SET007 Axioms: SET007+814.ax
%------------------------------------------------------------------------------
% File : SET007+814 : TPTP v9.0.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Formulas and Identities of Trigonometric 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 : sin_cos4 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 62 ( 2 unt; 0 def)
% Number of atoms : 216 ( 98 equ)
% Maximal formula atoms : 7 ( 3 avg)
% Number of connectives : 228 ( 74 ~; 0 |; 36 &)
% ( 0 <=>; 118 =>; 0 <=; 0 <~>)
% Maximal formula depth : 12 ( 6 avg)
% Maximal term depth : 9 ( 2 avg)
% Number of predicates : 4 ( 2 usr; 1 prp; 0-2 aty)
% Number of functors : 16 ( 16 usr; 5 con; 0-2 aty)
% Number of variables : 112 ( 112 !; 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_cos4,axiom,
! [A] :
( v1_xreal_0(A)
=> k1_sin_cos4(A) = k7_xcmplx_0(k19_sin_cos(A),k22_sin_cos(A)) ) ).
fof(d2_sin_cos4,axiom,
! [A] :
( v1_xreal_0(A)
=> k2_sin_cos4(A) = k7_xcmplx_0(k22_sin_cos(A),k19_sin_cos(A)) ) ).
fof(d3_sin_cos4,axiom,
! [A] :
( v1_xreal_0(A)
=> k3_sin_cos4(A) = k7_xcmplx_0(np__1,k19_sin_cos(A)) ) ).
fof(d4_sin_cos4,axiom,
! [A] :
( v1_xreal_0(A)
=> k4_sin_cos4(A) = k7_xcmplx_0(np__1,k22_sin_cos(A)) ) ).
fof(t1_sin_cos4,axiom,
! [A] :
( v1_xreal_0(A)
=> k1_sin_cos4(A) = k7_xcmplx_0(np__1,k2_sin_cos4(A)) ) ).
fof(t2_sin_cos4,axiom,
! [A] :
( v1_xreal_0(A)
=> k1_sin_cos4(k4_xcmplx_0(A)) = k4_xcmplx_0(k1_sin_cos4(A)) ) ).
fof(t3_sin_cos4,axiom,
! [A] :
( v1_xreal_0(A)
=> k3_sin_cos4(k4_xcmplx_0(A)) = k4_xcmplx_0(k7_xcmplx_0(np__1,k19_sin_cos(A))) ) ).
fof(t4_sin_cos4,axiom,
! [A] :
( v1_xreal_0(A)
=> k2_sin_cos4(k4_xcmplx_0(A)) = k4_xcmplx_0(k2_sin_cos4(A)) ) ).
fof(t5_sin_cos4,axiom,
! [A] :
( v1_xreal_0(A)
=> ( k22_sin_cos(A) != np__0
=> k3_xcmplx_0(k22_sin_cos(A),k4_sin_cos4(A)) = np__1 ) ) ).
fof(t6_sin_cos4,axiom,
! [A] :
( v1_xreal_0(A)
=> k3_xcmplx_0(k19_sin_cos(A),k19_sin_cos(A)) = k6_xcmplx_0(np__1,k3_xcmplx_0(k22_sin_cos(A),k22_sin_cos(A))) ) ).
fof(t7_sin_cos4,axiom,
! [A] :
( v1_xreal_0(A)
=> k3_xcmplx_0(k22_sin_cos(A),k22_sin_cos(A)) = k6_xcmplx_0(np__1,k3_xcmplx_0(k19_sin_cos(A),k19_sin_cos(A))) ) ).
fof(t8_sin_cos4,axiom,
! [A] :
( v1_xreal_0(A)
=> ( k22_sin_cos(A) != np__0
=> k19_sin_cos(A) = k3_xcmplx_0(k22_sin_cos(A),k1_sin_cos4(A)) ) ) ).
fof(t9_sin_cos4,axiom,
$true ).
