq_1(k1_numbers,k1_numbers,D,E) != np__0 ) ) ) )
=> ( r2_fdiff_1(k2_rfunct_1(k1_numbers,k1_numbers,C,D),B)
& ! [E] :
( m1_subset_1(E,k1_numbers)
=> ( r2_hidden(E,B)
=> k2_seq_1(k1_numbers,k1_numbers,k2_fdiff_1(k2_rfunct_1(k1_numbers,k1_numbers,C,D),B),E) = k6_real_1(k4_real_1(np__2,A),k7_square_1(k3_real_1(E,A))) ) ) ) ) ) ) ) ) ).
fof(t3_fdiff_5,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ! [C] :
( ( v3_rcomp_1(C)
& m1_subset_1(C,k1_zfmisc_1(k1_numbers)) )
=> ! [D] :
( ( v1_funct_1(D)
& m2_relset_1(D,k1_numbers,k1_numbers) )
=> ! [E] :
( ( v1_funct_1(E)
& m2_relset_1(E,k1_numbers,k1_numbers) )
=> ( ( r1_tarski(C,k1_relat_1(k2_rfunct_1(k1_numbers,k1_numbers,D,E)))
& ! [F] :
( m1_subset_1(F,k1_numbers)
=> ( r2_hidden(F,C)
=> ( k2_seq_1(k1_numbers,k1_numbers,D,F) = k5_real_1(F,A)
& k2_seq_1(k1_numbers,k1_numbers,E,F) = k5_real_1(F,B)
& k2_seq_1(k1_numbers,k1_numbers,E,F) != np__0 ) ) ) )
=> ( r2_fdiff_1(k2_rfunct_1(k1_numbers,k1_numbers,D,E),C)
& ! [F] :
( m1_subset_1(F,k1_numbers)
=> ( r2_hidden(F,C)
=> k2_seq_1(k1_numbers,k1_numbers,k2_fdiff_1(k2_rfunct_1(k1_numbers,k1_numbers,D,E),C),F) = k6_real_1(k5_real_1(A,B),k7_square_1(k5_real_1(F,B))) ) ) ) ) ) ) ) ) ) ).
fof(t4_fdiff_5,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,k1_relat_1(B))
& ! [C] :
( m1_subset_1(C,k1_numbers)
=> ( r2_hidden(C,A)
=> ( k2_seq_1(k1_numbers,k1_numbers,B,C) = C
& k2_seq_1(k1_numbers,k1_numbers,B,C) != np__0 ) ) ) )
=> ( r2_fdiff_1(k4_rfunct_1(k1_numbers,k1_numbers,B),A)
& ! [C] :
( m1_subset_1(C,k1_numbers)
=> ( r2_hidden(C,A)
=> k2_seq_1(k1_numbers,k1_numbers,k2_fdiff_1(k4_rfunct_1(k1_numbers,k1_numbers,B),A),C) = k1_real_1(k6_real_1(np__1,k7_square_1(C))) ) ) ) ) ) ) ).
fof(t5_fdiff_5,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,k1_relat_1(k1_partfun1(k1_numbers,k1_numbers,k1_numbers,k1_numbers,k4_rfunct_1(k1_numbers,k1_numbers,B),k18_sin_cos)))
& ! [C] :
( m1_subset_1(C,k1_numbers)
=> ( r2_hidden(C,A)
=> ( k2_seq_1(k1_numbers,k1_numbers,B,C) = C
& k2_seq_1(k1_numbers,k1_numbers,B,C) != np__0 ) ) ) )
=> ( r2_fdiff_1(k1_partfun1(k1_numbers,k1_numbers,k1_numbers,k1_numbers,k4_rfunct_1(k1_numbers,k1_numbers,B),k18_sin_cos),A)
& ! [C] :
( m1_subset_1(C,k1_numbers)
=> ( r2_hidden(C,A)
=> k2_seq_1(k1_numbers,k1_numbers,k2_fdiff_1(k1_partfun1(k1_numbers,k1_numbers,k1_numbers,k1_numbers,k4_rfunct_1(k1_numbers,k1_numbers,B),k18_sin_cos),A),C) = k1_real_1(k4_real_1(k6_real_1(np__1,k7_square_1(C)),k2_seq_1(k1_numbers,k1_numbers,k21_sin_cos,k6_real_1(np__1,C)))) ) ) ) ) ) ) ).
