SET007 Axioms: SET007+195.ax
%------------------------------------------------------------------------------
% File : SET007+195 : TPTP v9.0.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Trigonometric Functions on Complex Space
% 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_cos3 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 71 ( 5 unt; 0 def)
% Number of atoms : 201 ( 85 equ)
% Maximal formula atoms : 6 ( 2 avg)
% Number of connectives : 130 ( 0 ~; 0 |; 30 &)
% ( 4 <=>; 96 =>; 0 <=; 0 <~>)
% Maximal formula depth : 7 ( 4 avg)
% Maximal term depth : 8 ( 2 avg)
% Number of predicates : 6 ( 5 usr; 0 prp; 1-3 aty)
% Number of functors : 38 ( 38 usr; 18 con; 0-4 aty)
% Number of variables : 96 ( 96 !; 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(d1_sin_cos3,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k2_numbers,k2_numbers)
& m2_relset_1(A,k2_numbers,k2_numbers) )
=> ( A = k1_sin_cos3
<=> ! [B] :
( m1_subset_1(B,k2_numbers)
=> k8_funct_2(k2_numbers,k2_numbers,A,B) = k7_xcmplx_0(k11_complex1(k17_sin_cos(k9_complex1(k7_complex1,B)),k17_sin_cos(k10_complex1(k9_complex1(k7_complex1,B)))),k3_xcmplx_0(np__2,k7_complex1)) ) ) ) ).
fof(d2_sin_cos3,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k2_numbers,k2_numbers)
& m2_relset_1(A,k2_numbers,k2_numbers) )
=> ( A = k2_sin_cos3
<=> ! [B] :
( m1_subset_1(B,k2_numbers)
=> k8_funct_2(k2_numbers,k2_numbers,A,B) = k7_xcmplx_0(k8_complex1(k17_sin_cos(k9_complex1(k7_complex1,B)),k17_sin_cos(k10_complex1(k9_complex1(k7_complex1,B)))),np__2) ) ) ) ).
fof(d3_sin_cos3,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k2_numbers,k2_numbers)
& m2_relset_1(A,k2_numbers,k2_numbers) )
=> ( A = k3_sin_cos3
<=> ! [B] :
( m1_subset_1(B,k2_numbers)
=> k8_funct_2(k2_numbers,k2_numbers,A,B) = k7_xcmplx_0(k11_complex1(k17_sin_cos(B),k17_sin_cos(k10_complex1(B))),np__2) ) ) ) ).
fof(d4_sin_cos3,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k2_numbers,k2_numbers)
& m2_relset_1(A,k2_numbers,k2_numbers) )
=> ( A = k4_sin_cos3
<=> ! [B] :
( m1_subset_1(B,k2_numbers)
=> k8_funct_2(k2_numbers,k2_numbers,A,B) = k7_xcmplx_0(k8_complex1(k17_sin_cos(B),k17_sin_cos(k10_complex1(B))),np__2) ) ) ) ).
fof(t1_sin_cos3,axiom,
! [A] :
( m1_subset_1(A,k2_numbers)
=> k8_complex1(k9_complex1(k4_finseq_4(k2_numbers,k2_numbers,k1_sin_cos3,A),k4_finseq_4(k2_numbers,k2_numbers,k1_sin_cos3,A)),k9_complex1(k4_finseq_4(k2_numbers,k2_numbers,k2_sin_cos3,A),k4_finseq_4(k2_numbers,k2_numbers,k2_sin_cos3,A))) = np__1 ) ).
fof(t2_sin_cos3,axiom,
! [A] :
( m1_subset_1(A,k2_numbers)
=> k10_complex1(k4_finseq_4(k2_numbers,k2_numbers,k1_sin_cos3,A)) = k4_finseq_4(k2_numbers,k2_numbers,k1_sin_cos3,k10_complex1(A)) ) ).
fof(t3_sin_cos3,axiom,
! [A] :
( m1_subset_1(A,k2_numbers)
=> k4_finseq_4(k2_numbers,k2_numbers,k2_sin_cos3,A) = k4_finseq_4(k2_numbers,k2_numbers,k2_sin_cos3,k10_complex1(A)) ) ).
fof(t4_sin_cos3,axiom,
! [A] :
( m1_subset_1(A,k2_numbers)
=> ! [B] :
( m1_subset_1(B,k2_numbers)
=> k4_finseq_4(k2_numbers,k2_numbers,k1_sin_cos3,k8_complex1(A,B)) = k8_complex1(k9_complex1(k4_finseq_4(k2_numbers,k2_numbers,k1_sin_cos3,A),k4_finseq_4(k2_numbers,k2_numbers,k2_sin_cos3,B)),k9_complex1(k4_finseq_4(k2_numbers,k2_numbers,k2_sin_cos3,A),k4_finseq_4(k2_numbers,k2_numbers,k1_sin_cos3,B))) ) ) ).
