## SET007 Axioms: SET007+924.ax

```%------------------------------------------------------------------------------
% File     : SET007+924 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Formulas and Identities of Hyperbolic 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_cos8 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   33 (   0 unit)
%            Number of atoms       :  135 (  75 equality)
%            Maximal formula depth :   10 (   6 average)
%            Number of connectives :  109 (   7 ~  ;   0  |;  41  &)
%                                         (   0 <=>;  61 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :    4 (   0 propositional; 1-3 arity)
%            Number of functors    :   44 (  21 constant; 0-2 arity)
%            Number of variables   :   52 (   0 singleton;  52 !;   0 ?)
%            Maximal term depth    :    9 (   3 average)
% 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_sin_cos8,axiom,(
! [A] :
( v1_xreal_0(A)
=> ( k9_sin_cos2(A) = k6_real_1(k3_sin_cos2(A),k6_sin_cos2(A))
& k9_sin_cos2(np__0) = np__0 ) ) )).

fof(t2_sin_cos8,axiom,(
! [A] :
( v1_xreal_0(A)
=> ( k3_sin_cos2(A) = k6_real_1(np__1,k3_sin_cos5(A))
& k6_sin_cos2(A) = k6_real_1(np__1,k2_sin_cos5(A))
& k9_sin_cos2(A) = k6_real_1(np__1,k1_sin_cos5(A)) ) ) )).

fof(t3_sin_cos8,axiom,(
! [A] :
( v1_xreal_0(A)
=> ( r1_xreal_0(k2_sin_cos5(A),np__1)
& ~ r1_xreal_0(k2_sin_cos5(A),np__0)
& k2_sin_cos5(np__0) = np__1 ) ) )).

fof(t4_sin_cos8,axiom,(
! [A] :
( v1_xreal_0(A)
=> ( r1_xreal_0(np__0,A)
=> r1_xreal_0(np__0,k9_sin_cos2(A)) ) ) )).

fof(t5_sin_cos8,axiom,(
! [A] :
( v1_xreal_0(A)
=> ( k6_sin_cos2(A) = k6_real_1(np__1,k9_square_1(k5_real_1(np__1,k7_square_1(k9_sin_cos2(A)))))
& k3_sin_cos2(A) = k6_real_1(k9_sin_cos2(A),k9_square_1(k5_real_1(np__1,k7_square_1(k9_sin_cos2(A))))) ) ) )).

fof(t6_sin_cos8,axiom,(
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( k3_prepower(k3_real_1(k6_sin_cos2(A),k3_sin_cos2(A)),B) = k3_real_1(k6_sin_cos2(k3_xcmplx_0(B,A)),k3_sin_cos2(k3_xcmplx_0(B,A)))
& k3_prepower(k5_real_1(k6_sin_cos2(A),k3_sin_cos2(A)),B) = k5_real_1(k6_sin_cos2(k3_xcmplx_0(B,A)),k3_sin_cos2(k3_xcmplx_0(B,A))) ) ) ) )).

fof(t7_sin_cos8,axiom,(
! [A] :
( v1_xreal_0(A)
=> ( k27_sin_cos(A) = k3_real_1(k6_sin_cos2(A),k3_sin_cos2(A))
& k27_sin_cos(k4_xcmplx_0(A)) = k5_real_1(k6_sin_cos2(A),k3_sin_cos2(A))
& k27_sin_cos(A) = k6_real_1(k3_real_1(k6_sin_cos2(k7_xcmplx_0(A,np__2)),k3_sin_cos2(k7_xcmplx_0(A,np__2))),k5_real_1(k6_sin_cos2(k7_xcmplx_0(A,np__2)),k3_sin_cos2(k7_xcmplx_0(A,np__2))))
& k27_sin_cos(k4_xcmplx_0(A)) = k6_real_1(k5_real_1(k6_sin_cos2(k7_xcmplx_0(A,np__2)),k3_sin_cos2(k7_xcmplx_0(A,np__2))),k3_real_1(k6_sin_cos2(k7_xcmplx_0(A,np__2)),k3_sin_cos2(k7_xcmplx_0(A,np__2))))
& k27_sin_cos(A) = k6_real_1(k3_real_1(np__1,k9_sin_cos2(k7_xcmplx_0(A,np__2))),k5_real_1(np__1,k9_sin_cos2(k7_xcmplx_0(A,np__2))))
& k27_sin_cos(k4_xcmplx_0(A)) = k6_real_1(k5_real_1(np__1,k9_sin_cos2(k7_xcmplx_0(A,np__2))),k3_real_1(np__1,k9_sin_cos2(k7_xcmplx_0(A,np__2)))) ) ) )).