fof(t6_fdiff_5,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,k1_relat_1(k1_partfun1(k1_numbers,k1_numbers,k1_numbers,k1_numbers,k4_rfunct_1(k1_numbers,k1_numbers,B),k21_sin_cos)))
& ! [C] :
( m1_subset_1(C,k1_numbers)
=> ( r2_hidden(C,A)
=> ( k2_seq_1(k1_numbers,k1_numbers,B,C) = C
& k2_seq_1(k1_numbers,k1_numbers,B,C) != np__0 ) ) ) )
=> ( r2_fdiff_1(k1_partfun1(k1_numbers,k1_numbers,k1_numbers,k1_numbers,k4_rfunct_1(k1_numbers,k1_numbers,B),k21_sin_cos),A)
& ! [C] :
( m1_subset_1(C,k1_numbers)
=> ( r2_hidden(C,A)
=> k2_seq_1(k1_numbers,k1_numbers,k2_fdiff_1(k1_partfun1(k1_numbers,k1_numbers,k1_numbers,k1_numbers,k4_rfunct_1(k1_numbers,k1_numbers,B),k21_sin_cos),A),C) = k4_real_1(k6_real_1(np__1,k7_square_1(C)),k2_seq_1(k1_numbers,k1_numbers,k18_sin_cos,k6_real_1(np__1,C))) ) ) ) ) ) ) ).
fof(t7_fdiff_5,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,k1_relat_1(k8_seq_1(k1_numbers,k1_numbers,k1_partfun2(k1_numbers,A),k1_partfun1(k1_numbers,k1_numbers,k1_numbers,k1_numbers,k4_rfunct_1(k1_numbers,k1_numbers,B),k18_sin_cos))))
& ! [C] :
( m1_subset_1(C,k1_numbers)
=> ( r2_hidden(C,A)
=> ( k2_seq_1(k1_numbers,k1_numbers,B,C) = C
& k2_seq_1(k1_numbers,k1_numbers,B,C) != np__0 ) ) ) )
=> ( r2_fdiff_1(k8_seq_1(k1_numbers,k1_numbers,k1_partfun2(k1_numbers,A),k1_partfun1(k1_numbers,k1_numbers,k1_numbers,k1_numbers,k4_rfunct_1(k1_numbers,k1_numbers,B),k18_sin_cos)),A)
& ! [C] :
( m1_subset_1(C,k1_numbers)
=> ( r2_hidden(C,A)
=> k2_seq_1(k1_numbers,k1_numbers,k2_fdiff_1(k8_seq_1(k1_numbers,k1_numbers,k1_partfun2(k1_numbers,A),k1_partfun1(k1_numbers,k1_numbers,k1_numbers,k1_numbers,k4_rfunct_1(k1_numbers,k1_numbers,B),k18_sin_cos)),A),C) = k5_real_1(k2_seq_1(k1_numbers,k1_numbers,k18_sin_cos,k6_real_1(np__1,C)),k4_real_1(k6_real_1(np__1,C),k2_seq_1(k1_numbers,k1_numbers,k21_sin_cos,k6_real_1(np__1,C)))) ) ) ) ) ) ) ).
fof(t8_fdiff_5,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,k1_relat_1(k8_seq_1(k1_numbers,k1_numbers,k1_partfun2(k1_numbers,A),k1_partfun1(k1_numbers,k1_numbers,k1_numbers,k1_numbers,k4_rfunct_1(k1_numbers,k1_numbers,B),k21_sin_cos))))
& ! [C] :
( m1_subset_1(C,k1_numbers)
=> ( r2_hidden(C,A)
=> ( k2_seq_1(k1_numbers,k1_numbers,B,C) = C
& k2_seq_1(k1_numbers,k1_numbers,B,C) != np__0 ) ) ) )
=> ( r2_fdiff_1(k8_seq_1(k1_numbers,k1_numbers,k1_partfun2(k1_numbers,A),k1_partfun1(k1_numbers,k1_numbers,k1_numbers,k1_numbers,k4_rfunct_1(k1_numbers,k1_numbers,B),k21_sin_cos)),A)
& ! [C] :
( m1_subset_1(C,k1_numbers)
=> ( r2_hidden(C,A)
=> k2_seq_1(k1_numbers,k1_numbers,k2_fdiff_1(k8_seq_1(k1_numbers,k1_numbers,k1_partfun2(k1_numbers,A),k1_partfun1(k1_numbers,k1_numbers,k1_numbers,k1_numbers,k4_rfunct_1(k1_numbers,k1_numbers,B),k21_sin_cos)),A),C) = k3_real_1(k2_seq_1(k1_numbers,k1_numbers,k21_sin_cos,k6_real_1(np__1,C)),k4_real_1(k6_real_1(np__1,C),k2_seq_1(k1_numbers,k1_numbers,k18_sin_cos,k6_real_1(np__1,C)))) ) ) ) ) ) ) ).