fof(t10_sin_cos4,axiom,
$true ).
fof(t11_sin_cos4,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ~ ( k22_sin_cos(A) != np__0
& k22_sin_cos(B) != np__0
& k1_sin_cos4(k2_xcmplx_0(A,B)) != k7_xcmplx_0(k2_xcmplx_0(k1_sin_cos4(A),k1_sin_cos4(B)),k6_xcmplx_0(np__1,k3_xcmplx_0(k1_sin_cos4(A),k1_sin_cos4(B)))) ) ) ) ).
fof(t12_sin_cos4,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ~ ( k22_sin_cos(A) != np__0
& k22_sin_cos(B) != np__0
& k1_sin_cos4(k6_xcmplx_0(A,B)) != k7_xcmplx_0(k6_xcmplx_0(k1_sin_cos4(A),k1_sin_cos4(B)),k2_xcmplx_0(np__1,k3_xcmplx_0(k1_sin_cos4(A),k1_sin_cos4(B)))) ) ) ) ).
fof(t13_sin_cos4,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ~ ( k19_sin_cos(A) != np__0
& k19_sin_cos(B) != np__0
& k2_sin_cos4(k2_xcmplx_0(A,B)) != k7_xcmplx_0(k6_xcmplx_0(k3_xcmplx_0(k2_sin_cos4(A),k2_sin_cos4(B)),np__1),k2_xcmplx_0(k2_sin_cos4(B),k2_sin_cos4(A))) ) ) ) ).
fof(t14_sin_cos4,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ~ ( k19_sin_cos(A) != np__0
& k19_sin_cos(B) != np__0
& k2_sin_cos4(k6_xcmplx_0(A,B)) != k7_xcmplx_0(k2_xcmplx_0(k3_xcmplx_0(k2_sin_cos4(A),k2_sin_cos4(B)),np__1),k6_xcmplx_0(k2_sin_cos4(B),k2_sin_cos4(A))) ) ) ) ).
fof(t15_sin_cos4,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v1_xreal_0(C)
=> ~ ( k22_sin_cos(A) != np__0
& k22_sin_cos(B) != np__0
& k22_sin_cos(C) != np__0
& k19_sin_cos(k2_xcmplx_0(k2_xcmplx_0(A,B),C)) != k3_xcmplx_0(k3_xcmplx_0(k3_xcmplx_0(k22_sin_cos(A),k22_sin_cos(B)),k22_sin_cos(C)),k6_xcmplx_0(k2_xcmplx_0(k2_xcmplx_0(k1_sin_cos4(A),k1_sin_cos4(B)),k1_sin_cos4(C)),k3_xcmplx_0(k3_xcmplx_0(k1_sin_cos4(A),k1_sin_cos4(B)),k1_sin_cos4(C)))) ) ) ) ) ).
fof(t16_sin_cos4,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v1_xreal_0(C)
=> ~ ( k22_sin_cos(A) != np__0
& k22_sin_cos(B) != np__0
& k22_sin_cos(C) != np__0
& k22_sin_cos(k2_xcmplx_0(k2_xcmplx_0(A,B),C)) != k3_xcmplx_0(k3_xcmplx_0(k3_xcmplx_0(k22_sin_cos(A),k22_sin_cos(B)),k22_sin_cos(C)),k6_xcmplx_0(k6_xcmplx_0(k6_xcmplx_0(np__1,k3_xcmplx_0(k1_sin_cos4(B),k1_sin_cos4(C))),k3_xcmplx_0(k1_sin_cos4(C),k1_sin_cos4(A))),k3_xcmplx_0(k1_sin_cos4(A),k1_sin_cos4(B)))) ) ) ) ) ).