fof(t8_sin_cos8,axiom,(
! [A] :
( v1_xreal_0(A)
=> ( A != np__0
=> ( k27_sin_cos(A) = k6_real_1(k3_real_1(k1_sin_cos5(k7_xcmplx_0(A,np__2)),np__1),k5_real_1(k1_sin_cos5(k7_xcmplx_0(A,np__2)),np__1))
& k27_sin_cos(k4_xcmplx_0(A)) = k6_real_1(k5_real_1(k1_sin_cos5(k7_xcmplx_0(A,np__2)),np__1),k3_real_1(k1_sin_cos5(k7_xcmplx_0(A,np__2)),np__1)) ) ) ) )).

fof(t9_sin_cos8,axiom,(
! [A] :
( v1_xreal_0(A)
=> k6_real_1(k3_real_1(k6_sin_cos2(A),k3_sin_cos2(A)),k5_real_1(k6_sin_cos2(A),k3_sin_cos2(A))) = k6_real_1(k3_real_1(np__1,k9_sin_cos2(A)),k5_real_1(np__1,k9_sin_cos2(A))) ) )).

fof(t10_sin_cos8,axiom,(
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ( A != np__0
=> ( k3_real_1(k1_sin_cos5(A),k9_sin_cos2(B)) = k6_real_1(k6_sin_cos2(k2_xcmplx_0(A,B)),k4_real_1(k3_sin_cos2(A),k6_sin_cos2(B)))
& k5_real_1(k1_sin_cos5(A),k9_sin_cos2(B)) = k6_real_1(k6_sin_cos2(k6_xcmplx_0(A,B)),k4_real_1(k3_sin_cos2(A),k6_sin_cos2(B))) ) ) ) ) )).

fof(t11_sin_cos8,axiom,(
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ( k4_real_1(k3_sin_cos2(A),k3_sin_cos2(B)) = k4_real_1(k6_real_1(np__1,np__2),k5_real_1(k6_sin_cos2(k2_xcmplx_0(A,B)),k6_sin_cos2(k6_xcmplx_0(A,B))))
& k4_real_1(k3_sin_cos2(A),k6_sin_cos2(B)) = k4_real_1(k6_real_1(np__1,np__2),k3_real_1(k3_sin_cos2(k2_xcmplx_0(A,B)),k3_sin_cos2(k6_xcmplx_0(A,B))))
& k4_real_1(k6_sin_cos2(A),k3_sin_cos2(B)) = k4_real_1(k6_real_1(np__1,np__2),k5_real_1(k3_sin_cos2(k2_xcmplx_0(A,B)),k3_sin_cos2(k6_xcmplx_0(A,B))))
& k4_real_1(k6_sin_cos2(A),k6_sin_cos2(B)) = k4_real_1(k6_real_1(np__1,np__2),k3_real_1(k6_sin_cos2(k2_xcmplx_0(A,B)),k6_sin_cos2(k6_xcmplx_0(A,B)))) ) ) ) )).

fof(t12_sin_cos8,axiom,(
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> k5_real_1(k7_square_1(k3_sin_cos2(A)),k7_square_1(k6_sin_cos2(B))) = k5_real_1(k4_real_1(k3_sin_cos2(k2_xcmplx_0(A,B)),k3_sin_cos2(k6_xcmplx_0(A,B))),np__1) ) ) )).

fof(t13_sin_cos8,axiom,(
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ( k5_real_1(k7_square_1(k5_real_1(k3_sin_cos2(A),k3_sin_cos2(B))),k7_square_1(k5_real_1(k6_sin_cos2(A),k6_sin_cos2(B)))) = k4_real_1(np__4,k7_square_1(k3_sin_cos2(k7_xcmplx_0(k6_xcmplx_0(A,B),np__2))))
& k5_real_1(k7_square_1(k3_real_1(k6_sin_cos2(A),k6_sin_cos2(B))),k7_square_1(k3_real_1(k3_sin_cos2(A),k3_sin_cos2(B)))) = k4_real_1(np__4,k7_square_1(k6_sin_cos2(k7_xcmplx_0(k6_xcmplx_0(A,B),np__2)))) ) ) ) )).