fof(t9_fdiff_5,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,k1_relat_1(k8_seq_1(k1_numbers,k1_numbers,k1_partfun1(k1_numbers,k1_numbers,k1_numbers,k1_numbers,k4_rfunct_1(k1_numbers,k1_numbers,B),k18_sin_cos),k1_partfun1(k1_numbers,k1_numbers,k1_numbers,k1_numbers,k4_rfunct_1(k1_numbers,k1_numbers,B),k21_sin_cos))))
& ! [C] :
( m1_subset_1(C,k1_numbers)
=> ( r2_hidden(C,A)
=> ( k2_seq_1(k1_numbers,k1_numbers,B,C) = C
& k2_seq_1(k1_numbers,k1_numbers,B,C) != np__0 ) ) ) )
=> ( r2_fdiff_1(k8_seq_1(k1_numbers,k1_numbers,k1_partfun1(k1_numbers,k1_numbers,k1_numbers,k1_numbers,k4_rfunct_1(k1_numbers,k1_numbers,B),k18_sin_cos),k1_partfun1(k1_numbers,k1_numbers,k1_numbers,k1_numbers,k4_rfunct_1(k1_numbers,k1_numbers,B),k21_sin_cos)),A)
& ! [C] :
( m1_subset_1(C,k1_numbers)
=> ( r2_hidden(C,A)
=> k2_seq_1(k1_numbers,k1_numbers,k2_fdiff_1(k8_seq_1(k1_numbers,k1_numbers,k1_partfun1(k1_numbers,k1_numbers,k1_numbers,k1_numbers,k4_rfunct_1(k1_numbers,k1_numbers,B),k18_sin_cos),k1_partfun1(k1_numbers,k1_numbers,k1_numbers,k1_numbers,k4_rfunct_1(k1_numbers,k1_numbers,B),k21_sin_cos)),A),C) = k4_real_1(k6_real_1(np__1,k7_square_1(C)),k5_real_1(k7_square_1(k2_seq_1(k1_numbers,k1_numbers,k18_sin_cos,k6_real_1(np__1,C))),k7_square_1(k2_seq_1(k1_numbers,k1_numbers,k21_sin_cos,k6_real_1(np__1,C))))) ) ) ) ) ) ) ).
fof(t10_fdiff_5,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_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) )
=> ( ( r1_tarski(B,k1_relat_1(k8_seq_1(k1_numbers,k1_numbers,k1_partfun1(k1_numbers,k1_numbers,k1_numbers,k1_numbers,C,k18_sin_cos),k1_partfun1(k1_numbers,k1_numbers,k1_numbers,k1_numbers,k18_sin_cos,k1_taylor_1(A)))))
& r1_xreal_0(np__1,A)
& ! [D] :
( m1_subset_1(D,k1_numbers)
=> ( r2_hidden(D,B)
=> k2_seq_1(k1_numbers,k1_numbers,C,D) = k4_real_1(A,D) ) ) )
=> ( r2_fdiff_1(k8_seq_1(k1_numbers,k1_numbers,k1_partfun1(k1_numbers,k1_numbers,k1_numbers,k1_numbers,C,k18_sin_cos),k1_partfun1(k1_numbers,k1_numbers,k1_numbers,k1_numbers,k18_sin_cos,k1_taylor_1(A))),B)
& ! [D] :
( m1_subset_1(D,k1_numbers)
=> ( r2_hidden(D,B)
=> k2_seq_1(k1_numbers,k1_numbers,k2_fdiff_1(k8_seq_1(k1_numbers,k1_numbers,k1_partfun1(k1_numbers,k1_numbers,k1_numbers,k1_numbers,C,k18_sin_cos),k1_partfun1(k1_numbers,k1_numbers,k1_numbers,k1_numbers,k18_sin_cos,k1_taylor_1(A))),B),D) = k4_real_1(k4_real_1(A,k7_prepower(k2_seq_1(k1_numbers,k1_numbers,k18_sin_cos,D),k5_real_1(A,np__1))),k2_seq_1(k1_numbers,k1_numbers,k18_sin_cos,k4_real_1(k1_nat_1(A,np__1),D))) ) ) ) ) ) ) ) ).
fof(t11_fdiff_5,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_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) )
=> ( ( r1_tarski(B,k1_relat_1(k8_seq_1(k1_numbers,k1_numbers,k1_partfun1(k1_numbers,k1_numbers,k1_numbers,k1_numbers,C,k21_sin_cos),k1_partfun1(k1_numbers,k1_numbers,k1_numbers,k1_numbers,k18_sin_cos,k1_taylor_1(A)))))
& r1_xreal_0(np__1,A)
& ! [D] :
( m1_subset_1(D,k1_numbers)
=> ( r2_hidden(D,B)
=> k2_seq_1(k1_numbers,k1_numbers,C,D) = k4_real_1(A,D) ) ) )
=> ( r2_fdiff_1(k8_seq_1(k1_numbers,k1_numbers,k1_partfun1(k1_numbers,k1_numbers,k1_numbers,k1_numbers,C,k21_sin_cos),k1_partfun1(k1_numbers,k1_numbers,k1_numbers,k1_numbers,k18_sin_cos,k1_taylor_1(A))),B)
& ! [D] :
( m1_subset_1(D,k1_numbers)
=> ( r2_hidden(D,B)
=> k2_seq_1(k1_numbers,k1_numbers,k2_fdiff_1(k8_seq_1(k1_numbers,k1_numbers,k1_partfun1(k1_numbers,k1_numbers,k1_numbers,k1_numbers,C,k21_sin_cos),k1_partfun1(k1_numbers,k1_numbers,k1_numbers,k1_numbers,k18_sin_cos,k1_taylor_1(A))),B),D) = k4_real_1(k4_real_1(A,k7_prepower(k2_seq_1(k1_numbers,k1_numbers,k18_sin_cos,D),k5_real_1(A,np__1))),k2_seq_1(k1_numbers,k1_numbers,k21_sin_cos,k4_real_1(k1_nat_1(A,np__1),D))) ) ) ) ) ) ) ) ).
