SET007 Axioms: SET007+193.ax
%------------------------------------------------------------------------------
% File : SET007+193 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Trigonometric Functions and Existence of Circle Ratio
% 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 : sin_cos [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 160 ( 13 unt; 0 def)
% Number of atoms : 680 ( 192 equ)
% Maximal formula atoms : 12 ( 4 avg)
% Number of connectives : 556 ( 36 ~; 0 |; 223 &)
% ( 21 <=>; 276 =>; 0 <=; 0 <~>)
% Maximal formula depth : 15 ( 6 avg)
% Maximal term depth : 8 ( 2 avg)
% Number of predicates : 25 ( 23 usr; 1 prp; 0-3 aty)
% Number of functors : 92 ( 92 usr; 18 con; 0-4 aty)
% Number of variables : 256 ( 253 !; 3 ?)
% 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_sin_cos,axiom,
! [A,B] :
( ( m1_subset_1(A,k5_numbers)
& m1_subset_1(B,k5_numbers) )
=> ( v1_xcmplx_0(k1_sin_cos(A,B))
& v1_xreal_0(k1_sin_cos(A,B)) ) ) ).
fof(fc2_sin_cos,axiom,
! [A] :
( m1_subset_1(A,k2_numbers)
=> ( ~ v1_xboole_0(k6_sin_cos(A))
& v1_relat_1(k6_sin_cos(A))
& v1_funct_1(k6_sin_cos(A))
& v1_funct_2(k6_sin_cos(A),k5_numbers,k2_numbers)
& v1_comseq_3(k6_sin_cos(A))
& v2_comseq_3(k6_sin_cos(A))
& v1_comseq_2(k6_sin_cos(A))
& v2_comseq_2(k6_sin_cos(A))
& v1_partfun1(k6_sin_cos(A),k5_numbers,k2_numbers) ) ) ).
fof(fc3_sin_cos,axiom,
! [A] :
( v1_xreal_0(A)
=> ( v1_xcmplx_0(k19_sin_cos(A))
& v1_xreal_0(k19_sin_cos(A)) ) ) ).
fof(fc4_sin_cos,axiom,
! [A] :
( v1_xreal_0(A)
=> ( v1_xcmplx_0(k22_sin_cos(A))
& v1_xreal_0(k22_sin_cos(A)) ) ) ).
fof(fc5_sin_cos,axiom,
! [A] :
( v1_xreal_0(A)
=> ( v1_xcmplx_0(k27_sin_cos(A))
& v1_xreal_0(k27_sin_cos(A)) ) ) ).
fof(d1_sin_cos,axiom,
$true ).
fof(d2_sin_cos,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( ( r1_xreal_0(A,B)
=> k1_sin_cos(A,B) = np__1 )
& ( ~ r1_xreal_0(A,B)
=> k1_sin_cos(A,B) = np__0 ) ) ) ) ).
fof(d3_sin_cos,axiom,
$true ).
fof(d4_sin_cos,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k2_numbers)
& m2_relset_1(A,k5_numbers,k2_numbers) )
=> ( A = k2_sin_cos
<=> ( k1_comseq_1(A,np__0) = np__1
& ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> k1_comseq_1(A,k1_nat_1(B,np__1)) = k3_xcmplx_0(k1_comseq_1(A,B),k1_nat_1(B,np__1)) ) ) ) ) ).
fof(d5_sin_cos,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& m2_relset_1(A,k5_numbers,k1_numbers) )
=> ( A = k3_sin_cos
<=> ( k2_seq_1(k5_numbers,k1_numbers,A,np__0) = np__1
& ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> k2_seq_1(k5_numbers,k1_numbers,A,k1_nat_1(B,np__1)) = k4_real_1(k2_seq_1(k5_numbers,k1_numbers,A,B),k1_nat_1(B,np__1)) ) ) ) ) ).
fof(d6_sin_cos,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> k4_sin_cos(A) = k1_comseq_1(k2_sin_cos,A) ) ).
fof(d7_sin_cos,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> k5_sin_cos(A) = k2_seq_1(k5_numbers,k1_numbers,k3_sin_cos,A) ) ).
fof(d8_sin_cos,axiom,
! [A] :
( m1_subset_1(A,k2_numbers)
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k2_numbers)
& m2_relset_1(B,k5_numbers,k2_numbers) )
=> ( B = k6_sin_cos(A)
<=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> k1_comseq_1(B,C) = k13_complex1(k2_comseq_3(A,C),k4_sin_cos(C)) ) ) ) ) ).
fof(d9_sin_cos,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_numbers)
& m2_relset_1(B,k5_numbers,k1_numbers) )
=> ( B = k7_sin_cos(A)
<=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> k2_seq_1(k5_numbers,k1_numbers,B,C) = k7_xcmplx_0(k2_newton(A,C),k5_sin_cos(C)) ) ) ) ) ).
fof(t1_sin_cos,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ( k4_sin_cos(np__0) = np__1
& k4_sin_cos(A) != np__0
& k4_sin_cos(k1_nat_1(A,np__1)) = k3_xcmplx_0(k4_sin_cos(A),k1_nat_1(A,np__1)) ) ) ).
fof(t2_sin_cos,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ( k5_sin_cos(A) != np__0
& k5_sin_cos(k1_nat_1(A,np__1)) = k4_real_1(k5_sin_cos(A),k1_nat_1(A,np__1)) ) ) ).
fof(t3_sin_cos,axiom,
( ! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ( ~ r1_xreal_0(A,np__0)
=> k3_xcmplx_0(k4_sin_cos(k5_binarith(A,np__1)),A) = k4_sin_cos(A) ) )
& ! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( r1_xreal_0(B,A)
=> k3_xcmplx_0(k4_sin_cos(k5_binarith(A,B)),k5_real_1(k1_nat_1(A,np__1),B)) = k4_sin_cos(k5_binarith(k1_nat_1(A,np__1),B)) ) ) ) ) ).
fof(d10_sin_cos,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k2_numbers)
& m2_relset_1(B,k5_numbers,k2_numbers) )
=> ( B = k8_sin_cos(A)
<=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( ( r1_xreal_0(C,A)
=> k1_comseq_1(B,C) = k13_complex1(k4_sin_cos(A),k9_complex1(k4_sin_cos(C),k4_sin_cos(k5_binarith(A,C)))) )
& ( ~ r1_xreal_0(C,A)
=> k1_comseq_1(B,C) = np__0 ) ) ) ) ) ) ).
fof(d11_sin_cos,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k2_numbers)
& m2_relset_1(B,k5_numbers,k2_numbers) )
=> ( B = k9_sin_cos(A)
<=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( ( r1_xreal_0(C,A)
=> k1_comseq_1(B,C) = k13_complex1(k6_complex1,k9_complex1(k4_sin_cos(C),k4_sin_cos(k5_binarith(A,C)))) )
& ( ~ r1_xreal_0(C,A)
=> k1_comseq_1(B,C) = np__0 ) ) ) ) ) ) ).
fof(d12_sin_cos,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k2_numbers)
& m2_relset_1(A,k5_numbers,k2_numbers) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k2_numbers)
& m2_relset_1(B,k5_numbers,k2_numbers) )
=> ( B = k10_sin_cos(A)
<=> ( k1_comseq_1(B,np__0) = np__0
& ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> k1_comseq_1(B,k1_nat_1(C,np__1)) = k1_comseq_1(A,C) ) ) ) ) ) ).
fof(d13_sin_cos,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,k2_numbers)
=> ! [C] :
( m1_subset_1(C,k2_numbers)
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k5_numbers,k2_numbers)
& m2_relset_1(D,k5_numbers,k2_numbers) )
=> ( D = k11_sin_cos(A,B,C)
<=> ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> ( ( r1_xreal_0(E,A)
=> k1_comseq_1(D,E) = k9_complex1(k9_complex1(k1_comseq_1(k8_sin_cos(A),E),k2_comseq_3(B,E)),k2_comseq_3(C,k5_binarith(A,E))) )
& ( ~ r1_xreal_0(E,A)
=> k1_comseq_1(D,E) = np__0 ) ) ) ) ) ) ) ) ).