fof(t5_sin_cos3,axiom,
! [A] :
( m1_subset_1(A,k2_numbers)
=> ! [B] :
( m1_subset_1(B,k2_numbers)
=> k4_finseq_4(k2_numbers,k2_numbers,k1_sin_cos3,k11_complex1(A,B)) = k11_complex1(k9_complex1(k4_finseq_4(k2_numbers,k2_numbers,k1_sin_cos3,A),k4_finseq_4(k2_numbers,k2_numbers,k2_sin_cos3,B)),k9_complex1(k4_finseq_4(k2_numbers,k2_numbers,k2_sin_cos3,A),k4_finseq_4(k2_numbers,k2_numbers,k1_sin_cos3,B))) ) ) ).
fof(t6_sin_cos3,axiom,
! [A] :
( m1_subset_1(A,k2_numbers)
=> ! [B] :
( m1_subset_1(B,k2_numbers)
=> k4_finseq_4(k2_numbers,k2_numbers,k2_sin_cos3,k8_complex1(A,B)) = k11_complex1(k9_complex1(k4_finseq_4(k2_numbers,k2_numbers,k2_sin_cos3,A),k4_finseq_4(k2_numbers,k2_numbers,k2_sin_cos3,B)),k9_complex1(k4_finseq_4(k2_numbers,k2_numbers,k1_sin_cos3,A),k4_finseq_4(k2_numbers,k2_numbers,k1_sin_cos3,B))) ) ) ).
fof(t7_sin_cos3,axiom,
! [A] :
( m1_subset_1(A,k2_numbers)
=> ! [B] :
( m1_subset_1(B,k2_numbers)
=> k4_finseq_4(k2_numbers,k2_numbers,k2_sin_cos3,k11_complex1(A,B)) = k8_complex1(k9_complex1(k4_finseq_4(k2_numbers,k2_numbers,k2_sin_cos3,A),k4_finseq_4(k2_numbers,k2_numbers,k2_sin_cos3,B)),k9_complex1(k4_finseq_4(k2_numbers,k2_numbers,k1_sin_cos3,A),k4_finseq_4(k2_numbers,k2_numbers,k1_sin_cos3,B))) ) ) ).
fof(t8_sin_cos3,axiom,
! [A] :
( m1_subset_1(A,k2_numbers)
=> k11_complex1(k9_complex1(k4_finseq_4(k2_numbers,k2_numbers,k4_sin_cos3,A),k4_finseq_4(k2_numbers,k2_numbers,k4_sin_cos3,A)),k9_complex1(k4_finseq_4(k2_numbers,k2_numbers,k3_sin_cos3,A),k4_finseq_4(k2_numbers,k2_numbers,k3_sin_cos3,A))) = np__1 ) ).
fof(t9_sin_cos3,axiom,
! [A] :
( m1_subset_1(A,k2_numbers)
=> k10_complex1(k4_finseq_4(k2_numbers,k2_numbers,k3_sin_cos3,A)) = k4_finseq_4(k2_numbers,k2_numbers,k3_sin_cos3,k10_complex1(A)) ) ).
fof(t10_sin_cos3,axiom,
! [A] :
( m1_subset_1(A,k2_numbers)
=> k4_finseq_4(k2_numbers,k2_numbers,k4_sin_cos3,A) = k4_finseq_4(k2_numbers,k2_numbers,k4_sin_cos3,k10_complex1(A)) ) ).
fof(t11_sin_cos3,axiom,
! [A] :
( m1_subset_1(A,k2_numbers)
=> ! [B] :
( m1_subset_1(B,k2_numbers)
=> k4_finseq_4(k2_numbers,k2_numbers,k3_sin_cos3,k8_complex1(A,B)) = k8_complex1(k9_complex1(k4_finseq_4(k2_numbers,k2_numbers,k3_sin_cos3,A),k4_finseq_4(k2_numbers,k2_numbers,k4_sin_cos3,B)),k9_complex1(k4_finseq_4(k2_numbers,k2_numbers,k4_sin_cos3,A),k4_finseq_4(k2_numbers,k2_numbers,k3_sin_cos3,B))) ) ) ).
fof(t12_sin_cos3,axiom,
! [A] :
( m1_subset_1(A,k2_numbers)
=> ! [B] :
( m1_subset_1(B,k2_numbers)
=> k4_finseq_4(k2_numbers,k2_numbers,k3_sin_cos3,k11_complex1(A,B)) = k11_complex1(k9_complex1(k4_finseq_4(k2_numbers,k2_numbers,k3_sin_cos3,A),k4_finseq_4(k2_numbers,k2_numbers,k4_sin_cos3,B)),k9_complex1(k4_finseq_4(k2_numbers,k2_numbers,k4_sin_cos3,A),k4_finseq_4(k2_numbers,k2_numbers,k3_sin_cos3,B))) ) ) ).
fof(t13_sin_cos3,axiom,
! [A] :
( m1_subset_1(A,k2_numbers)
=> ! [B] :
( m1_subset_1(B,k2_numbers)
=> k4_finseq_4(k2_numbers,k2_numbers,k4_sin_cos3,k11_complex1(A,B)) = k11_complex1(k9_complex1(k4_finseq_4(k2_numbers,k2_numbers,k4_sin_cos3,A),k4_finseq_4(k2_numbers,k2_numbers,k4_sin_cos3,B)),k9_complex1(k4_finseq_4(k2_numbers,k2_numbers,k3_sin_cos3,A),k4_finseq_4(k2_numbers,k2_numbers,k3_sin_cos3,B))) ) ) ).