fof(t12_fdiff_5,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_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) )
=> ( ( r1_tarski(B,k1_relat_1(k8_seq_1(k1_numbers,k1_numbers,k1_partfun1(k1_numbers,k1_numbers,k1_numbers,k1_numbers,C,k21_sin_cos),k1_partfun1(k1_numbers,k1_numbers,k1_numbers,k1_numbers,k21_sin_cos,k1_taylor_1(A)))))
& r1_xreal_0(np__1,A)
& ! [D] :
( m1_subset_1(D,k1_numbers)
=> ( r2_hidden(D,B)
=> k2_seq_1(k1_numbers,k1_numbers,C,D) = k4_real_1(A,D) ) ) )
=> ( r2_fdiff_1(k8_seq_1(k1_numbers,k1_numbers,k1_partfun1(k1_numbers,k1_numbers,k1_numbers,k1_numbers,C,k21_sin_cos),k1_partfun1(k1_numbers,k1_numbers,k1_numbers,k1_numbers,k21_sin_cos,k1_taylor_1(A))),B)
& ! [D] :
( m1_subset_1(D,k1_numbers)
=> ( r2_hidden(D,B)
=> k2_seq_1(k1_numbers,k1_numbers,k2_fdiff_1(k8_seq_1(k1_numbers,k1_numbers,k1_partfun1(k1_numbers,k1_numbers,k1_numbers,k1_numbers,C,k21_sin_cos),k1_partfun1(k1_numbers,k1_numbers,k1_numbers,k1_numbers,k21_sin_cos,k1_taylor_1(A))),B),D) = k1_real_1(k4_real_1(k4_real_1(A,k7_prepower(k2_seq_1(k1_numbers,k1_numbers,k21_sin_cos,D),k5_real_1(A,np__1))),k2_seq_1(k1_numbers,k1_numbers,k18_sin_cos,k4_real_1(k1_nat_1(A,np__1),D)))) ) ) ) ) ) ) ) ).
fof(t13_fdiff_5,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_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) )
=> ( ( r1_tarski(B,k1_relat_1(k8_seq_1(k1_numbers,k1_numbers,k1_partfun1(k1_numbers,k1_numbers,k1_numbers,k1_numbers,C,k18_sin_cos),k1_partfun1(k1_numbers,k1_numbers,k1_numbers,k1_numbers,k21_sin_cos,k1_taylor_1(A)))))
& r1_xreal_0(np__1,A)
& ! [D] :
( m1_subset_1(D,k1_numbers)
=> ( r2_hidden(D,B)
=> k2_seq_1(k1_numbers,k1_numbers,C,D) = k4_real_1(A,D) ) ) )
=> ( r2_fdiff_1(k8_seq_1(k1_numbers,k1_numbers,k1_partfun1(k1_numbers,k1_numbers,k1_numbers,k1_numbers,C,k18_sin_cos),k1_partfun1(k1_numbers,k1_numbers,k1_numbers,k1_numbers,k21_sin_cos,k1_taylor_1(A))),B)
& ! [D] :
( m1_subset_1(D,k1_numbers)
=> ( r2_hidden(D,B)
=> k2_seq_1(k1_numbers,k1_numbers,k2_fdiff_1(k8_seq_1(k1_numbers,k1_numbers,k1_partfun1(k1_numbers,k1_numbers,k1_numbers,k1_numbers,C,k18_sin_cos),k1_partfun1(k1_numbers,k1_numbers,k1_numbers,k1_numbers,k21_sin_cos,k1_taylor_1(A))),B),D) = k4_real_1(k4_real_1(A,k7_prepower(k2_seq_1(k1_numbers,k1_numbers,k21_sin_cos,D),k5_real_1(A,np__1))),k2_seq_1(k1_numbers,k1_numbers,k21_sin_cos,k4_real_1(k1_nat_1(A,np__1),D))) ) ) ) ) ) ) ) ).