fof(t17_sin_cos4,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v1_xreal_0(C)
=> ~ ( k22_sin_cos(A) != np__0
& k22_sin_cos(B) != np__0
& k22_sin_cos(C) != np__0
& k1_sin_cos4(k2_xcmplx_0(k2_xcmplx_0(A,B),C)) != k7_xcmplx_0(k6_xcmplx_0(k2_xcmplx_0(k2_xcmplx_0(k1_sin_cos4(A),k1_sin_cos4(B)),k1_sin_cos4(C)),k3_xcmplx_0(k3_xcmplx_0(k1_sin_cos4(A),k1_sin_cos4(B)),k1_sin_cos4(C))),k6_xcmplx_0(k6_xcmplx_0(k6_xcmplx_0(np__1,k3_xcmplx_0(k1_sin_cos4(B),k1_sin_cos4(C))),k3_xcmplx_0(k1_sin_cos4(C),k1_sin_cos4(A))),k3_xcmplx_0(k1_sin_cos4(A),k1_sin_cos4(B)))) ) ) ) ) ).
fof(t18_sin_cos4,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v1_xreal_0(C)
=> ~ ( k19_sin_cos(A) != np__0
& k19_sin_cos(B) != np__0
& k19_sin_cos(C) != np__0
& k2_sin_cos4(k2_xcmplx_0(k2_xcmplx_0(A,B),C)) != k7_xcmplx_0(k6_xcmplx_0(k6_xcmplx_0(k6_xcmplx_0(k3_xcmplx_0(k3_xcmplx_0(k2_sin_cos4(A),k2_sin_cos4(B)),k2_sin_cos4(C)),k2_sin_cos4(A)),k2_sin_cos4(B)),k2_sin_cos4(C)),k6_xcmplx_0(k2_xcmplx_0(k2_xcmplx_0(k3_xcmplx_0(k2_sin_cos4(B),k2_sin_cos4(C)),k3_xcmplx_0(k2_sin_cos4(C),k2_sin_cos4(A))),k3_xcmplx_0(k2_sin_cos4(A),k2_sin_cos4(B))),np__1)) ) ) ) ) ).
fof(t19_sin_cos4,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> k2_xcmplx_0(k19_sin_cos(A),k19_sin_cos(B)) = k3_xcmplx_0(np__2,k3_xcmplx_0(k22_sin_cos(k7_xcmplx_0(k6_xcmplx_0(A,B),np__2)),k19_sin_cos(k7_xcmplx_0(k2_xcmplx_0(A,B),np__2)))) ) ) ).
fof(t20_sin_cos4,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> k6_xcmplx_0(k19_sin_cos(A),k19_sin_cos(B)) = k3_xcmplx_0(np__2,k3_xcmplx_0(k22_sin_cos(k7_xcmplx_0(k2_xcmplx_0(A,B),np__2)),k19_sin_cos(k7_xcmplx_0(k6_xcmplx_0(A,B),np__2)))) ) ) ).
fof(t21_sin_cos4,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> k2_xcmplx_0(k22_sin_cos(A),k22_sin_cos(B)) = k3_xcmplx_0(np__2,k3_xcmplx_0(k22_sin_cos(k7_xcmplx_0(k2_xcmplx_0(A,B),np__2)),k22_sin_cos(k7_xcmplx_0(k6_xcmplx_0(A,B),np__2)))) ) ) ).
fof(t22_sin_cos4,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> k6_xcmplx_0(k22_sin_cos(A),k22_sin_cos(B)) = k4_xcmplx_0(k3_xcmplx_0(np__2,k3_xcmplx_0(k19_sin_cos(k7_xcmplx_0(k2_xcmplx_0(A,B),np__2)),k19_sin_cos(k7_xcmplx_0(k6_xcmplx_0(A,B),np__2))))) ) ) ).
fof(t23_sin_cos4,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ~ ( k22_sin_cos(A) != np__0
& k22_sin_cos(B) != np__0
& k2_xcmplx_0(k1_sin_cos4(A),k1_sin_cos4(B)) != k7_xcmplx_0(k19_sin_cos(k2_xcmplx_0(A,B)),k3_xcmplx_0(k22_sin_cos(A),k22_sin_cos(B))) ) ) ) ).