fof(d14_sin_cos,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,k2_numbers)
=> ! [C] :
( m1_subset_1(C,k2_numbers)
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k5_numbers,k2_numbers)
& m2_relset_1(D,k5_numbers,k2_numbers) )
=> ( D = k12_sin_cos(A,B,C)
<=> ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> ( ( r1_xreal_0(E,A)
=> k1_comseq_1(D,E) = k9_complex1(k9_complex1(k1_comseq_1(k9_sin_cos(A),E),k2_comseq_3(B,E)),k2_comseq_3(C,k5_binarith(A,E))) )
& ( ~ r1_xreal_0(E,A)
=> k1_comseq_1(D,E) = np__0 ) ) ) ) ) ) ) ) ).
fof(d15_sin_cos,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,k2_numbers)
=> ! [C] :
( m1_subset_1(C,k2_numbers)
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k5_numbers,k2_numbers)
& m2_relset_1(D,k5_numbers,k2_numbers) )
=> ( D = k13_sin_cos(A,B,C)
<=> ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> ( ( r1_xreal_0(E,A)
=> k1_comseq_1(D,E) = k9_complex1(k1_comseq_1(k6_sin_cos(B),E),k1_comseq_1(k7_comseq_3(k6_sin_cos(C)),k5_binarith(A,E))) )
& ( ~ r1_xreal_0(E,A)
=> k1_comseq_1(D,E) = np__0 ) ) ) ) ) ) ) ) ).
fof(d16_sin_cos,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k5_numbers,k1_numbers)
& m2_relset_1(D,k5_numbers,k1_numbers) )
=> ( D = k14_sin_cos(A,B,C)
<=> ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> ( ( r1_xreal_0(E,C)
=> k2_seq_1(k5_numbers,k1_numbers,D,E) = k4_real_1(k2_seq_1(k5_numbers,k1_numbers,k7_sin_cos(A),E),k5_real_1(k2_seq_1(k5_numbers,k1_numbers,k1_series_1(k7_sin_cos(B)),C),k2_seq_1(k5_numbers,k1_numbers,k1_series_1(k7_sin_cos(B)),k5_binarith(C,E)))) )
& ( ~ r1_xreal_0(E,C)
=> k2_seq_1(k5_numbers,k1_numbers,D,E) = np__0 ) ) ) ) ) ) ) ) ).
fof(d17_sin_cos,axiom,
! [A] :
( m1_subset_1(A,k2_numbers)
=> ! [B] :
( m1_subset_1(B,k2_numbers)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k5_numbers,k2_numbers)
& m2_relset_1(D,k5_numbers,k2_numbers) )
=> ( D = k15_sin_cos(A,B,C)
<=> ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> ( ( r1_xreal_0(E,C)
=> k1_comseq_1(D,E) = k9_complex1(k1_comseq_1(k6_sin_cos(A),E),k11_complex1(k1_comseq_1(k7_comseq_3(k6_sin_cos(B)),C),k1_comseq_1(k7_comseq_3(k6_sin_cos(B)),k5_binarith(C,E)))) )
& ( ~ r1_xreal_0(E,C)
=> k1_comseq_1(D,E) = np__0 ) ) ) ) ) ) ) ) ).
fof(t4_sin_cos,axiom,
! [A] :
( m1_subset_1(A,k2_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( k1_comseq_1(k6_sin_cos(A),k1_nat_1(B,np__1)) = k7_xcmplx_0(k9_complex1(k1_comseq_1(k6_sin_cos(A),B),A),k2_xcmplx_0(k1_nat_1(B,np__1),k3_xcmplx_0(np__0,k7_complex1)))
& k1_comseq_1(k6_sin_cos(A),np__0) = np__1
& k17_complex1(k1_comseq_1(k6_sin_cos(A),B)) = k2_seq_1(k5_numbers,k1_numbers,k7_sin_cos(k17_complex1(A)),B) ) ) ) ).
fof(t5_sin_cos,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k2_numbers)
& m2_relset_1(B,k5_numbers,k2_numbers) )
=> ( ~ r1_xreal_0(A,np__0)
=> k1_comseq_1(k10_sin_cos(B),A) = k1_comseq_1(B,k5_binarith(A,np__1)) ) ) ) ).
fof(t6_sin_cos,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k2_numbers)
& m2_relset_1(B,k5_numbers,k2_numbers) )
=> k1_comseq_1(k7_comseq_3(B),A) = k8_complex1(k1_comseq_1(k7_comseq_3(k10_sin_cos(B)),A),k1_comseq_1(B,A)) ) ) ).
fof(t7_sin_cos,axiom,
! [A] :
( m1_subset_1(A,k2_numbers)
=> ! [B] :
( m1_subset_1(B,k2_numbers)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> k2_comseq_3(k8_complex1(A,B),C) = k1_comseq_1(k7_comseq_3(k11_sin_cos(C,A,B)),C) ) ) ) ).
fof(t8_sin_cos,axiom,
! [A] :
( m1_subset_1(A,k2_numbers)
=> ! [B] :
( m1_subset_1(B,k2_numbers)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> k12_sin_cos(C,A,B) = k4_comseq_1(k5_numbers,k11_sin_cos(C,A,B),k13_complex1(k6_complex1,k4_sin_cos(C))) ) ) ) ).
fof(t9_sin_cos,axiom,
! [A] :
( m1_subset_1(A,k2_numbers)
=> ! [B] :
( m1_subset_1(B,k2_numbers)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> k13_complex1(k2_comseq_3(k8_complex1(A,B),C),k4_sin_cos(C)) = k1_comseq_1(k7_comseq_3(k12_sin_cos(C,A,B)),C) ) ) ) ).
fof(t10_sin_cos,axiom,
( v2_comseq_3(k6_sin_cos(k5_complex1))
& k8_comseq_3(k6_sin_cos(k5_complex1)) = k6_complex1 ) ).
fof(t11_sin_cos,axiom,
! [A] :
( m1_subset_1(A,k2_numbers)
=> ! [B] :
( m1_subset_1(B,k2_numbers)
=> ( k1_comseq_1(k6_sin_cos(A),np__0) = np__1
& k1_comseq_1(k11_sin_cos(np__0,A,B),np__0) = np__1 ) ) ) ).
fof(t12_sin_cos,axiom,
! [A] :
( m1_subset_1(A,k2_numbers)
=> ! [B] :
( m1_subset_1(B,k2_numbers)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ( r1_xreal_0(C,D)
=> k1_comseq_1(k13_sin_cos(k1_nat_1(D,np__1),A,B),C) = k8_complex1(k1_comseq_1(k13_sin_cos(D,A,B),C),k1_comseq_1(k12_sin_cos(k1_nat_1(D,np__1),A,B),C)) ) ) ) ) ) ).
fof(t13_sin_cos,axiom,
! [A] :
( m1_subset_1(A,k2_numbers)
=> ! [B] :
( m1_subset_1(B,k2_numbers)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> k1_comseq_1(k7_comseq_3(k13_sin_cos(k1_nat_1(C,np__1),A,B)),C) = k8_complex1(k1_comseq_1(k7_comseq_3(k13_sin_cos(C,A,B)),C),k1_comseq_1(k7_comseq_3(k12_sin_cos(k1_nat_1(C,np__1),A,B)),C)) ) ) ) ).
fof(t14_sin_cos,axiom,
! [A] :
( m1_subset_1(A,k2_numbers)
=> ! [B] :
( m1_subset_1(B,k2_numbers)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> k1_comseq_1(k6_sin_cos(A),C) = k1_comseq_1(k12_sin_cos(C,A,B),C) ) ) ) ).