fof(t14_sin_cos8,axiom,(
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> k6_real_1(k3_real_1(k3_sin_cos2(A),k3_sin_cos2(B)),k5_real_1(k3_sin_cos2(A),k3_sin_cos2(B))) = k4_real_1(k9_sin_cos2(k7_xcmplx_0(k2_xcmplx_0(A,B),np__2)),k1_sin_cos5(k7_xcmplx_0(k6_xcmplx_0(A,B),np__2))) ) ) )).

fof(t15_sin_cos8,axiom,(
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> k6_real_1(k3_real_1(k6_sin_cos2(A),k6_sin_cos2(B)),k5_real_1(k6_sin_cos2(A),k6_sin_cos2(B))) = k4_real_1(k1_sin_cos5(k7_xcmplx_0(k2_xcmplx_0(A,B),np__2)),k1_sin_cos5(k7_xcmplx_0(k6_xcmplx_0(A,B),np__2))) ) ) )).

fof(t16_sin_cos8,axiom,(
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ( k6_xcmplx_0(A,B) != np__0
=> k6_real_1(k3_real_1(k3_sin_cos2(A),k3_sin_cos2(B)),k3_real_1(k6_sin_cos2(A),k6_sin_cos2(B))) = k6_real_1(k5_real_1(k6_sin_cos2(A),k6_sin_cos2(B)),k5_real_1(k3_sin_cos2(A),k3_sin_cos2(B))) ) ) ) )).

fof(t17_sin_cos8,axiom,(
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ( k2_xcmplx_0(A,B) != np__0
=> k6_real_1(k5_real_1(k3_sin_cos2(A),k3_sin_cos2(B)),k3_real_1(k6_sin_cos2(A),k6_sin_cos2(B))) = k6_real_1(k5_real_1(k6_sin_cos2(A),k6_sin_cos2(B)),k3_real_1(k3_sin_cos2(A),k3_sin_cos2(B))) ) ) ) )).

fof(t18_sin_cos8,axiom,(
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ( k6_real_1(k3_real_1(k3_sin_cos2(A),k3_sin_cos2(B)),k3_real_1(k6_sin_cos2(A),k6_sin_cos2(B))) = k9_sin_cos2(k2_xcmplx_0(k7_xcmplx_0(A,np__2),k7_xcmplx_0(B,np__2)))
& k6_real_1(k5_real_1(k3_sin_cos2(A),k3_sin_cos2(B)),k3_real_1(k6_sin_cos2(A),k6_sin_cos2(B))) = k9_sin_cos2(k6_xcmplx_0(k7_xcmplx_0(A,np__2),k7_xcmplx_0(B,np__2))) ) ) ) )).

fof(t19_sin_cos8,axiom,(
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> k6_real_1(k3_real_1(k9_sin_cos2(A),k9_sin_cos2(B)),k5_real_1(k9_sin_cos2(A),k9_sin_cos2(B))) = k6_real_1(k3_sin_cos2(k2_xcmplx_0(A,B)),k3_sin_cos2(k6_xcmplx_0(A,B))) ) ) )).

fof(t20_sin_cos8,axiom,(
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> k6_real_1(k3_real_1(k3_real_1(k3_sin_cos2(k6_xcmplx_0(A,B)),k3_sin_cos2(A)),k3_sin_cos2(k2_xcmplx_0(A,B))),k3_real_1(k3_real_1(k6_sin_cos2(k6_xcmplx_0(A,B)),k6_sin_cos2(A)),k6_sin_cos2(k2_xcmplx_0(A,B)))) = k9_sin_cos2(A) ) ) )).

fof(t21_sin_cos8,axiom,(
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v1_xreal_0(C)
=> ( k3_sin_cos2(k2_xcmplx_0(k2_xcmplx_0(A,B),C)) = k4_real_1(k4_real_1(k4_real_1(k3_real_1(k3_real_1(k3_real_1(k9_sin_cos2(A),k9_sin_cos2(B)),k9_sin_cos2(C)),k4_real_1(k4_real_1(k9_sin_cos2(A),k9_sin_cos2(B)),k9_sin_cos2(C))),k6_sin_cos2(A)),k6_sin_cos2(B)),k6_sin_cos2(C))
& k6_sin_cos2(k2_xcmplx_0(k2_xcmplx_0(A,B),C)) = k4_real_1(k4_real_1(k4_real_1(k3_real_1(k3_real_1(k3_real_1(np__1,k4_real_1(k9_sin_cos2(A),k9_sin_cos2(B))),k4_real_1(k9_sin_cos2(B),k9_sin_cos2(C))),k4_real_1(k9_sin_cos2(C),k9_sin_cos2(A))),k6_sin_cos2(A)),k6_sin_cos2(B)),k6_sin_cos2(C))
& k9_sin_cos2(k2_xcmplx_0(k2_xcmplx_0(A,B),C)) = k6_real_1(k3_real_1(k3_real_1(k3_real_1(k9_sin_cos2(A),k9_sin_cos2(B)),k9_sin_cos2(C)),k4_real_1(k4_real_1(k9_sin_cos2(A),k9_sin_cos2(B)),k9_sin_cos2(C))),k3_real_1(k3_real_1(k3_real_1(np__1,k4_real_1(k9_sin_cos2(B),k9_sin_cos2(C))),k4_real_1(k9_sin_cos2(C),k9_sin_cos2(A))),k4_real_1(k9_sin_cos2(A),k9_sin_cos2(B)))) ) ) ) ) )).