fof(t24_sin_cos4,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ~ ( k22_sin_cos(A) != np__0
& k22_sin_cos(B) != np__0
& k6_xcmplx_0(k1_sin_cos4(A),k1_sin_cos4(B)) != k7_xcmplx_0(k19_sin_cos(k6_xcmplx_0(A,B)),k3_xcmplx_0(k22_sin_cos(A),k22_sin_cos(B))) ) ) ) ).
fof(t25_sin_cos4,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ~ ( k22_sin_cos(A) != np__0
& k19_sin_cos(B) != np__0
& k2_xcmplx_0(k1_sin_cos4(A),k2_sin_cos4(B)) != k7_xcmplx_0(k22_sin_cos(k6_xcmplx_0(A,B)),k3_xcmplx_0(k22_sin_cos(A),k19_sin_cos(B))) ) ) ) ).
fof(t26_sin_cos4,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ~ ( k22_sin_cos(A) != np__0
& k19_sin_cos(B) != np__0
& k6_xcmplx_0(k1_sin_cos4(A),k2_sin_cos4(B)) != k4_xcmplx_0(k7_xcmplx_0(k22_sin_cos(k2_xcmplx_0(A,B)),k3_xcmplx_0(k22_sin_cos(A),k19_sin_cos(B)))) ) ) ) ).
fof(t27_sin_cos4,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ~ ( k19_sin_cos(A) != np__0
& k19_sin_cos(B) != np__0
& k2_xcmplx_0(k2_sin_cos4(A),k2_sin_cos4(B)) != k7_xcmplx_0(k19_sin_cos(k2_xcmplx_0(A,B)),k3_xcmplx_0(k19_sin_cos(A),k19_sin_cos(B))) ) ) ) ).
fof(t28_sin_cos4,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ~ ( k19_sin_cos(A) != np__0
& k19_sin_cos(B) != np__0
& k6_xcmplx_0(k2_sin_cos4(A),k2_sin_cos4(B)) != k4_xcmplx_0(k7_xcmplx_0(k19_sin_cos(k6_xcmplx_0(A,B)),k3_xcmplx_0(k19_sin_cos(A),k19_sin_cos(B)))) ) ) ) ).
fof(t29_sin_cos4,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> k2_xcmplx_0(k19_sin_cos(k2_xcmplx_0(A,B)),k19_sin_cos(k6_xcmplx_0(A,B))) = k3_xcmplx_0(np__2,k3_xcmplx_0(k19_sin_cos(A),k22_sin_cos(B))) ) ) ).
fof(t30_sin_cos4,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> k6_xcmplx_0(k19_sin_cos(k2_xcmplx_0(A,B)),k19_sin_cos(k6_xcmplx_0(A,B))) = k3_xcmplx_0(np__2,k3_xcmplx_0(k22_sin_cos(A),k19_sin_cos(B))) ) ) ).
fof(t31_sin_cos4,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> k2_xcmplx_0(k22_sin_cos(k2_xcmplx_0(A,B)),k22_sin_cos(k6_xcmplx_0(A,B))) = k3_xcmplx_0(np__2,k3_xcmplx_0(k22_sin_cos(A),k22_sin_cos(B))) ) ) ).
fof(t32_sin_cos4,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> k6_xcmplx_0(k22_sin_cos(k2_xcmplx_0(A,B)),k22_sin_cos(k6_xcmplx_0(A,B))) = k4_xcmplx_0(k3_xcmplx_0(np__2,k3_xcmplx_0(k19_sin_cos(A),k19_sin_cos(B)))) ) ) ).
fof(t33_sin_cos4,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> k3_xcmplx_0(k19_sin_cos(A),k19_sin_cos(B)) = k4_xcmplx_0(k3_xcmplx_0(k7_xcmplx_0(np__1,np__2),k6_xcmplx_0(k22_sin_cos(k2_xcmplx_0(A,B)),k22_sin_cos(k6_xcmplx_0(A,B))))) ) ) ).