fof(t15_sin_cos,axiom,
! [A] :
( m1_subset_1(A,k2_numbers)
=> ! [B] :
( m1_subset_1(B,k2_numbers)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> k1_comseq_1(k7_comseq_3(k6_sin_cos(k8_complex1(A,B))),C) = k1_comseq_1(k7_comseq_3(k13_sin_cos(C,A,B)),C) ) ) ) ).
fof(t16_sin_cos,axiom,
! [A] :
( m1_subset_1(A,k2_numbers)
=> ! [B] :
( m1_subset_1(B,k2_numbers)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> k11_complex1(k9_complex1(k1_comseq_1(k7_comseq_3(k6_sin_cos(A)),C),k1_comseq_1(k7_comseq_3(k6_sin_cos(B)),C)),k1_comseq_1(k7_comseq_3(k6_sin_cos(k8_complex1(A,B))),C)) = k1_comseq_1(k7_comseq_3(k15_sin_cos(A,B,C)),C) ) ) ) ).
fof(t17_sin_cos,axiom,
! [A] :
( m1_subset_1(A,k2_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( r1_xreal_0(k17_complex1(k1_comseq_1(k7_comseq_3(k6_sin_cos(A)),B)),k2_seq_1(k5_numbers,k1_numbers,k1_series_1(k7_sin_cos(k17_complex1(A))),B))
& r1_xreal_0(k2_seq_1(k5_numbers,k1_numbers,k1_series_1(k7_sin_cos(k17_complex1(A))),B),k2_series_1(k7_sin_cos(k17_complex1(A))))
& r1_xreal_0(k17_complex1(k1_comseq_1(k7_comseq_3(k6_sin_cos(A)),B)),k2_series_1(k7_sin_cos(k17_complex1(A)))) ) ) ) ).
fof(t18_sin_cos,axiom,
! [A] :
( m1_subset_1(A,k2_numbers)
=> r1_xreal_0(np__1,k2_series_1(k7_sin_cos(k17_complex1(A)))) ) ).
fof(t19_sin_cos,axiom,
! [A] :
( m1_subset_1(A,k2_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> r1_xreal_0(np__0,k2_seq_1(k5_numbers,k1_numbers,k7_sin_cos(k17_complex1(A)),B)) ) ) ).
fof(t20_sin_cos,axiom,
! [A] :
( m1_subset_1(A,k2_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( k18_complex1(k2_seq_1(k5_numbers,k1_numbers,k1_series_1(k7_sin_cos(k17_complex1(A))),B)) = k2_seq_1(k5_numbers,k1_numbers,k1_series_1(k7_sin_cos(k17_complex1(A))),B)
& ( r1_xreal_0(B,C)
=> k18_complex1(k5_real_1(k2_seq_1(k5_numbers,k1_numbers,k1_series_1(k7_sin_cos(k17_complex1(A))),C),k2_seq_1(k5_numbers,k1_numbers,k1_series_1(k7_sin_cos(k17_complex1(A))),B))) = k5_real_1(k2_seq_1(k5_numbers,k1_numbers,k1_series_1(k7_sin_cos(k17_complex1(A))),C),k2_seq_1(k5_numbers,k1_numbers,k1_series_1(k7_sin_cos(k17_complex1(A))),B)) ) ) ) ) ) ).
fof(t21_sin_cos,axiom,
! [A] :
( m1_subset_1(A,k2_numbers)
=> ! [B] :
( m1_subset_1(B,k2_numbers)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> k18_complex1(k2_seq_1(k5_numbers,k1_numbers,k1_series_1(k9_comseq_1(k5_numbers,k15_sin_cos(A,B,C))),D)) = k2_seq_1(k5_numbers,k1_numbers,k1_series_1(k9_comseq_1(k5_numbers,k15_sin_cos(A,B,C))),D) ) ) ) ) ).
fof(t22_sin_cos,axiom,
! [A] :
( m1_subset_1(A,k2_numbers)
=> ! [B] :
( m1_subset_1(B,k2_numbers)
=> ! [C] :
( v1_xreal_0(C)
=> ~ ( ~ 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,k18_complex1(k2_seq_1(k5_numbers,k1_numbers,k1_series_1(k9_comseq_1(k5_numbers,k15_sin_cos(A,B,E))),E))) ) ) ) ) ) ) ).
fof(t23_sin_cos,axiom,
! [A] :
( m1_subset_1(A,k2_numbers)
=> ! [B] :
( m1_subset_1(B,k2_numbers)
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,k2_numbers)
& m2_relset_1(C,k5_numbers,k2_numbers) )
=> ( ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> k1_comseq_1(C,D) = k1_comseq_1(k7_comseq_3(k15_sin_cos(A,B,D)),D) )
=> ( v2_comseq_2(C)
& k2_comseq_2(C) = np__0 ) ) ) ) ) ).
fof(d18_sin_cos,axiom,
! [A] :
( ( v1_funct_1(A)
& m2_relset_1(A,k2_numbers,k2_numbers) )
=> ( A = k16_sin_cos
<=> ( k1_relat_1(A) = k2_numbers
& ! [B] :
( m1_subset_1(B,k2_numbers)
=> k1_funct_1(A,B) = k8_comseq_3(k6_sin_cos(B)) ) ) ) ) ).
fof(d19_sin_cos,axiom,
! [A] :
( m1_subset_1(A,k2_numbers)
=> k17_sin_cos(A) = k1_funct_1(k16_sin_cos,A) ) ).
fof(t24_sin_cos,axiom,
! [A] :
( m1_subset_1(A,k2_numbers)
=> ! [B] :
( m1_subset_1(B,k2_numbers)
=> k17_sin_cos(k8_complex1(A,B)) = k9_complex1(k17_sin_cos(A),k17_sin_cos(B)) ) ) ).
fof(d20_sin_cos,axiom,
! [A] :
( ( v1_funct_1(A)
& m2_relset_1(A,k1_numbers,k1_numbers) )
=> ( A = k18_sin_cos
<=> ( k1_relat_1(A) = k1_numbers
& ! [B] :
( m1_subset_1(B,k1_numbers)
=> k2_seq_1(k1_numbers,k1_numbers,A,B) = k4_complex1(k8_comseq_3(k6_sin_cos(k5_arytm_0(np__0,B)))) ) ) ) ) ).
fof(d21_sin_cos,axiom,
! [A] :
( v1_xreal_0(A)
=> k19_sin_cos(A) = k2_seq_1(k1_numbers,k1_numbers,k18_sin_cos,A) ) ).
fof(t25_sin_cos,axiom,
( v1_funct_1(k18_sin_cos)
& v1_funct_2(k18_sin_cos,k1_numbers,k1_numbers)
& m2_relset_1(k18_sin_cos,k1_numbers,k1_numbers) ) ).
fof(d22_sin_cos,axiom,
! [A] :
( ( v1_funct_1(A)
& m2_relset_1(A,k1_numbers,k1_numbers) )
=> ( A = k21_sin_cos
<=> ( k1_relat_1(A) = k1_numbers
& ! [B] :
( m1_subset_1(B,k1_numbers)
=> k2_seq_1(k1_numbers,k1_numbers,A,B) = k3_complex1(k8_comseq_3(k6_sin_cos(k5_arytm_0(np__0,B)))) ) ) ) ) ).
fof(d23_sin_cos,axiom,
! [A] :
( v1_xreal_0(A)
=> k22_sin_cos(A) = k2_seq_1(k1_numbers,k1_numbers,k21_sin_cos,A) ) ).
fof(t26_sin_cos,axiom,
( v1_funct_1(k21_sin_cos)
& v1_funct_2(k21_sin_cos,k1_numbers,k1_numbers)
& m2_relset_1(k21_sin_cos,k1_numbers,k1_numbers) ) ).
fof(t27_sin_cos,axiom,
( k1_relat_1(k18_sin_cos) = k1_numbers
& k1_relat_1(k21_sin_cos) = k1_numbers ) ).