fof(t22_sin_cos8,axiom,(
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v1_xreal_0(C)
=> k3_real_1(k3_real_1(k3_real_1(k6_sin_cos2(k3_xcmplx_0(np__2,A)),k6_sin_cos2(k3_xcmplx_0(np__2,B))),k6_sin_cos2(k3_xcmplx_0(np__2,C))),k6_sin_cos2(k3_xcmplx_0(np__2,k2_xcmplx_0(k2_xcmplx_0(A,B),C)))) = k4_real_1(k4_real_1(k4_real_1(np__4,k6_sin_cos2(k2_xcmplx_0(B,C))),k6_sin_cos2(k2_xcmplx_0(C,A))),k6_sin_cos2(k2_xcmplx_0(A,B))) ) ) ) )).

fof(t23_sin_cos8,axiom,(
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v1_xreal_0(C)
=> k3_real_1(k3_real_1(k3_real_1(k4_real_1(k4_real_1(k3_sin_cos2(A),k3_sin_cos2(B)),k3_sin_cos2(k6_xcmplx_0(B,A))),k4_real_1(k4_real_1(k3_sin_cos2(B),k3_sin_cos2(C)),k3_sin_cos2(k6_xcmplx_0(C,B)))),k4_real_1(k4_real_1(k3_sin_cos2(C),k3_sin_cos2(A)),k3_sin_cos2(k6_xcmplx_0(A,C)))),k4_real_1(k4_real_1(k3_sin_cos2(k6_xcmplx_0(B,A)),k3_sin_cos2(k6_xcmplx_0(C,B))),k3_sin_cos2(k6_xcmplx_0(A,C)))) = np__0 ) ) ) )).

fof(t24_sin_cos8,axiom,(
! [A] :
( v1_xreal_0(A)
=> ( r1_xreal_0(np__0,A)
=> k3_sin_cos2(k7_xcmplx_0(A,np__2)) = k9_square_1(k6_real_1(k5_real_1(k6_sin_cos2(A),np__1),np__2)) ) ) )).

fof(t25_sin_cos8,axiom,(
! [A] :
( v1_xreal_0(A)
=> ( ~ r1_xreal_0(np__0,A)
=> k3_sin_cos2(k7_xcmplx_0(A,np__2)) = k1_real_1(k9_square_1(k6_real_1(k5_real_1(k6_sin_cos2(A),np__1),np__2))) ) ) )).

fof(t26_sin_cos8,axiom,(
! [A] :
( v1_xreal_0(A)
=> ( k3_sin_cos2(k3_xcmplx_0(np__2,A)) = k4_real_1(k4_real_1(np__2,k3_sin_cos2(A)),k6_sin_cos2(A))
& k6_sin_cos2(k3_xcmplx_0(np__2,A)) = k5_real_1(k4_real_1(np__2,k7_square_1(k6_sin_cos2(A))),np__1)
& k9_sin_cos2(k3_xcmplx_0(np__2,A)) = k6_real_1(k4_real_1(np__2,k9_sin_cos2(A)),k3_real_1(np__1,k7_square_1(k9_sin_cos2(A)))) ) ) )).