fof(t34_sin_cos4,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> k3_xcmplx_0(k19_sin_cos(A),k22_sin_cos(B)) = k3_xcmplx_0(k7_xcmplx_0(np__1,np__2),k2_xcmplx_0(k19_sin_cos(k2_xcmplx_0(A,B)),k19_sin_cos(k6_xcmplx_0(A,B)))) ) ) ).
fof(t35_sin_cos4,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> k3_xcmplx_0(k22_sin_cos(A),k19_sin_cos(B)) = k3_xcmplx_0(k7_xcmplx_0(np__1,np__2),k6_xcmplx_0(k19_sin_cos(k2_xcmplx_0(A,B)),k19_sin_cos(k6_xcmplx_0(A,B)))) ) ) ).
fof(t36_sin_cos4,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> k3_xcmplx_0(k22_sin_cos(A),k22_sin_cos(B)) = k3_xcmplx_0(k7_xcmplx_0(np__1,np__2),k2_xcmplx_0(k22_sin_cos(k2_xcmplx_0(A,B)),k22_sin_cos(k6_xcmplx_0(A,B)))) ) ) ).
fof(t37_sin_cos4,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v1_xreal_0(C)
=> k3_xcmplx_0(k3_xcmplx_0(k19_sin_cos(A),k19_sin_cos(B)),k19_sin_cos(C)) = k3_xcmplx_0(k7_xcmplx_0(np__1,np__4),k6_xcmplx_0(k2_xcmplx_0(k2_xcmplx_0(k19_sin_cos(k6_xcmplx_0(k2_xcmplx_0(A,B),C)),k19_sin_cos(k6_xcmplx_0(k2_xcmplx_0(B,C),A))),k19_sin_cos(k6_xcmplx_0(k2_xcmplx_0(C,A),B))),k19_sin_cos(k2_xcmplx_0(k2_xcmplx_0(A,B),C)))) ) ) ) ).
fof(t38_sin_cos4,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v1_xreal_0(C)
=> k3_xcmplx_0(k3_xcmplx_0(k19_sin_cos(A),k19_sin_cos(B)),k22_sin_cos(C)) = k3_xcmplx_0(k7_xcmplx_0(np__1,np__4),k6_xcmplx_0(k2_xcmplx_0(k2_xcmplx_0(k4_xcmplx_0(k22_sin_cos(k6_xcmplx_0(k2_xcmplx_0(A,B),C))),k22_sin_cos(k6_xcmplx_0(k2_xcmplx_0(B,C),A))),k22_sin_cos(k6_xcmplx_0(k2_xcmplx_0(C,A),B))),k22_sin_cos(k2_xcmplx_0(k2_xcmplx_0(A,B),C)))) ) ) ) ).
fof(t39_sin_cos4,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v1_xreal_0(C)
=> k3_xcmplx_0(k3_xcmplx_0(k19_sin_cos(A),k22_sin_cos(B)),k22_sin_cos(C)) = k3_xcmplx_0(k7_xcmplx_0(np__1,np__4),k2_xcmplx_0(k2_xcmplx_0(k6_xcmplx_0(k19_sin_cos(k6_xcmplx_0(k2_xcmplx_0(A,B),C)),k19_sin_cos(k6_xcmplx_0(k2_xcmplx_0(B,C),A))),k19_sin_cos(k6_xcmplx_0(k2_xcmplx_0(C,A),B))),k19_sin_cos(k2_xcmplx_0(k2_xcmplx_0(A,B),C)))) ) ) ) ).