fof(t28_sin_cos,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> k17_sin_cos(k5_arytm_0(np__0,A)) = k2_xcmplx_0(k23_sin_cos(A),k3_xcmplx_0(k20_sin_cos(A),k7_complex1)) ) ).
fof(t29_sin_cos,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> k15_complex1(k17_sin_cos(k5_arytm_0(np__0,A))) = k17_sin_cos(k10_complex1(k5_arytm_0(np__0,A))) ) ).
fof(t30_sin_cos,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ( k17_complex1(k17_sin_cos(k5_arytm_0(np__0,A))) = np__1
& ! [B] :
( v1_xreal_0(B)
=> ( r1_xreal_0(k18_complex1(k19_sin_cos(B)),np__1)
& r1_xreal_0(k18_complex1(k22_sin_cos(B)),np__1) ) ) ) ) ).
fof(t31_sin_cos,axiom,
! [A] :
( v1_xreal_0(A)
=> ( k3_real_1(k7_square_1(k2_seq_1(k1_numbers,k1_numbers,k21_sin_cos,A)),k7_square_1(k2_seq_1(k1_numbers,k1_numbers,k18_sin_cos,A))) = np__1
& k3_real_1(k4_real_1(k2_seq_1(k1_numbers,k1_numbers,k21_sin_cos,A),k2_seq_1(k1_numbers,k1_numbers,k21_sin_cos,A)),k4_real_1(k2_seq_1(k1_numbers,k1_numbers,k18_sin_cos,A),k2_seq_1(k1_numbers,k1_numbers,k18_sin_cos,A))) = np__1 ) ) ).
fof(t32_sin_cos,axiom,
! [A] :
( v1_xreal_0(A)
=> ( k2_xcmplx_0(k5_square_1(k22_sin_cos(A)),k5_square_1(k19_sin_cos(A))) = np__1
& k2_xcmplx_0(k3_xcmplx_0(k22_sin_cos(A),k22_sin_cos(A)),k3_xcmplx_0(k19_sin_cos(A),k19_sin_cos(A))) = np__1 ) ) ).
fof(t33_sin_cos,axiom,
! [A] :
( v1_xreal_0(A)
=> ( k2_seq_1(k1_numbers,k1_numbers,k21_sin_cos,np__0) = np__1
& k2_seq_1(k1_numbers,k1_numbers,k18_sin_cos,np__0) = np__0
& k2_seq_1(k1_numbers,k1_numbers,k21_sin_cos,k4_xcmplx_0(A)) = k2_seq_1(k1_numbers,k1_numbers,k21_sin_cos,A)
& k2_seq_1(k1_numbers,k1_numbers,k18_sin_cos,k4_xcmplx_0(A)) = k1_real_1(k2_seq_1(k1_numbers,k1_numbers,k18_sin_cos,A)) ) ) ).
fof(t34_sin_cos,axiom,
! [A] :
( v1_xreal_0(A)
=> ( k23_sin_cos(np__0) = np__1
& k20_sin_cos(np__0) = np__0
& k22_sin_cos(k4_xcmplx_0(A)) = k22_sin_cos(A)
& k19_sin_cos(k4_xcmplx_0(A)) = k4_xcmplx_0(k19_sin_cos(A)) ) ) ).
fof(d24_sin_cos,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_numbers)
& m2_relset_1(B,k5_numbers,k1_numbers) )
=> ( B = k24_sin_cos(A)
<=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> k2_seq_1(k5_numbers,k1_numbers,B,C) = k7_xcmplx_0(k3_xcmplx_0(k3_prepower(k1_real_1(np__1),C),k2_newton(A,k1_nat_1(k2_nat_1(np__2,C),np__1))),k5_sin_cos(k1_nat_1(k2_nat_1(np__2,C),np__1))) ) ) ) ) ).
fof(d25_sin_cos,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_numbers)
& m2_relset_1(B,k5_numbers,k1_numbers) )
=> ( B = k25_sin_cos(A)
<=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> k2_seq_1(k5_numbers,k1_numbers,B,C) = k7_xcmplx_0(k3_xcmplx_0(k3_prepower(k1_real_1(np__1),C),k2_newton(A,k2_nat_1(np__2,C))),k5_sin_cos(k2_nat_1(np__2,C))) ) ) ) ) ).
fof(t35_sin_cos,axiom,
! [A] :
( m1_subset_1(A,k2_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( k2_comseq_3(A,k2_nat_1(np__2,B)) = k2_comseq_3(k2_comseq_3(A,B),np__2)
& k2_comseq_3(A,k2_nat_1(np__2,B)) = k2_comseq_3(k2_comseq_3(A,np__2),B) ) ) ) ).
fof(t36_sin_cos,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ( k2_comseq_3(k5_arytm_0(np__0,B),k2_nat_1(np__2,A)) = k5_arytm_0(k4_real_1(k3_prepower(k1_real_1(np__1),A),k3_prepower(B,k2_nat_1(np__2,A))),np__0)
& k2_comseq_3(k5_arytm_0(np__0,B),k1_nat_1(k2_nat_1(np__2,A),np__1)) = k5_arytm_0(np__0,k4_real_1(k3_prepower(k1_real_1(np__1),A),k3_prepower(B,k1_nat_1(k2_nat_1(np__2,A),np__1)))) ) ) ) ).
fof(t37_sin_cos,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> k4_sin_cos(A) = k5_arytm_0(k5_sin_cos(A),np__0) ) ).
fof(t38_sin_cos,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ( k2_seq_1(k5_numbers,k1_numbers,k1_series_1(k24_sin_cos(B)),A) = k2_seq_1(k5_numbers,k1_numbers,k1_series_1(k4_comseq_3(k6_sin_cos(k5_arytm_0(np__0,B)))),k1_nat_1(k2_nat_1(np__2,A),np__1))
& k2_seq_1(k5_numbers,k1_numbers,k1_series_1(k25_sin_cos(B)),A) = k2_seq_1(k5_numbers,k1_numbers,k1_series_1(k3_comseq_3(k6_sin_cos(k5_arytm_0(np__0,B)))),k2_nat_1(np__2,A)) ) ) ) ).
fof(t39_sin_cos,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ( v4_seq_2(k1_series_1(k24_sin_cos(A)))
& k2_series_1(k24_sin_cos(A)) = k4_complex1(k8_comseq_3(k6_sin_cos(k5_arytm_0(np__0,A))))
& v4_seq_2(k1_series_1(k25_sin_cos(A)))
& k2_series_1(k25_sin_cos(A)) = k3_complex1(k8_comseq_3(k6_sin_cos(k5_arytm_0(np__0,A)))) ) ) ).
fof(t40_sin_cos,axiom,
! [A] :
( v1_xreal_0(A)
=> ( k2_seq_1(k1_numbers,k1_numbers,k21_sin_cos,A) = k2_series_1(k25_sin_cos(A))
& k2_seq_1(k1_numbers,k1_numbers,k18_sin_cos,A) = k2_series_1(k24_sin_cos(A)) ) ) ).
fof(t41_sin_cos,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,k1_numbers)
& m2_relset_1(C,k5_numbers,k1_numbers) )
=> ( ( v4_seq_2(C)
& k2_seq_2(C) = B
& ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> r1_xreal_0(A,k2_seq_1(k5_numbers,k1_numbers,C,D)) ) )
=> r1_xreal_0(A,B) ) ) ) ) ).
fof(t42_sin_cos,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( ~ r1_xreal_0(B,A)
=> ( ~ r1_xreal_0(k5_sin_cos(C),np__0)
& r1_xreal_0(k5_sin_cos(A),k5_sin_cos(B)) ) ) ) ) ) ).
fof(t43_sin_cos,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( ( r1_xreal_0(np__0,A)
& r1_xreal_0(A,np__1)
& r1_xreal_0(B,C) )
=> r1_xreal_0(k2_newton(A,C),k2_newton(A,B)) ) ) ) ) ).