fof(t14_sin_cos3,axiom,
! [A] :
( m1_subset_1(A,k2_numbers)
=> ! [B] :
( m1_subset_1(B,k2_numbers)
=> k4_finseq_4(k2_numbers,k2_numbers,k4_sin_cos3,k8_complex1(A,B)) = k8_complex1(k9_complex1(k4_finseq_4(k2_numbers,k2_numbers,k4_sin_cos3,A),k4_finseq_4(k2_numbers,k2_numbers,k4_sin_cos3,B)),k9_complex1(k4_finseq_4(k2_numbers,k2_numbers,k3_sin_cos3,A),k4_finseq_4(k2_numbers,k2_numbers,k3_sin_cos3,B))) ) ) ).
fof(t15_sin_cos3,axiom,
! [A] :
( m1_subset_1(A,k2_numbers)
=> k4_finseq_4(k2_numbers,k2_numbers,k1_sin_cos3,k9_complex1(k7_complex1,A)) = k9_complex1(k7_complex1,k4_finseq_4(k2_numbers,k2_numbers,k3_sin_cos3,A)) ) ).
fof(t16_sin_cos3,axiom,
! [A] :
( m1_subset_1(A,k2_numbers)
=> k4_finseq_4(k2_numbers,k2_numbers,k2_sin_cos3,k9_complex1(k7_complex1,A)) = k4_finseq_4(k2_numbers,k2_numbers,k4_sin_cos3,A) ) ).
fof(t17_sin_cos3,axiom,
! [A] :
( m1_subset_1(A,k2_numbers)
=> k4_finseq_4(k2_numbers,k2_numbers,k3_sin_cos3,k9_complex1(k7_complex1,A)) = k9_complex1(k7_complex1,k4_finseq_4(k2_numbers,k2_numbers,k1_sin_cos3,A)) ) ).
fof(t18_sin_cos3,axiom,
! [A] :
( m1_subset_1(A,k2_numbers)
=> k4_finseq_4(k2_numbers,k2_numbers,k4_sin_cos3,k9_complex1(k7_complex1,A)) = k4_finseq_4(k2_numbers,k2_numbers,k2_sin_cos3,A) ) ).
fof(t19_sin_cos3,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> k17_sin_cos(k5_arytm_0(A,B)) = k5_arytm_0(k4_real_1(k2_seq_1(k1_numbers,k1_numbers,k26_sin_cos,A),k2_seq_1(k1_numbers,k1_numbers,k21_sin_cos,B)),k4_real_1(k2_seq_1(k1_numbers,k1_numbers,k26_sin_cos,A),k2_seq_1(k1_numbers,k1_numbers,k18_sin_cos,B))) ) ) ).
fof(t20_sin_cos3,axiom,
k17_sin_cos(k5_complex1) = np__1 ).
fof(t21_sin_cos3,axiom,
k4_finseq_4(k2_numbers,k2_numbers,k1_sin_cos3,k5_complex1) = np__0 ).
fof(t22_sin_cos3,axiom,
k4_finseq_4(k2_numbers,k2_numbers,k3_sin_cos3,k5_complex1) = np__0 ).
fof(t23_sin_cos3,axiom,
k4_finseq_4(k2_numbers,k2_numbers,k2_sin_cos3,k5_complex1) = np__1 ).
fof(t24_sin_cos3,axiom,
k4_finseq_4(k2_numbers,k2_numbers,k4_sin_cos3,k5_complex1) = np__1 ).
fof(t25_sin_cos3,axiom,
! [A] :
( m1_subset_1(A,k2_numbers)
=> k17_sin_cos(A) = k8_complex1(k4_finseq_4(k2_numbers,k2_numbers,k4_sin_cos3,A),k4_finseq_4(k2_numbers,k2_numbers,k3_sin_cos3,A)) ) ).
fof(t26_sin_cos3,axiom,
! [A] :
( m1_subset_1(A,k2_numbers)
=> k17_sin_cos(k10_complex1(A)) = k11_complex1(k4_finseq_4(k2_numbers,k2_numbers,k4_sin_cos3,A),k4_finseq_4(k2_numbers,k2_numbers,k3_sin_cos3,A)) ) ).
fof(t27_sin_cos3,axiom,
! [A] :
( m1_subset_1(A,k2_numbers)
=> ( k17_sin_cos(k8_complex1(A,k9_complex1(k5_arytm_0(k4_real_1(np__2,k32_sin_cos),np__0),k7_complex1))) = k17_sin_cos(A)
& k17_sin_cos(k8_complex1(A,k5_arytm_0(np__0,k4_real_1(np__2,k32_sin_cos)))) = k17_sin_cos(A) ) ) ).
fof(t28_sin_cos3,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ( k17_sin_cos(k5_arytm_0(np__0,k4_real_1(k4_real_1(np__2,k32_sin_cos),A))) = np__1
& k17_sin_cos(k9_complex1(k5_arytm_0(k4_real_1(k4_real_1(np__2,k32_sin_cos),A),np__0),k7_complex1)) = np__1 ) ) ).