fof(t27_sin_cos8,axiom,(
! [A] :
( v1_xreal_0(A)
=> ( k3_sin_cos2(k3_xcmplx_0(np__2,A)) = k6_real_1(k4_real_1(np__2,k9_sin_cos2(A)),k5_real_1(np__1,k7_square_1(k9_sin_cos2(A))))
& k3_sin_cos2(k3_xcmplx_0(np__3,A)) = k4_real_1(k3_sin_cos2(A),k5_real_1(k4_real_1(np__4,k7_square_1(k6_sin_cos2(A))),np__1))
& k3_sin_cos2(k3_xcmplx_0(np__3,A)) = k5_real_1(k4_real_1(np__3,k3_sin_cos2(A)),k4_real_1(k4_real_1(np__2,k3_sin_cos2(A)),k5_real_1(np__1,k6_sin_cos2(k3_xcmplx_0(np__2,A)))))
& k6_sin_cos2(k3_xcmplx_0(np__2,A)) = k3_real_1(np__1,k4_real_1(np__2,k7_square_1(k3_sin_cos2(A))))
& k6_sin_cos2(k3_xcmplx_0(np__2,A)) = k3_real_1(k7_square_1(k6_sin_cos2(A)),k7_square_1(k3_sin_cos2(A)))
& k6_sin_cos2(k3_xcmplx_0(np__2,A)) = k6_real_1(k3_real_1(np__1,k7_square_1(k9_sin_cos2(A))),k5_real_1(np__1,k7_square_1(k9_sin_cos2(A))))
& k6_sin_cos2(k3_xcmplx_0(np__3,A)) = k4_real_1(k6_sin_cos2(A),k3_real_1(k4_real_1(np__4,k7_square_1(k3_sin_cos2(A))),np__1))
& k9_sin_cos2(k3_xcmplx_0(np__3,A)) = k6_real_1(k3_real_1(k4_real_1(np__3,k9_sin_cos2(A)),k3_prepower(k9_sin_cos2(A),np__3)),k3_real_1(np__1,k4_real_1(np__3,k7_square_1(k9_sin_cos2(A))))) ) ) )).

fof(t28_sin_cos8,axiom,(
! [A] :
( v1_xreal_0(A)
=> k6_real_1(k3_real_1(k3_real_1(k3_sin_cos2(k3_xcmplx_0(np__5,A)),k4_real_1(np__2,k3_sin_cos2(k3_xcmplx_0(np__3,A)))),k3_sin_cos2(A)),k3_real_1(k3_real_1(k3_sin_cos2(k3_xcmplx_0(np__7,A)),k4_real_1(np__2,k3_sin_cos2(k3_xcmplx_0(np__5,A)))),k3_sin_cos2(k3_xcmplx_0(np__3,A)))) = k6_real_1(k3_sin_cos2(k3_xcmplx_0(np__3,A)),k3_sin_cos2(k3_xcmplx_0(np__5,A))) ) )).

fof(t29_sin_cos8,axiom,(
! [A] :
( v1_xreal_0(A)
=> ( r1_xreal_0(np__0,A)
=> k9_sin_cos2(k7_xcmplx_0(A,np__2)) = k9_square_1(k6_real_1(k5_real_1(k6_sin_cos2(A),np__1),k3_real_1(k6_sin_cos2(A),np__1))) ) ) )).

fof(t30_sin_cos8,axiom,(
! [A] :
( v1_xreal_0(A)
=> ( ~ r1_xreal_0(np__0,A)
=> k9_sin_cos2(k7_xcmplx_0(A,np__2)) = k1_real_1(k9_square_1(k6_real_1(k5_real_1(k6_sin_cos2(A),np__1),k3_real_1(k6_sin_cos2(A),np__1)))) ) ) )).

fof(t31_sin_cos8,axiom,(
! [A] :
( v1_xreal_0(A)
=> ( k3_prepower(k3_sin_cos2(A),np__3) = k6_real_1(k5_real_1(k3_sin_cos2(k3_xcmplx_0(np__3,A)),k4_real_1(np__3,k3_sin_cos2(A))),np__4)
& k3_prepower(k3_sin_cos2(A),np__4) = k6_real_1(k3_real_1(k5_real_1(k6_sin_cos2(k3_xcmplx_0(np__4,A)),k4_real_1(np__4,k6_sin_cos2(k3_xcmplx_0(np__2,A)))),np__3),np__8)
& k3_prepower(k3_sin_cos2(A),np__5) = k6_real_1(k3_real_1(k5_real_1(k3_sin_cos2(k3_xcmplx_0(np__5,A)),k4_real_1(np__5,k3_sin_cos2(k3_xcmplx_0(np__3,A)))),k4_real_1(np__10,k3_sin_cos2(A))),np__16)
& k3_prepower(k3_sin_cos2(A),np__6) = k6_real_1(k5_real_1(k3_real_1(k5_real_1(k6_sin_cos2(k3_xcmplx_0(np__6,A)),k4_real_1(np__6,k6_sin_cos2(k3_xcmplx_0(np__4,A)))),k4_real_1(np__15,k6_sin_cos2(k3_xcmplx_0(np__2,A)))),np__10),np__32)
& k3_prepower(k3_sin_cos2(A),np__7) = k6_real_1(k5_real_1(k3_real_1(k5_real_1(k3_sin_cos2(k3_xcmplx_0(np__7,A)),k4_real_1(np__7,k3_sin_cos2(k3_xcmplx_0(np__5,A)))),k4_real_1(np__21,k3_sin_cos2(k3_xcmplx_0(np__3,A)))),k4_real_1(np__35,k3_sin_cos2(A))),np__64)
& k3_prepower(k3_sin_cos2(A),np__8) = k6_real_1(k3_real_1(k5_real_1(k3_real_1(k5_real_1(k6_sin_cos2(k3_xcmplx_0(np__8,A)),k4_real_1(np__8,k6_sin_cos2(k3_xcmplx_0(np__6,A)))),k4_real_1(np__28,k6_sin_cos2(k3_xcmplx_0(np__4,A)))),k4_real_1(np__56,k6_sin_cos2(k3_xcmplx_0(np__2,A)))),np__35),np__128) ) ) )).