fof(t40_sin_cos4,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v1_xreal_0(C)
=> k3_xcmplx_0(k3_xcmplx_0(k22_sin_cos(A),k22_sin_cos(B)),k22_sin_cos(C)) = k3_xcmplx_0(k7_xcmplx_0(np__1,np__4),k2_xcmplx_0(k2_xcmplx_0(k2_xcmplx_0(k22_sin_cos(k6_xcmplx_0(k2_xcmplx_0(A,B),C)),k22_sin_cos(k6_xcmplx_0(k2_xcmplx_0(B,C),A))),k22_sin_cos(k6_xcmplx_0(k2_xcmplx_0(C,A),B))),k22_sin_cos(k2_xcmplx_0(k2_xcmplx_0(A,B),C)))) ) ) ) ).
fof(t41_sin_cos4,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> k3_xcmplx_0(k19_sin_cos(k2_xcmplx_0(A,B)),k19_sin_cos(k6_xcmplx_0(A,B))) = k6_xcmplx_0(k3_xcmplx_0(k19_sin_cos(A),k19_sin_cos(A)),k3_xcmplx_0(k19_sin_cos(B),k19_sin_cos(B))) ) ) ).
fof(t42_sin_cos4,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> k3_xcmplx_0(k19_sin_cos(k2_xcmplx_0(A,B)),k19_sin_cos(k6_xcmplx_0(A,B))) = k6_xcmplx_0(k3_xcmplx_0(k22_sin_cos(B),k22_sin_cos(B)),k3_xcmplx_0(k22_sin_cos(A),k22_sin_cos(A))) ) ) ).
fof(t43_sin_cos4,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> k3_xcmplx_0(k19_sin_cos(k2_xcmplx_0(A,B)),k22_sin_cos(k6_xcmplx_0(A,B))) = k2_xcmplx_0(k3_xcmplx_0(k19_sin_cos(A),k22_sin_cos(A)),k3_xcmplx_0(k19_sin_cos(B),k22_sin_cos(B))) ) ) ).
fof(t44_sin_cos4,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> k3_xcmplx_0(k22_sin_cos(k2_xcmplx_0(A,B)),k19_sin_cos(k6_xcmplx_0(A,B))) = k6_xcmplx_0(k3_xcmplx_0(k19_sin_cos(A),k22_sin_cos(A)),k3_xcmplx_0(k19_sin_cos(B),k22_sin_cos(B))) ) ) ).
fof(t45_sin_cos4,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> k3_xcmplx_0(k22_sin_cos(k2_xcmplx_0(A,B)),k22_sin_cos(k6_xcmplx_0(A,B))) = k6_xcmplx_0(k3_xcmplx_0(k22_sin_cos(A),k22_sin_cos(A)),k3_xcmplx_0(k19_sin_cos(B),k19_sin_cos(B))) ) ) ).
fof(t46_sin_cos4,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> k3_xcmplx_0(k22_sin_cos(k2_xcmplx_0(A,B)),k22_sin_cos(k6_xcmplx_0(A,B))) = k6_xcmplx_0(k3_xcmplx_0(k22_sin_cos(B),k22_sin_cos(B)),k3_xcmplx_0(k19_sin_cos(A),k19_sin_cos(A))) ) ) ).
fof(t47_sin_cos4,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ~ ( k22_sin_cos(A) != np__0
& k22_sin_cos(B) != np__0
& k7_xcmplx_0(k19_sin_cos(k2_xcmplx_0(A,B)),k19_sin_cos(k6_xcmplx_0(A,B))) != k7_xcmplx_0(k2_xcmplx_0(k1_sin_cos4(A),k1_sin_cos4(B)),k6_xcmplx_0(k1_sin_cos4(A),k1_sin_cos4(B))) ) ) ) ).
fof(t48_sin_cos4,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ~ ( k22_sin_cos(A) != np__0
& k22_sin_cos(B) != np__0
& k7_xcmplx_0(k22_sin_cos(k2_xcmplx_0(A,B)),k22_sin_cos(k6_xcmplx_0(A,B))) != k7_xcmplx_0(k6_xcmplx_0(np__1,k3_xcmplx_0(k1_sin_cos4(A),k1_sin_cos4(B))),k2_xcmplx_0(np__1,k3_xcmplx_0(k1_sin_cos4(A),k1_sin_cos4(B)))) ) ) ) ).