fof(t44_sin_cos,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> k2_comseq_3(k5_arytm_0(B,np__0),A) = k5_arytm_0(k3_prepower(B,A),np__0) ) ) ).
fof(t45_sin_cos,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> k13_complex1(k2_comseq_3(k5_arytm_0(B,np__0),A),k4_sin_cos(A)) = k5_arytm_0(k6_real_1(k3_prepower(B,A),k5_sin_cos(A)),np__0) ) ) ).
fof(t46_sin_cos,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> k4_complex1(k8_comseq_3(k6_sin_cos(k5_arytm_0(A,np__0)))) = np__0 ) ).
fof(t47_sin_cos,axiom,
( ~ r1_xreal_0(k2_seq_1(k1_numbers,k1_numbers,k21_sin_cos,np__1),np__0)
& ~ r1_xreal_0(k2_seq_1(k1_numbers,k1_numbers,k18_sin_cos,np__1),np__0)
& ~ r1_xreal_0(k2_seq_1(k1_numbers,k1_numbers,k18_sin_cos,np__1),k2_seq_1(k1_numbers,k1_numbers,k21_sin_cos,np__1)) ) ).
fof(t48_sin_cos,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> k7_sin_cos(A) = k3_comseq_3(k6_sin_cos(k5_arytm_0(A,np__0))) ) ).
fof(t49_sin_cos,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ( v1_series_1(k7_sin_cos(A))
& k2_series_1(k7_sin_cos(A)) = k3_complex1(k8_comseq_3(k6_sin_cos(k5_arytm_0(A,np__0)))) ) ) ).
fof(t50_sin_cos,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> k2_series_1(k7_sin_cos(k2_xcmplx_0(A,B))) = k4_real_1(k2_series_1(k7_sin_cos(A)),k2_series_1(k7_sin_cos(B))) ) ) ).
fof(d26_sin_cos,axiom,
! [A] :
( ( v1_funct_1(A)
& m2_relset_1(A,k1_numbers,k1_numbers) )
=> ( A = k26_sin_cos
<=> ( k1_relat_1(A) = k1_numbers
& ! [B] :
( v1_xreal_0(B)
=> k2_seq_1(k1_numbers,k1_numbers,A,B) = k2_series_1(k7_sin_cos(B)) ) ) ) ) ).
fof(d27_sin_cos,axiom,
! [A] :
( v1_xreal_0(A)
=> k27_sin_cos(A) = k2_seq_1(k1_numbers,k1_numbers,k26_sin_cos,A) ) ).
fof(t51_sin_cos,axiom,
k1_relat_1(k26_sin_cos) = k1_numbers ).
fof(t52_sin_cos,axiom,
$true ).
fof(t53_sin_cos,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> k2_seq_1(k1_numbers,k1_numbers,k26_sin_cos,A) = k3_complex1(k8_comseq_3(k6_sin_cos(k5_arytm_0(A,np__0)))) ) ).
fof(t54_sin_cos,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> k17_sin_cos(k5_arytm_0(A,np__0)) = k5_arytm_0(k28_sin_cos(A),np__0) ) ).
fof(t55_sin_cos,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> k27_sin_cos(k2_xcmplx_0(A,B)) = k3_xcmplx_0(k27_sin_cos(A),k27_sin_cos(B)) ) ) ).
fof(t56_sin_cos,axiom,
k28_sin_cos(np__0) = np__1 ).
fof(t57_sin_cos,axiom,
! [A] :
( v1_xreal_0(A)
=> ( ~ r1_xreal_0(A,np__0)
=> r1_xreal_0(np__1,k2_seq_1(k1_numbers,k1_numbers,k26_sin_cos,A)) ) ) ).
fof(t58_sin_cos,axiom,
! [A] :
( v1_xreal_0(A)
=> ( ~ r1_xreal_0(np__0,A)
=> ( ~ r1_xreal_0(k2_seq_1(k1_numbers,k1_numbers,k26_sin_cos,A),np__0)
& r1_xreal_0(k2_seq_1(k1_numbers,k1_numbers,k26_sin_cos,A),np__1) ) ) ) ).
fof(t59_sin_cos,axiom,
! [A] :
( v1_xreal_0(A)
=> ~ r1_xreal_0(k2_seq_1(k1_numbers,k1_numbers,k26_sin_cos,A),np__0) ) ).
fof(t60_sin_cos,axiom,
! [A] :
( v1_xreal_0(A)
=> ~ r1_xreal_0(k27_sin_cos(A),np__0) ) ).
fof(d28_sin_cos,axiom,
! [A] :
( m1_subset_1(A,k2_numbers)
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k2_numbers)
& m2_relset_1(B,k5_numbers,k2_numbers) )
=> ( B = k29_sin_cos(A)
<=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> k1_comseq_1(B,C) = k13_complex1(k2_comseq_3(A,k1_nat_1(C,np__1)),k4_sin_cos(k1_nat_1(C,np__2))) ) ) ) ) ).
fof(d29_sin_cos,axiom,
! [A] :
( m1_subset_1(A,k2_numbers)
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k2_numbers)
& m2_relset_1(B,k5_numbers,k2_numbers) )
=> ( B = k30_sin_cos(A)
<=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> k1_comseq_1(B,C) = k13_complex1(k2_comseq_3(A,C),k4_sin_cos(k1_nat_1(C,np__2))) ) ) ) ) ).
fof(t61_sin_cos,axiom,
! [A] :
( m1_subset_1(A,k2_numbers)
=> v2_comseq_3(k29_sin_cos(A)) ) ).
fof(t62_sin_cos,axiom,
! [A] :
( m1_subset_1(A,k2_numbers)
=> k9_complex1(A,k8_comseq_3(k29_sin_cos(A))) = k11_complex1(k11_complex1(k8_comseq_3(k6_sin_cos(A)),k6_complex1),A) ) ).
fof(t63_sin_cos,axiom,
! [A] :
( v1_xreal_0(A)
=> ~ ( ~ r1_xreal_0(A,np__0)
& ! [B] :
( m1_subset_1(B,k1_numbers)
=> ~ ( ~ r1_xreal_0(B,np__0)
& ! [C] :
( m1_subset_1(C,k2_numbers)
=> ~ ( ~ r1_xreal_0(B,k17_complex1(C))
& r1_xreal_0(A,k17_complex1(k8_comseq_3(k29_sin_cos(C)))) ) ) ) ) ) ) ).
fof(t64_sin_cos,axiom,
! [A] :
( m1_subset_1(A,k2_numbers)
=> ! [B] :
( m1_subset_1(B,k2_numbers)
=> k11_complex1(k8_comseq_3(k6_sin_cos(k8_complex1(B,A))),k8_comseq_3(k6_sin_cos(B))) = k8_complex1(k9_complex1(k8_comseq_3(k6_sin_cos(B)),A),k9_complex1(k9_complex1(A,k8_comseq_3(k29_sin_cos(A))),k8_comseq_3(k6_sin_cos(B)))) ) ) ).
fof(t65_sin_cos,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> k5_real_1(k2_seq_1(k1_numbers,k1_numbers,k21_sin_cos,k3_real_1(A,B)),k2_seq_1(k1_numbers,k1_numbers,k21_sin_cos,A)) = k5_real_1(k1_real_1(k4_real_1(B,k2_seq_1(k1_numbers,k1_numbers,k18_sin_cos,A))),k4_real_1(B,k4_complex1(k9_complex1(k8_comseq_3(k29_sin_cos(k5_arytm_0(np__0,B))),k5_arytm_0(k2_seq_1(k1_numbers,k1_numbers,k21_sin_cos,A),k2_seq_1(k1_numbers,k1_numbers,k18_sin_cos,A)))))) ) ) ).