fof(t29_sin_cos3,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ( k17_sin_cos(k5_arytm_0(np__0,k1_real_1(k4_real_1(k4_real_1(np__2,k32_sin_cos),A)))) = np__1
& k17_sin_cos(k9_complex1(k5_arytm_0(k1_real_1(k4_real_1(k4_real_1(np__2,k32_sin_cos),A)),np__0),k7_complex1)) = np__1 ) ) ).
fof(t30_sin_cos3,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ( k17_sin_cos(k5_arytm_0(np__0,k4_real_1(k1_nat_1(k2_nat_1(np__2,A),np__1),k32_sin_cos))) = k5_arytm_0(k1_real_1(np__1),np__0)
& k17_sin_cos(k9_complex1(k5_arytm_0(k4_real_1(k1_nat_1(k2_nat_1(np__2,A),np__1),k32_sin_cos),np__0),k7_complex1)) = k5_arytm_0(k1_real_1(np__1),np__0) ) ) ).
fof(t31_sin_cos3,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ( k17_sin_cos(k5_arytm_0(np__0,k1_real_1(k4_real_1(k1_nat_1(k2_nat_1(np__2,A),np__1),k32_sin_cos)))) = k5_arytm_0(k1_real_1(np__1),np__0)
& k17_sin_cos(k9_complex1(k5_arytm_0(k1_real_1(k4_real_1(k1_nat_1(k2_nat_1(np__2,A),np__1),k32_sin_cos)),np__0),k7_complex1)) = k5_arytm_0(k1_real_1(np__1),np__0) ) ) ).
fof(t32_sin_cos3,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ( k17_sin_cos(k5_arytm_0(np__0,k4_real_1(k3_real_1(k2_nat_1(np__2,A),k6_real_1(np__1,np__2)),k32_sin_cos))) = k5_arytm_0(np__0,np__1)
& k17_sin_cos(k9_complex1(k5_arytm_0(k4_real_1(k3_real_1(k2_nat_1(np__2,A),k6_real_1(np__1,np__2)),k32_sin_cos),np__0),k7_complex1)) = k5_arytm_0(np__0,np__1) ) ) ).
fof(t33_sin_cos3,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ( k17_sin_cos(k5_arytm_0(np__0,k1_real_1(k4_real_1(k3_real_1(k2_nat_1(np__2,A),k6_real_1(np__1,np__2)),k32_sin_cos)))) = k5_arytm_0(np__0,k1_real_1(np__1))
& k17_sin_cos(k9_complex1(k5_arytm_0(k1_real_1(k4_real_1(k3_real_1(k2_nat_1(np__2,A),k6_real_1(np__1,np__2)),k32_sin_cos)),np__0),k7_complex1)) = k5_arytm_0(np__0,k1_real_1(np__1)) ) ) ).
fof(t34_sin_cos3,axiom,
! [A] :
( m1_subset_1(A,k2_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> k4_finseq_4(k2_numbers,k2_numbers,k1_sin_cos3,k2_xcmplx_0(A,k4_real_1(k2_nat_1(np__2,B),k32_sin_cos))) = k4_finseq_4(k2_numbers,k2_numbers,k1_sin_cos3,A) ) ) ).
fof(t35_sin_cos3,axiom,
! [A] :
( m1_subset_1(A,k2_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> k4_finseq_4(k2_numbers,k2_numbers,k2_sin_cos3,k2_xcmplx_0(A,k4_real_1(k2_nat_1(np__2,B),k32_sin_cos))) = k4_finseq_4(k2_numbers,k2_numbers,k2_sin_cos3,A) ) ) ).
fof(t36_sin_cos3,axiom,
! [A] :
( m1_subset_1(A,k2_numbers)
=> k17_sin_cos(k9_complex1(k7_complex1,A)) = k8_complex1(k4_finseq_4(k2_numbers,k2_numbers,k2_sin_cos3,A),k9_complex1(k7_complex1,k4_finseq_4(k2_numbers,k2_numbers,k1_sin_cos3,A))) ) ).
fof(t37_sin_cos3,axiom,
! [A] :
( m1_subset_1(A,k2_numbers)
=> k17_sin_cos(k10_complex1(k9_complex1(k7_complex1,A))) = k11_complex1(k4_finseq_4(k2_numbers,k2_numbers,k2_sin_cos3,A),k9_complex1(k7_complex1,k4_finseq_4(k2_numbers,k2_numbers,k1_sin_cos3,A))) ) ).
fof(t38_sin_cos3,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> k4_finseq_4(k2_numbers,k2_numbers,k1_sin_cos3,k5_arytm_0(A,np__0)) = k2_seq_1(k1_numbers,k1_numbers,k18_sin_cos,A) ) ).
fof(t39_sin_cos3,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> k4_finseq_4(k2_numbers,k2_numbers,k2_sin_cos3,k5_arytm_0(A,np__0)) = k2_seq_1(k1_numbers,k1_numbers,k21_sin_cos,A) ) ).