fof(t49_sin_cos4,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> k7_xcmplx_0(k2_xcmplx_0(k19_sin_cos(A),k19_sin_cos(B)),k6_xcmplx_0(k19_sin_cos(A),k19_sin_cos(B))) = k3_xcmplx_0(k1_sin_cos4(k7_xcmplx_0(k2_xcmplx_0(A,B),np__2)),k2_sin_cos4(k7_xcmplx_0(k6_xcmplx_0(A,B),np__2))) ) ) ).
fof(t50_sin_cos4,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ( k22_sin_cos(k7_xcmplx_0(k6_xcmplx_0(A,B),np__2)) != np__0
=> k7_xcmplx_0(k2_xcmplx_0(k19_sin_cos(A),k19_sin_cos(B)),k2_xcmplx_0(k22_sin_cos(A),k22_sin_cos(B))) = k1_sin_cos4(k7_xcmplx_0(k2_xcmplx_0(A,B),np__2)) ) ) ) ).
fof(t51_sin_cos4,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ( k22_sin_cos(k7_xcmplx_0(k2_xcmplx_0(A,B),np__2)) != np__0
=> k7_xcmplx_0(k6_xcmplx_0(k19_sin_cos(A),k19_sin_cos(B)),k2_xcmplx_0(k22_sin_cos(A),k22_sin_cos(B))) = k1_sin_cos4(k7_xcmplx_0(k6_xcmplx_0(A,B),np__2)) ) ) ) ).
fof(t52_sin_cos4,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ( k19_sin_cos(k7_xcmplx_0(k2_xcmplx_0(A,B),np__2)) != np__0
=> k7_xcmplx_0(k2_xcmplx_0(k19_sin_cos(A),k19_sin_cos(B)),k6_xcmplx_0(k22_sin_cos(B),k22_sin_cos(A))) = k2_sin_cos4(k7_xcmplx_0(k6_xcmplx_0(A,B),np__2)) ) ) ) ).
fof(t53_sin_cos4,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ( k19_sin_cos(k7_xcmplx_0(k6_xcmplx_0(A,B),np__2)) != np__0
=> k7_xcmplx_0(k6_xcmplx_0(k19_sin_cos(A),k19_sin_cos(B)),k6_xcmplx_0(k22_sin_cos(B),k22_sin_cos(A))) = k2_sin_cos4(k7_xcmplx_0(k2_xcmplx_0(A,B),np__2)) ) ) ) ).
fof(t54_sin_cos4,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> k7_xcmplx_0(k2_xcmplx_0(k22_sin_cos(A),k22_sin_cos(B)),k6_xcmplx_0(k22_sin_cos(A),k22_sin_cos(B))) = k3_xcmplx_0(k2_sin_cos4(k7_xcmplx_0(k2_xcmplx_0(A,B),np__2)),k2_sin_cos4(k7_xcmplx_0(k6_xcmplx_0(B,A),np__2))) ) ) ).
fof(dt_k1_sin_cos4,axiom,
! [A] :
( v1_xreal_0(A)
=> m1_subset_1(k1_sin_cos4(A),k1_numbers) ) ).
fof(dt_k2_sin_cos4,axiom,
! [A] :
( v1_xreal_0(A)
=> m1_subset_1(k2_sin_cos4(A),k1_numbers) ) ).
fof(dt_k3_sin_cos4,axiom,
! [A] :
( v1_xreal_0(A)
=> m1_subset_1(k3_sin_cos4(A),k1_numbers) ) ).
fof(dt_k4_sin_cos4,axiom,
! [A] :
( v1_xreal_0(A)
=> m1_subset_1(k4_sin_cos4(A),k1_numbers) ) ).
%------------------------------------------------------------------------------