fof(t66_sin_cos,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> k5_real_1(k2_seq_1(k1_numbers,k1_numbers,k18_sin_cos,k3_real_1(A,B)),k2_seq_1(k1_numbers,k1_numbers,k18_sin_cos,A)) = k3_real_1(k4_real_1(B,k2_seq_1(k1_numbers,k1_numbers,k21_sin_cos,A)),k4_real_1(B,k3_complex1(k9_complex1(k8_comseq_3(k29_sin_cos(k5_arytm_0(np__0,B))),k5_arytm_0(k2_seq_1(k1_numbers,k1_numbers,k21_sin_cos,A),k2_seq_1(k1_numbers,k1_numbers,k18_sin_cos,A)))))) ) ) ).
fof(t67_sin_cos,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> k5_real_1(k2_seq_1(k1_numbers,k1_numbers,k26_sin_cos,k3_real_1(A,B)),k2_seq_1(k1_numbers,k1_numbers,k26_sin_cos,A)) = k3_real_1(k4_real_1(B,k2_seq_1(k1_numbers,k1_numbers,k26_sin_cos,A)),k4_real_1(k4_real_1(B,k2_seq_1(k1_numbers,k1_numbers,k26_sin_cos,A)),k3_complex1(k8_comseq_3(k29_sin_cos(k5_arytm_0(B,np__0)))))) ) ) ).
fof(t68_sin_cos,axiom,
! [A] :
( v1_xreal_0(A)
=> ( r1_fdiff_1(k21_sin_cos,A)
& k1_fdiff_1(k21_sin_cos,A) = k1_real_1(k2_seq_1(k1_numbers,k1_numbers,k18_sin_cos,A)) ) ) ).
fof(t69_sin_cos,axiom,
! [A] :
( v1_xreal_0(A)
=> ( r1_fdiff_1(k18_sin_cos,A)
& k1_fdiff_1(k18_sin_cos,A) = k2_seq_1(k1_numbers,k1_numbers,k21_sin_cos,A) ) ) ).
fof(t70_sin_cos,axiom,
! [A] :
( v1_xreal_0(A)
=> ( r1_fdiff_1(k26_sin_cos,A)
& k1_fdiff_1(k26_sin_cos,A) = k2_seq_1(k1_numbers,k1_numbers,k26_sin_cos,A) ) ) ).
fof(t71_sin_cos,axiom,
( r2_fdiff_1(k26_sin_cos,k1_numbers)
& ! [A] :
( v1_xreal_0(A)
=> ( r2_hidden(A,k1_numbers)
=> k1_fdiff_1(k26_sin_cos,A) = k2_seq_1(k1_numbers,k1_numbers,k26_sin_cos,A) ) ) ) ).
fof(t72_sin_cos,axiom,
( r2_fdiff_1(k21_sin_cos,k1_numbers)
& ! [A] :
( v1_xreal_0(A)
=> ( r2_hidden(A,k1_numbers)
=> k1_fdiff_1(k21_sin_cos,A) = k1_real_1(k2_seq_1(k1_numbers,k1_numbers,k18_sin_cos,A)) ) ) ) ).
fof(t73_sin_cos,axiom,
! [A] :
( v1_xreal_0(A)
=> ( r2_fdiff_1(k18_sin_cos,k1_numbers)
& k1_fdiff_1(k18_sin_cos,A) = k2_seq_1(k1_numbers,k1_numbers,k21_sin_cos,A) ) ) ).
fof(t74_sin_cos,axiom,
! [A] :
( v1_xreal_0(A)
=> ( r2_hidden(A,k1_rcomp_1(np__0,np__1))
=> ( ~ r1_xreal_0(k2_seq_1(k1_numbers,k1_numbers,k21_sin_cos,A),np__0)
& r1_xreal_0(k6_real_1(np__1,np__2),k2_seq_1(k1_numbers,k1_numbers,k21_sin_cos,A)) ) ) ) ).
fof(t75_sin_cos,axiom,
( r1_tarski(k1_rcomp_1(np__0,np__1),k1_relat_1(k2_rfunct_1(k1_numbers,k1_numbers,k18_sin_cos,k21_sin_cos)))
& r1_tarski(k2_rcomp_1(np__0,np__1),k1_relat_1(k2_rfunct_1(k1_numbers,k1_numbers,k18_sin_cos,k21_sin_cos))) ) ).
fof(t76_sin_cos,axiom,
r2_fcont_1(k2_rfunct_1(k1_numbers,k1_numbers,k18_sin_cos,k21_sin_cos),k1_rcomp_1(np__0,np__1)) ).
fof(t77_sin_cos,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ( ( r2_hidden(A,k2_rcomp_1(np__0,np__1))
& r2_hidden(B,k2_rcomp_1(np__0,np__1))
& k2_seq_1(k1_numbers,k1_numbers,k2_rfunct_1(k1_numbers,k1_numbers,k18_sin_cos,k21_sin_cos),A) = k2_seq_1(k1_numbers,k1_numbers,k2_rfunct_1(k1_numbers,k1_numbers,k18_sin_cos,k21_sin_cos),B) )
=> A = B ) ) ) ).
fof(d30_sin_cos,axiom,
! [A] :
( v1_xreal_0(A)
=> ( A = k31_sin_cos
<=> ( k2_seq_1(k1_numbers,k1_numbers,k2_rfunct_1(k1_numbers,k1_numbers,k18_sin_cos,k21_sin_cos),k7_xcmplx_0(A,np__4)) = np__1
& r2_hidden(A,k2_rcomp_1(np__0,np__4)) ) ) ) ).
fof(t78_sin_cos,axiom,
k2_seq_1(k1_numbers,k1_numbers,k18_sin_cos,k6_real_1(k32_sin_cos,np__4)) = k2_seq_1(k1_numbers,k1_numbers,k21_sin_cos,k6_real_1(k32_sin_cos,np__4)) ).
fof(t79_sin_cos,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ( k2_seq_1(k1_numbers,k1_numbers,k18_sin_cos,k2_xcmplx_0(A,B)) = k3_real_1(k4_real_1(k2_seq_1(k1_numbers,k1_numbers,k18_sin_cos,A),k2_seq_1(k1_numbers,k1_numbers,k21_sin_cos,B)),k4_real_1(k2_seq_1(k1_numbers,k1_numbers,k21_sin_cos,A),k2_seq_1(k1_numbers,k1_numbers,k18_sin_cos,B)))
& k2_seq_1(k1_numbers,k1_numbers,k21_sin_cos,k2_xcmplx_0(A,B)) = k5_real_1(k4_real_1(k2_seq_1(k1_numbers,k1_numbers,k21_sin_cos,A),k2_seq_1(k1_numbers,k1_numbers,k21_sin_cos,B)),k4_real_1(k2_seq_1(k1_numbers,k1_numbers,k18_sin_cos,A),k2_seq_1(k1_numbers,k1_numbers,k18_sin_cos,B))) ) ) ) ).
fof(t80_sin_cos,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ( k19_sin_cos(k2_xcmplx_0(A,B)) = k2_xcmplx_0(k3_xcmplx_0(k19_sin_cos(A),k22_sin_cos(B)),k3_xcmplx_0(k22_sin_cos(A),k19_sin_cos(B)))
& k22_sin_cos(k2_xcmplx_0(A,B)) = k6_xcmplx_0(k3_xcmplx_0(k22_sin_cos(A),k22_sin_cos(B)),k3_xcmplx_0(k19_sin_cos(A),k19_sin_cos(B))) ) ) ) ).