fof(t14_fdiff_5,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,k1_relat_1(k8_seq_1(k1_numbers,k1_numbers,k4_rfunct_1(k1_numbers,k1_numbers,B),k18_sin_cos)))
& ! [C] :
( m1_subset_1(C,k1_numbers)
=> ( r2_hidden(C,A)
=> ( k2_seq_1(k1_numbers,k1_numbers,B,C) = C
& k2_seq_1(k1_numbers,k1_numbers,B,C) != np__0 ) ) ) )
=> ( r2_fdiff_1(k8_seq_1(k1_numbers,k1_numbers,k4_rfunct_1(k1_numbers,k1_numbers,B),k18_sin_cos),A)
& ! [C] :
( m1_subset_1(C,k1_numbers)
=> ( r2_hidden(C,A)
=> k2_seq_1(k1_numbers,k1_numbers,k2_fdiff_1(k8_seq_1(k1_numbers,k1_numbers,k4_rfunct_1(k1_numbers,k1_numbers,B),k18_sin_cos),A),C) = k5_real_1(k4_real_1(k6_real_1(np__1,C),k2_seq_1(k1_numbers,k1_numbers,k21_sin_cos,C)),k4_real_1(k6_real_1(np__1,k7_square_1(C)),k2_seq_1(k1_numbers,k1_numbers,k18_sin_cos,C))) ) ) ) ) ) ) ).
fof(t15_fdiff_5,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,k1_relat_1(k8_seq_1(k1_numbers,k1_numbers,k4_rfunct_1(k1_numbers,k1_numbers,B),k21_sin_cos)))
& ! [C] :
( m1_subset_1(C,k1_numbers)
=> ( r2_hidden(C,A)
=> ( k2_seq_1(k1_numbers,k1_numbers,B,C) = C
& k2_seq_1(k1_numbers,k1_numbers,B,C) != np__0 ) ) ) )
=> ( r2_fdiff_1(k8_seq_1(k1_numbers,k1_numbers,k4_rfunct_1(k1_numbers,k1_numbers,B),k21_sin_cos),A)
& ! [C] :
( m1_subset_1(C,k1_numbers)
=> ( r2_hidden(C,A)
=> k2_seq_1(k1_numbers,k1_numbers,k2_fdiff_1(k8_seq_1(k1_numbers,k1_numbers,k4_rfunct_1(k1_numbers,k1_numbers,B),k21_sin_cos),A),C) = k5_real_1(k1_real_1(k4_real_1(k6_real_1(np__1,C),k2_seq_1(k1_numbers,k1_numbers,k18_sin_cos,C))),k4_real_1(k6_real_1(np__1,k7_square_1(C)),k2_seq_1(k1_numbers,k1_numbers,k21_sin_cos,C))) ) ) ) ) ) ) ).
fof(t16_fdiff_5,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,k1_relat_1(k6_seq_1(k1_numbers,k1_numbers,k18_sin_cos,k1_partfun1(k1_numbers,k1_numbers,k1_numbers,k1_numbers,B,k3_taylor_1(k6_real_1(np__1,np__2))))))
& ! [C] :
( m1_subset_1(C,k1_numbers)
=> ( r2_hidden(C,A)
=> ( k2_seq_1(k1_numbers,k1_numbers,B,C) = C
& ~ r1_xreal_0(k2_seq_1(k1_numbers,k1_numbers,B,C),np__0) ) ) ) )
=> ( r2_fdiff_1(k6_seq_1(k1_numbers,k1_numbers,k18_sin_cos,k1_partfun1(k1_numbers,k1_numbers,k1_numbers,k1_numbers,B,k3_taylor_1(k6_real_1(np__1,np__2)))),A)
& ! [C] :
( m1_subset_1(C,k1_numbers)
=> ( r2_hidden(C,A)
=> k2_seq_1(k1_numbers,k1_numbers,k2_fdiff_1(k6_seq_1(k1_numbers,k1_numbers,k18_sin_cos,k1_partfun1(k1_numbers,k1_numbers,k1_numbers,k1_numbers,B,k3_taylor_1(k6_real_1(np__1,np__2)))),A),C) = k3_real_1(k2_seq_1(k1_numbers,k1_numbers,k21_sin_cos,C),k4_real_1(k6_real_1(np__1,np__2),k13_prepower(C,k1_real_1(k6_real_1(np__1,np__2))))) ) ) ) ) ) ) ).