fof(t40_sin_cos3,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> k4_finseq_4(k2_numbers,k2_numbers,k3_sin_cos3,k5_arytm_0(A,np__0)) = k2_seq_1(k1_numbers,k1_numbers,k1_sin_cos2,A) ) ).
fof(t41_sin_cos3,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> k4_finseq_4(k2_numbers,k2_numbers,k4_sin_cos3,k5_arytm_0(A,np__0)) = k2_seq_1(k1_numbers,k1_numbers,k4_sin_cos2,A) ) ).
fof(t42_sin_cos3,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> k8_complex1(k5_arytm_0(A,np__0),k9_complex1(k7_complex1,k5_arytm_0(B,np__0))) = k5_arytm_0(A,B) ) ) ).
fof(t43_sin_cos3,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> k4_finseq_4(k2_numbers,k2_numbers,k1_sin_cos3,k5_arytm_0(A,B)) = k5_arytm_0(k4_real_1(k2_seq_1(k1_numbers,k1_numbers,k18_sin_cos,A),k2_seq_1(k1_numbers,k1_numbers,k4_sin_cos2,B)),k4_real_1(k2_seq_1(k1_numbers,k1_numbers,k21_sin_cos,A),k2_seq_1(k1_numbers,k1_numbers,k1_sin_cos2,B))) ) ) ).
fof(t44_sin_cos3,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> k4_finseq_4(k2_numbers,k2_numbers,k1_sin_cos3,k5_arytm_0(A,k1_real_1(B))) = k5_arytm_0(k4_real_1(k2_seq_1(k1_numbers,k1_numbers,k18_sin_cos,A),k2_seq_1(k1_numbers,k1_numbers,k4_sin_cos2,B)),k1_real_1(k4_real_1(k2_seq_1(k1_numbers,k1_numbers,k21_sin_cos,A),k2_seq_1(k1_numbers,k1_numbers,k1_sin_cos2,B)))) ) ) ).
fof(t45_sin_cos3,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> k4_finseq_4(k2_numbers,k2_numbers,k2_sin_cos3,k5_arytm_0(A,B)) = k5_arytm_0(k4_real_1(k2_seq_1(k1_numbers,k1_numbers,k21_sin_cos,A),k2_seq_1(k1_numbers,k1_numbers,k4_sin_cos2,B)),k1_real_1(k4_real_1(k2_seq_1(k1_numbers,k1_numbers,k18_sin_cos,A),k2_seq_1(k1_numbers,k1_numbers,k1_sin_cos2,B)))) ) ) ).
fof(t46_sin_cos3,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> k4_finseq_4(k2_numbers,k2_numbers,k2_sin_cos3,k5_arytm_0(A,k1_real_1(B))) = k5_arytm_0(k4_real_1(k2_seq_1(k1_numbers,k1_numbers,k21_sin_cos,A),k2_seq_1(k1_numbers,k1_numbers,k4_sin_cos2,B)),k4_real_1(k2_seq_1(k1_numbers,k1_numbers,k18_sin_cos,A),k2_seq_1(k1_numbers,k1_numbers,k1_sin_cos2,B))) ) ) ).
fof(t47_sin_cos3,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> k4_finseq_4(k2_numbers,k2_numbers,k3_sin_cos3,k5_arytm_0(A,B)) = k5_arytm_0(k4_real_1(k2_seq_1(k1_numbers,k1_numbers,k1_sin_cos2,A),k2_seq_1(k1_numbers,k1_numbers,k21_sin_cos,B)),k4_real_1(k2_seq_1(k1_numbers,k1_numbers,k4_sin_cos2,A),k2_seq_1(k1_numbers,k1_numbers,k18_sin_cos,B))) ) ) ).
fof(t48_sin_cos3,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> k4_finseq_4(k2_numbers,k2_numbers,k3_sin_cos3,k5_arytm_0(A,k1_real_1(B))) = k5_arytm_0(k4_real_1(k2_seq_1(k1_numbers,k1_numbers,k1_sin_cos2,A),k2_seq_1(k1_numbers,k1_numbers,k21_sin_cos,B)),k1_real_1(k4_real_1(k2_seq_1(k1_numbers,k1_numbers,k4_sin_cos2,A),k2_seq_1(k1_numbers,k1_numbers,k18_sin_cos,B)))) ) ) ).
fof(t49_sin_cos3,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> k4_finseq_4(k2_numbers,k2_numbers,k4_sin_cos3,k5_arytm_0(A,B)) = k5_arytm_0(k4_real_1(k2_seq_1(k1_numbers,k1_numbers,k4_sin_cos2,A),k2_seq_1(k1_numbers,k1_numbers,k21_sin_cos,B)),k4_real_1(k2_seq_1(k1_numbers,k1_numbers,k1_sin_cos2,A),k2_seq_1(k1_numbers,k1_numbers,k18_sin_cos,B))) ) ) ).