fof(t81_sin_cos,axiom,
( k2_seq_1(k1_numbers,k1_numbers,k21_sin_cos,k6_real_1(k32_sin_cos,np__2)) = np__0
& k2_seq_1(k1_numbers,k1_numbers,k18_sin_cos,k6_real_1(k32_sin_cos,np__2)) = np__1
& k2_seq_1(k1_numbers,k1_numbers,k21_sin_cos,k32_sin_cos) = k1_real_1(np__1)
& k2_seq_1(k1_numbers,k1_numbers,k18_sin_cos,k32_sin_cos) = np__0
& k2_seq_1(k1_numbers,k1_numbers,k21_sin_cos,k3_real_1(k32_sin_cos,k6_real_1(k32_sin_cos,np__2))) = np__0
& k2_seq_1(k1_numbers,k1_numbers,k18_sin_cos,k3_real_1(k32_sin_cos,k6_real_1(k32_sin_cos,np__2))) = k1_real_1(np__1)
& k2_seq_1(k1_numbers,k1_numbers,k21_sin_cos,k4_real_1(np__2,k32_sin_cos)) = np__1
& k2_seq_1(k1_numbers,k1_numbers,k18_sin_cos,k4_real_1(np__2,k32_sin_cos)) = np__0 ) ).
fof(t82_sin_cos,axiom,
( k23_sin_cos(k6_real_1(k32_sin_cos,np__2)) = np__0
& k20_sin_cos(k6_real_1(k32_sin_cos,np__2)) = np__1
& k23_sin_cos(k32_sin_cos) = k1_real_1(np__1)
& k20_sin_cos(k32_sin_cos) = np__0
& k23_sin_cos(k3_real_1(k32_sin_cos,k6_real_1(k32_sin_cos,np__2))) = np__0
& k20_sin_cos(k3_real_1(k32_sin_cos,k6_real_1(k32_sin_cos,np__2))) = k1_real_1(np__1)
& k23_sin_cos(k4_real_1(np__2,k32_sin_cos)) = np__1
& k20_sin_cos(k4_real_1(np__2,k32_sin_cos)) = np__0 ) ).
fof(t83_sin_cos,axiom,
! [A] :
( v1_xreal_0(A)
=> ( k2_seq_1(k1_numbers,k1_numbers,k18_sin_cos,k2_xcmplx_0(A,k4_real_1(np__2,k32_sin_cos))) = k2_seq_1(k1_numbers,k1_numbers,k18_sin_cos,A)
& k2_seq_1(k1_numbers,k1_numbers,k21_sin_cos,k2_xcmplx_0(A,k4_real_1(np__2,k32_sin_cos))) = k2_seq_1(k1_numbers,k1_numbers,k21_sin_cos,A)
& k2_seq_1(k1_numbers,k1_numbers,k18_sin_cos,k6_xcmplx_0(k6_real_1(k32_sin_cos,np__2),A)) = k2_seq_1(k1_numbers,k1_numbers,k21_sin_cos,A)
& k2_seq_1(k1_numbers,k1_numbers,k21_sin_cos,k6_xcmplx_0(k6_real_1(k32_sin_cos,np__2),A)) = k2_seq_1(k1_numbers,k1_numbers,k18_sin_cos,A)
& k2_seq_1(k1_numbers,k1_numbers,k18_sin_cos,k2_xcmplx_0(k6_real_1(k32_sin_cos,np__2),A)) = k2_seq_1(k1_numbers,k1_numbers,k21_sin_cos,A)
& k2_seq_1(k1_numbers,k1_numbers,k21_sin_cos,k2_xcmplx_0(k6_real_1(k32_sin_cos,np__2),A)) = k1_real_1(k2_seq_1(k1_numbers,k1_numbers,k18_sin_cos,A))
& k2_seq_1(k1_numbers,k1_numbers,k18_sin_cos,k2_xcmplx_0(k32_sin_cos,A)) = k1_real_1(k2_seq_1(k1_numbers,k1_numbers,k18_sin_cos,A))
& k2_seq_1(k1_numbers,k1_numbers,k21_sin_cos,k2_xcmplx_0(k32_sin_cos,A)) = k1_real_1(k2_seq_1(k1_numbers,k1_numbers,k21_sin_cos,A)) ) ) ).
fof(t84_sin_cos,axiom,
! [A] :
( v1_xreal_0(A)
=> ( k19_sin_cos(k2_xcmplx_0(A,k4_real_1(np__2,k32_sin_cos))) = k19_sin_cos(A)
& k22_sin_cos(k2_xcmplx_0(A,k4_real_1(np__2,k32_sin_cos))) = k22_sin_cos(A)
& k19_sin_cos(k6_xcmplx_0(k6_real_1(k32_sin_cos,np__2),A)) = k22_sin_cos(A)
& k22_sin_cos(k6_xcmplx_0(k6_real_1(k32_sin_cos,np__2),A)) = k19_sin_cos(A)
& k19_sin_cos(k2_xcmplx_0(k6_real_1(k32_sin_cos,np__2),A)) = k22_sin_cos(A)
& k22_sin_cos(k2_xcmplx_0(k6_real_1(k32_sin_cos,np__2),A)) = k4_xcmplx_0(k19_sin_cos(A))
& k19_sin_cos(k2_xcmplx_0(k32_sin_cos,A)) = k4_xcmplx_0(k19_sin_cos(A))
& k22_sin_cos(k2_xcmplx_0(k32_sin_cos,A)) = k4_xcmplx_0(k22_sin_cos(A)) ) ) ).
fof(t85_sin_cos,axiom,
! [A] :
( v1_xreal_0(A)
=> ~ ( r2_hidden(A,k2_rcomp_1(np__0,k6_real_1(k32_sin_cos,np__2)))
& r1_xreal_0(k2_seq_1(k1_numbers,k1_numbers,k21_sin_cos,A),np__0) ) ) ).
fof(t86_sin_cos,axiom,
! [A] :
( v1_xreal_0(A)
=> ~ ( r2_hidden(A,k2_rcomp_1(np__0,k6_real_1(k32_sin_cos,np__2)))
& r1_xreal_0(k22_sin_cos(A),np__0) ) ) ).
fof(s1_sin_cos,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ? [B] :
( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k2_numbers)
& m2_relset_1(B,k5_numbers,k2_numbers)
& ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( ( r1_xreal_0(C,A)
=> k1_comseq_1(B,C) = f1_s1_sin_cos(A,C) )
& ( ~ r1_xreal_0(C,A)
=> k1_comseq_1(B,C) = np__0 ) ) ) ) ) ).
fof(s2_sin_cos,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ? [B] :
( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_numbers)
& m2_relset_1(B,k5_numbers,k1_numbers)
& ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( ( r1_xreal_0(C,A)
=> k2_seq_1(k5_numbers,k1_numbers,B,C) = f1_s2_sin_cos(A,C) )
& ( ~ r1_xreal_0(C,A)
=> k2_seq_1(k5_numbers,k1_numbers,B,C) = np__0 ) ) ) ) ) ).
fof(dt_k1_sin_cos,axiom,
! [A,B] :
( ( m1_subset_1(A,k5_numbers)
& m1_subset_1(B,k5_numbers) )
=> m1_subset_1(k1_sin_cos(A,B),k2_numbers) ) ).
fof(dt_k2_sin_cos,axiom,
( v1_funct_1(k2_sin_cos)
& v1_funct_2(k2_sin_cos,k5_numbers,k2_numbers)
& m2_relset_1(k2_sin_cos,k5_numbers,k2_numbers) ) ).
fof(dt_k3_sin_cos,axiom,
( v1_funct_1(k3_sin_cos)
& v1_funct_2(k3_sin_cos,k5_numbers,k1_numbers)
& m2_relset_1(k3_sin_cos,k5_numbers,k1_numbers) ) ).
fof(dt_k4_sin_cos,axiom,
! [A] :
( m1_subset_1(A,k5_numbers)
=> m1_subset_1(k4_sin_cos(A),k2_numbers) ) ).
fof(dt_k5_sin_cos,axiom,
! [A] :
( m1_subset_1(A,k5_numbers)
=> m1_subset_1(k5_sin_cos(A),k1_numbers) ) ).
fof(redefinition_k5_sin_cos,axiom,
! [A] :
( m1_subset_1(A,k5_numbers)
=> k5_sin_cos(A) = k5_newton(A) ) ).