fof(t17_fdiff_5,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) )
=> ( ( r1_tarski(A,k1_relat_1(k8_seq_1(k1_numbers,k1_numbers,B,k1_partfun1(k1_numbers,k1_numbers,k1_numbers,k1_numbers,k4_rfunct_1(k1_numbers,k1_numbers,C),k18_sin_cos))))
& B = k1_taylor_1(np__2)
& ! [D] :
( m1_subset_1(D,k1_numbers)
=> ( r2_hidden(D,A)
=> ( k2_seq_1(k1_numbers,k1_numbers,C,D) = D
& k2_seq_1(k1_numbers,k1_numbers,C,D) != np__0 ) ) ) )
=> ( r2_fdiff_1(k8_seq_1(k1_numbers,k1_numbers,B,k1_partfun1(k1_numbers,k1_numbers,k1_numbers,k1_numbers,k4_rfunct_1(k1_numbers,k1_numbers,C),k18_sin_cos)),A)
& ! [D] :
( m1_subset_1(D,k1_numbers)
=> ( r2_hidden(D,A)
=> k2_seq_1(k1_numbers,k1_numbers,k2_fdiff_1(k8_seq_1(k1_numbers,k1_numbers,B,k1_partfun1(k1_numbers,k1_numbers,k1_numbers,k1_numbers,k4_rfunct_1(k1_numbers,k1_numbers,C),k18_sin_cos)),A),D) = k5_real_1(k4_real_1(k4_real_1(np__2,D),k2_seq_1(k1_numbers,k1_numbers,k18_sin_cos,k6_real_1(np__1,D))),k2_seq_1(k1_numbers,k1_numbers,k21_sin_cos,k6_real_1(np__1,D))) ) ) ) ) ) ) ) ).
fof(t18_fdiff_5,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) )
=> ( ( r1_tarski(A,k1_relat_1(k8_seq_1(k1_numbers,k1_numbers,B,k1_partfun1(k1_numbers,k1_numbers,k1_numbers,k1_numbers,k4_rfunct_1(k1_numbers,k1_numbers,C),k21_sin_cos))))
& B = k1_taylor_1(np__2)
& ! [D] :
( m1_subset_1(D,k1_numbers)
=> ( r2_hidden(D,A)
=> ( k2_seq_1(k1_numbers,k1_numbers,C,D) = D
& k2_seq_1(k1_numbers,k1_numbers,C,D) != np__0 ) ) ) )
=> ( r2_fdiff_1(k8_seq_1(k1_numbers,k1_numbers,B,k1_partfun1(k1_numbers,k1_numbers,k1_numbers,k1_numbers,k4_rfunct_1(k1_numbers,k1_numbers,C),k21_sin_cos)),A)
& ! [D] :
( m1_subset_1(D,k1_numbers)
=> ( r2_hidden(D,A)
=> k2_seq_1(k1_numbers,k1_numbers,k2_fdiff_1(k8_seq_1(k1_numbers,k1_numbers,B,k1_partfun1(k1_numbers,k1_numbers,k1_numbers,k1_numbers,k4_rfunct_1(k1_numbers,k1_numbers,C),k21_sin_cos)),A),D) = k3_real_1(k4_real_1(k4_real_1(np__2,D),k2_seq_1(k1_numbers,k1_numbers,k21_sin_cos,k6_real_1(np__1,D))),k2_seq_1(k1_numbers,k1_numbers,k18_sin_cos,k6_real_1(np__1,D))) ) ) ) ) ) ) ) ).
fof(t19_fdiff_5,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,k1_relat_1(k1_partfun1(k1_numbers,k1_numbers,k1_numbers,k1_numbers,B,k2_taylor_1(k8_power))))
& ! [C] :
( m1_subset_1(C,k1_numbers)
=> ( r2_hidden(C,A)
=> ( k2_seq_1(k1_numbers,k1_numbers,B,C) = C
& ~ r1_xreal_0(k2_seq_1(k1_numbers,k1_numbers,B,C),np__0) ) ) ) )
=> ( r2_fdiff_1(k1_partfun1(k1_numbers,k1_numbers,k1_numbers,k1_numbers,B,k2_taylor_1(k8_power)),A)
& ! [C] :
( m1_subset_1(C,k1_numbers)
=> ( r2_hidden(C,A)
=> k2_seq_1(k1_numbers,k1_numbers,k2_fdiff_1(k1_partfun1(k1_numbers,k1_numbers,k1_numbers,k1_numbers,B,k2_taylor_1(k8_power)),A),C) = k6_real_1(np__1,C) ) ) ) ) ) ) ).