fof(t50_sin_cos3,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> k4_finseq_4(k2_numbers,k2_numbers,k4_sin_cos3,k5_arytm_0(A,k1_real_1(B))) = k5_arytm_0(k4_real_1(k2_seq_1(k1_numbers,k1_numbers,k4_sin_cos2,A),k2_seq_1(k1_numbers,k1_numbers,k21_sin_cos,B)),k1_real_1(k4_real_1(k2_seq_1(k1_numbers,k1_numbers,k1_sin_cos2,A),k2_seq_1(k1_numbers,k1_numbers,k18_sin_cos,B)))) ) ) ).
fof(t51_sin_cos3,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,k2_numbers)
=> k2_comseq_3(k8_complex1(k4_finseq_4(k2_numbers,k2_numbers,k2_sin_cos3,B),k9_complex1(k7_complex1,k4_finseq_4(k2_numbers,k2_numbers,k1_sin_cos3,B))),A) = k8_complex1(k4_finseq_4(k2_numbers,k2_numbers,k2_sin_cos3,k9_complex1(k5_arytm_0(A,np__0),B)),k9_complex1(k7_complex1,k4_finseq_4(k2_numbers,k2_numbers,k1_sin_cos3,k9_complex1(k5_arytm_0(A,np__0),B)))) ) ) ).
fof(t52_sin_cos3,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,k2_numbers)
=> k2_comseq_3(k11_complex1(k4_finseq_4(k2_numbers,k2_numbers,k2_sin_cos3,B),k9_complex1(k7_complex1,k4_finseq_4(k2_numbers,k2_numbers,k1_sin_cos3,B))),A) = k11_complex1(k4_finseq_4(k2_numbers,k2_numbers,k2_sin_cos3,k9_complex1(k5_arytm_0(A,np__0),B)),k9_complex1(k7_complex1,k4_finseq_4(k2_numbers,k2_numbers,k1_sin_cos3,k9_complex1(k5_arytm_0(A,np__0),B)))) ) ) ).
fof(t53_sin_cos3,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,k2_numbers)
=> k17_sin_cos(k9_complex1(k9_complex1(k7_complex1,k5_arytm_0(A,np__0)),B)) = k2_comseq_3(k8_complex1(k4_finseq_4(k2_numbers,k2_numbers,k2_sin_cos3,B),k9_complex1(k7_complex1,k4_finseq_4(k2_numbers,k2_numbers,k1_sin_cos3,B))),A) ) ) ).
fof(t54_sin_cos3,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,k2_numbers)
=> k17_sin_cos(k10_complex1(k9_complex1(k9_complex1(k7_complex1,k5_arytm_0(A,np__0)),B))) = k2_comseq_3(k11_complex1(k4_finseq_4(k2_numbers,k2_numbers,k2_sin_cos3,B),k9_complex1(k7_complex1,k4_finseq_4(k2_numbers,k2_numbers,k1_sin_cos3,B))),A) ) ) ).
fof(t55_sin_cos3,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> k2_xcmplx_0(k3_xcmplx_0(k7_xcmplx_0(k5_arytm_0(np__1,k1_real_1(np__1)),np__2),k4_finseq_4(k2_numbers,k2_numbers,k3_sin_cos3,k5_arytm_0(A,B))),k3_xcmplx_0(k7_xcmplx_0(k5_arytm_0(np__1,np__1),np__2),k4_finseq_4(k2_numbers,k2_numbers,k3_sin_cos3,k5_arytm_0(A,k1_real_1(B))))) = k5_arytm_0(k3_real_1(k4_real_1(k2_seq_1(k1_numbers,k1_numbers,k1_sin_cos2,A),k2_seq_1(k1_numbers,k1_numbers,k21_sin_cos,B)),k4_real_1(k2_seq_1(k1_numbers,k1_numbers,k4_sin_cos2,A),k2_seq_1(k1_numbers,k1_numbers,k18_sin_cos,B))),np__0) ) ) ).
fof(t56_sin_cos3,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> k2_xcmplx_0(k3_xcmplx_0(k7_xcmplx_0(k5_arytm_0(np__1,k1_real_1(np__1)),np__2),k4_finseq_4(k2_numbers,k2_numbers,k4_sin_cos3,k5_arytm_0(A,B))),k3_xcmplx_0(k7_xcmplx_0(k5_arytm_0(np__1,np__1),np__2),k4_finseq_4(k2_numbers,k2_numbers,k4_sin_cos3,k5_arytm_0(A,k1_real_1(B))))) = k3_real_1(k4_real_1(k2_seq_1(k1_numbers,k1_numbers,k1_sin_cos2,A),k2_seq_1(k1_numbers,k1_numbers,k18_sin_cos,B)),k4_real_1(k2_seq_1(k1_numbers,k1_numbers,k4_sin_cos2,A),k2_seq_1(k1_numbers,k1_numbers,k21_sin_cos,B))) ) ) ).
fof(t57_sin_cos3,axiom,
! [A] :
( m1_subset_1(A,k2_numbers)
=> k9_complex1(k4_finseq_4(k2_numbers,k2_numbers,k3_sin_cos3,A),k4_finseq_4(k2_numbers,k2_numbers,k3_sin_cos3,A)) = k7_xcmplx_0(k6_xcmplx_0(k4_finseq_4(k2_numbers,k2_numbers,k4_sin_cos3,k3_xcmplx_0(np__2,A)),np__1),np__2) ) ).