fof(dt_k6_sin_cos,axiom,
! [A] :
( m1_subset_1(A,k2_numbers)
=> ( v1_funct_1(k6_sin_cos(A))
& v1_funct_2(k6_sin_cos(A),k5_numbers,k2_numbers)
& m2_relset_1(k6_sin_cos(A),k5_numbers,k2_numbers) ) ) ).
fof(dt_k7_sin_cos,axiom,
! [A] :
( v1_xreal_0(A)
=> ( v1_funct_1(k7_sin_cos(A))
& v1_funct_2(k7_sin_cos(A),k5_numbers,k1_numbers)
& m2_relset_1(k7_sin_cos(A),k5_numbers,k1_numbers) ) ) ).
fof(dt_k8_sin_cos,axiom,
! [A] :
( m1_subset_1(A,k5_numbers)
=> ( v1_funct_1(k8_sin_cos(A))
& v1_funct_2(k8_sin_cos(A),k5_numbers,k2_numbers)
& m2_relset_1(k8_sin_cos(A),k5_numbers,k2_numbers) ) ) ).
fof(dt_k9_sin_cos,axiom,
! [A] :
( m1_subset_1(A,k5_numbers)
=> ( v1_funct_1(k9_sin_cos(A))
& v1_funct_2(k9_sin_cos(A),k5_numbers,k2_numbers)
& m2_relset_1(k9_sin_cos(A),k5_numbers,k2_numbers) ) ) ).
fof(dt_k10_sin_cos,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k2_numbers)
& m1_relset_1(A,k5_numbers,k2_numbers) )
=> ( v1_funct_1(k10_sin_cos(A))
& v1_funct_2(k10_sin_cos(A),k5_numbers,k2_numbers)
& m2_relset_1(k10_sin_cos(A),k5_numbers,k2_numbers) ) ) ).
fof(dt_k11_sin_cos,axiom,
! [A,B,C] :
( ( m1_subset_1(A,k5_numbers)
& m1_subset_1(B,k2_numbers)
& m1_subset_1(C,k2_numbers) )
=> ( v1_funct_1(k11_sin_cos(A,B,C))
& v1_funct_2(k11_sin_cos(A,B,C),k5_numbers,k2_numbers)
& m2_relset_1(k11_sin_cos(A,B,C),k5_numbers,k2_numbers) ) ) ).
fof(dt_k12_sin_cos,axiom,
! [A,B,C] :
( ( m1_subset_1(A,k5_numbers)
& m1_subset_1(B,k2_numbers)
& m1_subset_1(C,k2_numbers) )
=> ( v1_funct_1(k12_sin_cos(A,B,C))
& v1_funct_2(k12_sin_cos(A,B,C),k5_numbers,k2_numbers)
& m2_relset_1(k12_sin_cos(A,B,C),k5_numbers,k2_numbers) ) ) ).
fof(dt_k13_sin_cos,axiom,
! [A,B,C] :
( ( m1_subset_1(A,k5_numbers)
& m1_subset_1(B,k2_numbers)
& m1_subset_1(C,k2_numbers) )
=> ( v1_funct_1(k13_sin_cos(A,B,C))
& v1_funct_2(k13_sin_cos(A,B,C),k5_numbers,k2_numbers)
& m2_relset_1(k13_sin_cos(A,B,C),k5_numbers,k2_numbers) ) ) ).
fof(dt_k14_sin_cos,axiom,
! [A,B,C] :
( ( v1_xreal_0(A)
& v1_xreal_0(B)
& m1_subset_1(C,k5_numbers) )
=> ( v1_funct_1(k14_sin_cos(A,B,C))
& v1_funct_2(k14_sin_cos(A,B,C),k5_numbers,k1_numbers)
& m2_relset_1(k14_sin_cos(A,B,C),k5_numbers,k1_numbers) ) ) ).
fof(dt_k15_sin_cos,axiom,
! [A,B,C] :
( ( m1_subset_1(A,k2_numbers)
& m1_subset_1(B,k2_numbers)
& m1_subset_1(C,k5_numbers) )
=> ( v1_funct_1(k15_sin_cos(A,B,C))
& v1_funct_2(k15_sin_cos(A,B,C),k5_numbers,k2_numbers)
& m2_relset_1(k15_sin_cos(A,B,C),k5_numbers,k2_numbers) ) ) ).
fof(dt_k16_sin_cos,axiom,
( v1_funct_1(k16_sin_cos)
& m2_relset_1(k16_sin_cos,k2_numbers,k2_numbers) ) ).
fof(dt_k17_sin_cos,axiom,
! [A] :
( m1_subset_1(A,k2_numbers)
=> m1_subset_1(k17_sin_cos(A),k2_numbers) ) ).
fof(dt_k18_sin_cos,axiom,
( v1_funct_1(k18_sin_cos)
& m2_relset_1(k18_sin_cos,k1_numbers,k1_numbers) ) ).
fof(dt_k19_sin_cos,axiom,
$true ).
fof(dt_k20_sin_cos,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> m1_subset_1(k20_sin_cos(A),k1_numbers) ) ).
fof(redefinition_k20_sin_cos,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> k20_sin_cos(A) = k19_sin_cos(A) ) ).
fof(dt_k21_sin_cos,axiom,
( v1_funct_1(k21_sin_cos)
& m2_relset_1(k21_sin_cos,k1_numbers,k1_numbers) ) ).
fof(dt_k22_sin_cos,axiom,
$true ).
fof(dt_k23_sin_cos,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> m1_subset_1(k23_sin_cos(A),k1_numbers) ) ).
fof(redefinition_k23_sin_cos,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> k23_sin_cos(A) = k22_sin_cos(A) ) ).
fof(dt_k24_sin_cos,axiom,
! [A] :
( v1_xreal_0(A)
=> ( v1_funct_1(k24_sin_cos(A))
& v1_funct_2(k24_sin_cos(A),k5_numbers,k1_numbers)
& m2_relset_1(k24_sin_cos(A),k5_numbers,k1_numbers) ) ) ).
fof(dt_k25_sin_cos,axiom,
! [A] :
( v1_xreal_0(A)
=> ( v1_funct_1(k25_sin_cos(A))
& v1_funct_2(k25_sin_cos(A),k5_numbers,k1_numbers)
& m2_relset_1(k25_sin_cos(A),k5_numbers,k1_numbers) ) ) ).
fof(dt_k26_sin_cos,axiom,
( v1_funct_1(k26_sin_cos)
& m2_relset_1(k26_sin_cos,k1_numbers,k1_numbers) ) ).
fof(dt_k27_sin_cos,axiom,
$true ).
fof(dt_k28_sin_cos,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> m1_subset_1(k28_sin_cos(A),k1_numbers) ) ).
fof(redefinition_k28_sin_cos,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> k28_sin_cos(A) = k27_sin_cos(A) ) ).
fof(dt_k29_sin_cos,axiom,
! [A] :
( m1_subset_1(A,k2_numbers)
=> ( v1_funct_1(k29_sin_cos(A))
& v1_funct_2(k29_sin_cos(A),k5_numbers,k2_numbers)
& m2_relset_1(k29_sin_cos(A),k5_numbers,k2_numbers) ) ) ).
fof(dt_k30_sin_cos,axiom,
! [A] :
( m1_subset_1(A,k2_numbers)
=> ( v1_funct_1(k30_sin_cos(A))
& v1_funct_2(k30_sin_cos(A),k5_numbers,k2_numbers)
& m2_relset_1(k30_sin_cos(A),k5_numbers,k2_numbers) ) ) ).
fof(dt_k31_sin_cos,axiom,
v1_xreal_0(k31_sin_cos) ).
fof(dt_k32_sin_cos,axiom,
m1_subset_1(k32_sin_cos,k1_numbers) ).
fof(redefinition_k32_sin_cos,axiom,
k32_sin_cos = k31_sin_cos ).
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