fof(t20_fdiff_5,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) )
=> ( ( r1_tarski(A,k1_relat_1(k8_seq_1(k1_numbers,k1_numbers,k1_partfun2(k1_numbers,A),B)))
& B = k1_partfun1(k1_numbers,k1_numbers,k1_numbers,k1_numbers,C,k2_taylor_1(k8_power))
& ! [D] :
( m1_subset_1(D,k1_numbers)
=> ( r2_hidden(D,A)
=> ( k2_seq_1(k1_numbers,k1_numbers,C,D) = D
& ~ r1_xreal_0(k2_seq_1(k1_numbers,k1_numbers,C,D),np__0) ) ) ) )
=> ( r2_fdiff_1(k8_seq_1(k1_numbers,k1_numbers,k1_partfun2(k1_numbers,A),B),A)
& ! [D] :
( m1_subset_1(D,k1_numbers)
=> ( r2_hidden(D,A)
=> k2_seq_1(k1_numbers,k1_numbers,k2_fdiff_1(k8_seq_1(k1_numbers,k1_numbers,k1_partfun2(k1_numbers,A),B),A),D) = k3_real_1(np__1,k2_seq_1(k1_numbers,k1_numbers,k2_taylor_1(k8_power),D)) ) ) ) ) ) ) ) ).
fof(t21_fdiff_5,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) )
=> ! [D] :
( ( v1_funct_1(D)
& m2_relset_1(D,k1_numbers,k1_numbers) )
=> ( ( r1_tarski(A,k1_relat_1(k8_seq_1(k1_numbers,k1_numbers,B,C)))
& B = k1_taylor_1(np__2)
& C = k1_partfun1(k1_numbers,k1_numbers,k1_numbers,k1_numbers,D,k2_taylor_1(k8_power))
& ! [E] :
( m1_subset_1(E,k1_numbers)
=> ( r2_hidden(E,A)
=> ( k2_seq_1(k1_numbers,k1_numbers,D,E) = E
& ~ r1_xreal_0(k2_seq_1(k1_numbers,k1_numbers,D,E),np__0) ) ) ) )
=> ( r2_fdiff_1(k8_seq_1(k1_numbers,k1_numbers,B,C),A)
& ! [E] :
( m1_subset_1(E,k1_numbers)
=> ( r2_hidden(E,A)
=> k2_seq_1(k1_numbers,k1_numbers,k2_fdiff_1(k8_seq_1(k1_numbers,k1_numbers,B,C),A),E) = k3_real_1(E,k4_real_1(k4_real_1(np__2,E),k2_seq_1(k1_numbers,k1_numbers,k2_taylor_1(k8_power),E))) ) ) ) ) ) ) ) ) ).
fof(t22_fdiff_5,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) )
=> ! [D] :
( ( v1_funct_1(D)
& m2_relset_1(D,k1_numbers,k1_numbers) )
=> ( ( r1_tarski(B,k1_relat_1(k2_rfunct_1(k1_numbers,k1_numbers,k6_seq_1(k1_numbers,k1_numbers,C,D),k7_seq_1(k1_numbers,k1_numbers,C,D))))
& ! [E] :
( m1_subset_1(E,k1_numbers)
=> ( r2_hidden(E,B)
=> k2_seq_1(k1_numbers,k1_numbers,C,E) = A ) )
& D = k1_taylor_1(np__2)
& ! [E] :
( m1_subset_1(E,k1_numbers)
=> ~ ( r2_hidden(E,B)
& r1_xreal_0(k2_seq_1(k1_numbers,k1_numbers,k7_seq_1(k1_numbers,k1_numbers,C,D),E),np__0) ) ) )
=> ( r2_fdiff_1(k2_rfunct_1(k1_numbers,k1_numbers,k6_seq_1(k1_numbers,k1_numbers,C,D),k7_seq_1(k1_numbers,k1_numbers,C,D)),B)
& ! [E] :
( m1_subset_1(E,k1_numbers)
=> ( r2_hidden(E,B)
=> k2_seq_1(k1_numbers,k1_numbers,k2_fdiff_1(k2_rfunct_1(k1_numbers,k1_numbers,k6_seq_1(k1_numbers,k1_numbers,C,D),k7_seq_1(k1_numbers,k1_numbers,C,D)),B),E) = k6_real_1(k4_real_1(k4_real_1(np__4,A),E),k3_prepower(k5_real_1(A,k3_prepower(E,np__2)),np__2)) ) ) ) ) ) ) ) ) ).