fof(t58_sin_cos3,axiom,
! [A] :
( m1_subset_1(A,k2_numbers)
=> k9_complex1(k4_finseq_4(k2_numbers,k2_numbers,k4_sin_cos3,A),k4_finseq_4(k2_numbers,k2_numbers,k4_sin_cos3,A)) = k7_xcmplx_0(k2_xcmplx_0(k4_finseq_4(k2_numbers,k2_numbers,k4_sin_cos3,k3_xcmplx_0(np__2,A)),np__1),np__2) ) ).
fof(t59_sin_cos3,axiom,
! [A] :
( m1_subset_1(A,k2_numbers)
=> ( k4_finseq_4(k2_numbers,k2_numbers,k3_sin_cos3,k3_xcmplx_0(np__2,A)) = k3_xcmplx_0(k3_xcmplx_0(np__2,k4_finseq_4(k2_numbers,k2_numbers,k3_sin_cos3,A)),k4_finseq_4(k2_numbers,k2_numbers,k4_sin_cos3,A))
& k4_finseq_4(k2_numbers,k2_numbers,k4_sin_cos3,k3_xcmplx_0(np__2,A)) = k6_xcmplx_0(k3_xcmplx_0(k3_xcmplx_0(np__2,k4_finseq_4(k2_numbers,k2_numbers,k4_sin_cos3,A)),k4_finseq_4(k2_numbers,k2_numbers,k4_sin_cos3,A)),np__1) ) ) ).
fof(t60_sin_cos3,axiom,
! [A] :
( m1_subset_1(A,k2_numbers)
=> ! [B] :
( m1_subset_1(B,k2_numbers)
=> ( k11_complex1(k9_complex1(k4_finseq_4(k2_numbers,k2_numbers,k3_sin_cos3,A),k4_finseq_4(k2_numbers,k2_numbers,k3_sin_cos3,A)),k9_complex1(k4_finseq_4(k2_numbers,k2_numbers,k3_sin_cos3,B),k4_finseq_4(k2_numbers,k2_numbers,k3_sin_cos3,B))) = k9_complex1(k4_finseq_4(k2_numbers,k2_numbers,k3_sin_cos3,k8_complex1(A,B)),k4_finseq_4(k2_numbers,k2_numbers,k3_sin_cos3,k11_complex1(A,B)))
& k11_complex1(k9_complex1(k4_finseq_4(k2_numbers,k2_numbers,k4_sin_cos3,A),k4_finseq_4(k2_numbers,k2_numbers,k4_sin_cos3,A)),k9_complex1(k4_finseq_4(k2_numbers,k2_numbers,k4_sin_cos3,B),k4_finseq_4(k2_numbers,k2_numbers,k4_sin_cos3,B))) = k9_complex1(k4_finseq_4(k2_numbers,k2_numbers,k3_sin_cos3,k8_complex1(A,B)),k4_finseq_4(k2_numbers,k2_numbers,k3_sin_cos3,k11_complex1(A,B)))
& k11_complex1(k9_complex1(k4_finseq_4(k2_numbers,k2_numbers,k3_sin_cos3,A),k4_finseq_4(k2_numbers,k2_numbers,k3_sin_cos3,A)),k9_complex1(k4_finseq_4(k2_numbers,k2_numbers,k3_sin_cos3,B),k4_finseq_4(k2_numbers,k2_numbers,k3_sin_cos3,B))) = k11_complex1(k9_complex1(k4_finseq_4(k2_numbers,k2_numbers,k4_sin_cos3,A),k4_finseq_4(k2_numbers,k2_numbers,k4_sin_cos3,A)),k9_complex1(k4_finseq_4(k2_numbers,k2_numbers,k4_sin_cos3,B),k4_finseq_4(k2_numbers,k2_numbers,k4_sin_cos3,B))) ) ) ) ).