fof(t32_sin_cos8,axiom,(
! [A] :
( v1_xreal_0(A)
=> ( k3_prepower(k6_sin_cos2(A),np__3) = k6_real_1(k3_real_1(k6_sin_cos2(k3_xcmplx_0(np__3,A)),k4_real_1(np__3,k6_sin_cos2(A))),np__4)
& k3_prepower(k6_sin_cos2(A),np__4) = k6_real_1(k3_real_1(k3_real_1(k6_sin_cos2(k3_xcmplx_0(np__4,A)),k4_real_1(np__4,k6_sin_cos2(k3_xcmplx_0(np__2,A)))),np__3),np__8)
& k3_prepower(k6_sin_cos2(A),np__5) = k6_real_1(k3_real_1(k3_real_1(k6_sin_cos2(k3_xcmplx_0(np__5,A)),k4_real_1(np__5,k6_sin_cos2(k3_xcmplx_0(np__3,A)))),k4_real_1(np__10,k6_sin_cos2(A))),np__16)
& k3_prepower(k6_sin_cos2(A),np__6) = k6_real_1(k3_real_1(k3_real_1(k3_real_1(k6_sin_cos2(k3_xcmplx_0(np__6,A)),k4_real_1(np__6,k6_sin_cos2(k3_xcmplx_0(np__4,A)))),k4_real_1(np__15,k6_sin_cos2(k3_xcmplx_0(np__2,A)))),np__10),np__32)
& k3_prepower(k6_sin_cos2(A),np__7) = k6_real_1(k3_real_1(k3_real_1(k3_real_1(k6_sin_cos2(k3_xcmplx_0(np__7,A)),k4_real_1(np__7,k6_sin_cos2(k3_xcmplx_0(np__5,A)))),k4_real_1(np__21,k6_sin_cos2(k3_xcmplx_0(np__3,A)))),k4_real_1(np__35,k6_sin_cos2(A))),np__64)
& k3_prepower(k6_sin_cos2(A),np__8) = k6_real_1(k3_real_1(k3_real_1(k3_real_1(k3_real_1(k6_sin_cos2(k3_xcmplx_0(np__8,A)),k4_real_1(np__8,k6_sin_cos2(k3_xcmplx_0(np__6,A)))),k4_real_1(np__28,k6_sin_cos2(k3_xcmplx_0(np__4,A)))),k4_real_1(np__56,k6_sin_cos2(k3_xcmplx_0(np__2,A)))),np__35),np__128) ) ) )).

fof(t33_sin_cos8,axiom,(
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ( k2_xcmplx_0(k6_sin_cos2(k3_xcmplx_0(np__2,A)),k22_sin_cos(k3_xcmplx_0(np__2,B))) = k2_xcmplx_0(np__2,k3_xcmplx_0(np__2,k6_xcmplx_0(k7_square_1(k3_sin_cos2(A)),k5_square_1(k19_sin_cos(B)))))
& k6_xcmplx_0(k6_sin_cos2(k3_xcmplx_0(np__2,A)),k22_sin_cos(k3_xcmplx_0(np__2,B))) = k3_xcmplx_0(np__2,k2_xcmplx_0(k7_square_1(k3_sin_cos2(A)),k5_square_1(k19_sin_cos(B)))) ) ) ) )).
%------------------------------------------------------------------------------
```