fof(t23_fdiff_5,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) )
=> ! [D] :
( ( v1_funct_1(D)
& m2_relset_1(D,k1_numbers,k1_numbers) )
=> ( ( r1_tarski(B,k1_relat_1(k1_partfun1(k1_numbers,k1_numbers,k1_numbers,k1_numbers,k2_rfunct_1(k1_numbers,k1_numbers,k6_seq_1(k1_numbers,k1_numbers,C,D),k7_seq_1(k1_numbers,k1_numbers,C,D)),k2_taylor_1(k8_power))))
& ! [E] :
( m1_subset_1(E,k1_numbers)
=> ( r2_hidden(E,B)
=> k2_seq_1(k1_numbers,k1_numbers,C,E) = A ) )
& D = k1_taylor_1(np__2)
& ! [E] :
( m1_subset_1(E,k1_numbers)
=> ~ ( r2_hidden(E,B)
& r1_xreal_0(k2_seq_1(k1_numbers,k1_numbers,k7_seq_1(k1_numbers,k1_numbers,C,D),E),np__0) ) )
& ! [E] :
( m1_subset_1(E,k1_numbers)
=> ~ ( r2_hidden(E,B)
& r1_xreal_0(k2_seq_1(k1_numbers,k1_numbers,k6_seq_1(k1_numbers,k1_numbers,C,D),E),np__0) ) ) )
=> ( r2_fdiff_1(k1_partfun1(k1_numbers,k1_numbers,k1_numbers,k1_numbers,k2_rfunct_1(k1_numbers,k1_numbers,k6_seq_1(k1_numbers,k1_numbers,C,D),k7_seq_1(k1_numbers,k1_numbers,C,D)),k2_taylor_1(k8_power)),B)
& ! [E] :
( m1_subset_1(E,k1_numbers)
=> ( r2_hidden(E,B)
=> k2_seq_1(k1_numbers,k1_numbers,k2_fdiff_1(k1_partfun1(k1_numbers,k1_numbers,k1_numbers,k1_numbers,k2_rfunct_1(k1_numbers,k1_numbers,k6_seq_1(k1_numbers,k1_numbers,C,D),k7_seq_1(k1_numbers,k1_numbers,C,D)),k2_taylor_1(k8_power)),B),E) = k6_real_1(k4_real_1(k4_real_1(np__4,A),E),k5_real_1(k3_prepower(A,np__2),k3_prepower(E,np__4))) ) ) ) ) ) ) ) ) ).
fof(t24_fdiff_5,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) )
=> ! [D] :
( ( v1_funct_1(D)
& m2_relset_1(D,k1_numbers,k1_numbers) )
=> ( ( r1_tarski(A,k1_relat_1(k8_seq_1(k1_numbers,k1_numbers,k4_rfunct_1(k1_numbers,k1_numbers,B),C)))
& ! [E] :
( m1_subset_1(E,k1_numbers)
=> ( r2_hidden(E,A)
=> k2_seq_1(k1_numbers,k1_numbers,B,E) = E ) )
& C = k1_partfun1(k1_numbers,k1_numbers,k1_numbers,k1_numbers,D,k2_taylor_1(k8_power))
& ! [E] :
( m1_subset_1(E,k1_numbers)
=> ( r2_hidden(E,A)
=> ( k2_seq_1(k1_numbers,k1_numbers,D,E) = E
& ~ r1_xreal_0(k2_seq_1(k1_numbers,k1_numbers,D,E),np__0) ) ) ) )
=> ( r2_fdiff_1(k8_seq_1(k1_numbers,k1_numbers,k4_rfunct_1(k1_numbers,k1_numbers,B),C),A)
& ! [E] :
( m1_subset_1(E,k1_numbers)
=> ( r2_hidden(E,A)
=> k2_seq_1(k1_numbers,k1_numbers,k2_fdiff_1(k8_seq_1(k1_numbers,k1_numbers,k4_rfunct_1(k1_numbers,k1_numbers,B),C),A),E) = k4_real_1(k6_real_1(np__1,k7_square_1(E)),k5_real_1(np__1,k2_seq_1(k1_numbers,k1_numbers,k2_taylor_1(k8_power),E))) ) ) ) ) ) ) ) ) ).
fof(t25_fdiff_5,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) )
=> ( ( r1_tarski(A,k1_relat_1(k4_rfunct_1(k1_numbers,k1_numbers,B)))
& B = k1_partfun1(k1_numbers,k1_numbers,k1_numbers,k1_numbers,C,k2_taylor_1(k8_power))
& ! [D] :
( m1_subset_1(D,k1_numbers)
=> ( r2_hidden(D,A)
=> ( k2_seq_1(k1_numbers,k1_numbers,C,D) = D
& ~ r1_xreal_0(k2_seq_1(k1_numbers,k1_numbers,C,D),np__0) ) ) )
& ! [D] :
( m1_subset_1(D,k1_numbers)
=> ~ ( r2_hidden(D,A)
& k2_seq_1(k1_numbers,k1_numbers,B,D) = np__0 ) ) )
=> ( r2_fdiff_1(k4_rfunct_1(k1_numbers,k1_numbers,B),A)
& ! [D] :
( m1_subset_1(D,k1_numbers)
=> ( r2_hidden(D,A)
=> k2_seq_1(k1_numbers,k1_numbers,k2_fdiff_1(k4_rfunct_1(k1_numbers,k1_numbers,B),A),D) = k1_real_1(k6_real_1(np__1,k4_real_1(D,k7_square_1(k2_seq_1(k1_numbers,k1_numbers,k2_taylor_1(k8_power),D))))) ) ) ) ) ) ) ) ).
%------------------------------------------------------------------------------