fof(t61_sin_cos3,axiom,
! [A] :
( m1_subset_1(A,k2_numbers)
=> ! [B] :
( m1_subset_1(B,k2_numbers)
=> ( k9_complex1(k4_finseq_4(k2_numbers,k2_numbers,k4_sin_cos3,k8_complex1(A,B)),k4_finseq_4(k2_numbers,k2_numbers,k4_sin_cos3,k11_complex1(A,B))) = k8_complex1(k9_complex1(k4_finseq_4(k2_numbers,k2_numbers,k3_sin_cos3,A),k4_finseq_4(k2_numbers,k2_numbers,k3_sin_cos3,A)),k9_complex1(k4_finseq_4(k2_numbers,k2_numbers,k4_sin_cos3,B),k4_finseq_4(k2_numbers,k2_numbers,k4_sin_cos3,B)))
& k9_complex1(k4_finseq_4(k2_numbers,k2_numbers,k4_sin_cos3,k8_complex1(A,B)),k4_finseq_4(k2_numbers,k2_numbers,k4_sin_cos3,k11_complex1(A,B))) = k8_complex1(k9_complex1(k4_finseq_4(k2_numbers,k2_numbers,k4_sin_cos3,A),k4_finseq_4(k2_numbers,k2_numbers,k4_sin_cos3,A)),k9_complex1(k4_finseq_4(k2_numbers,k2_numbers,k3_sin_cos3,B),k4_finseq_4(k2_numbers,k2_numbers,k3_sin_cos3,B)))
& k8_complex1(k9_complex1(k4_finseq_4(k2_numbers,k2_numbers,k3_sin_cos3,A),k4_finseq_4(k2_numbers,k2_numbers,k3_sin_cos3,A)),k9_complex1(k4_finseq_4(k2_numbers,k2_numbers,k4_sin_cos3,B),k4_finseq_4(k2_numbers,k2_numbers,k4_sin_cos3,B))) = k8_complex1(k9_complex1(k4_finseq_4(k2_numbers,k2_numbers,k4_sin_cos3,A),k4_finseq_4(k2_numbers,k2_numbers,k4_sin_cos3,A)),k9_complex1(k4_finseq_4(k2_numbers,k2_numbers,k3_sin_cos3,B),k4_finseq_4(k2_numbers,k2_numbers,k3_sin_cos3,B))) ) ) ) ).
fof(t62_sin_cos3,axiom,
! [A] :
( m1_subset_1(A,k2_numbers)
=> ! [B] :
( m1_subset_1(B,k2_numbers)
=> ( k8_complex1(k4_finseq_4(k2_numbers,k2_numbers,k3_sin_cos3,k3_xcmplx_0(np__2,A)),k4_finseq_4(k2_numbers,k2_numbers,k3_sin_cos3,k3_xcmplx_0(np__2,B))) = k3_xcmplx_0(k3_xcmplx_0(np__2,k4_finseq_4(k2_numbers,k2_numbers,k3_sin_cos3,k8_complex1(A,B))),k4_finseq_4(k2_numbers,k2_numbers,k4_sin_cos3,k11_complex1(A,B)))
& k11_complex1(k4_finseq_4(k2_numbers,k2_numbers,k3_sin_cos3,k3_xcmplx_0(np__2,A)),k4_finseq_4(k2_numbers,k2_numbers,k3_sin_cos3,k3_xcmplx_0(np__2,B))) = k3_xcmplx_0(k3_xcmplx_0(np__2,k4_finseq_4(k2_numbers,k2_numbers,k3_sin_cos3,k11_complex1(A,B))),k4_finseq_4(k2_numbers,k2_numbers,k4_sin_cos3,k8_complex1(A,B))) ) ) ) ).
fof(t63_sin_cos3,axiom,
! [A] :
( m1_subset_1(A,k2_numbers)
=> ! [B] :
( m1_subset_1(B,k2_numbers)
=> ( k8_complex1(k4_finseq_4(k2_numbers,k2_numbers,k4_sin_cos3,k3_xcmplx_0(np__2,A)),k4_finseq_4(k2_numbers,k2_numbers,k4_sin_cos3,k3_xcmplx_0(np__2,B))) = k3_xcmplx_0(k3_xcmplx_0(np__2,k4_finseq_4(k2_numbers,k2_numbers,k4_sin_cos3,k8_complex1(A,B))),k4_finseq_4(k2_numbers,k2_numbers,k4_sin_cos3,k11_complex1(A,B)))
& k11_complex1(k4_finseq_4(k2_numbers,k2_numbers,k4_sin_cos3,k3_xcmplx_0(np__2,A)),k4_finseq_4(k2_numbers,k2_numbers,k4_sin_cos3,k3_xcmplx_0(np__2,B))) = k3_xcmplx_0(k3_xcmplx_0(np__2,k4_finseq_4(k2_numbers,k2_numbers,k3_sin_cos3,k8_complex1(A,B))),k4_finseq_4(k2_numbers,k2_numbers,k3_sin_cos3,k11_complex1(A,B))) ) ) ) ).
fof(dt_k1_sin_cos3,axiom,
( v1_funct_1(k1_sin_cos3)
& v1_funct_2(k1_sin_cos3,k2_numbers,k2_numbers)
& m2_relset_1(k1_sin_cos3,k2_numbers,k2_numbers) ) ).
fof(dt_k2_sin_cos3,axiom,
( v1_funct_1(k2_sin_cos3)
& v1_funct_2(k2_sin_cos3,k2_numbers,k2_numbers)
& m2_relset_1(k2_sin_cos3,k2_numbers,k2_numbers) ) ).
fof(dt_k3_sin_cos3,axiom,
( v1_funct_1(k3_sin_cos3)
& v1_funct_2(k3_sin_cos3,k2_numbers,k2_numbers)
& m2_relset_1(k3_sin_cos3,k2_numbers,k2_numbers) ) ).
fof(dt_k4_sin_cos3,axiom,
( v1_funct_1(k4_sin_cos3)
& v1_funct_2(k4_sin_cos3,k2_numbers,k2_numbers)
& m2_relset_1(k4_sin_cos3,k2_numbers,k2_numbers) ) ).
%------------------------------------------------------------------------------