let Z be open Subset of REAL; ( not 0 in Z & Z c= dom (((id Z) ^) (#) cos) implies ( ((id Z) ^) (#) cos is_differentiable_on Z & ( for x being Real st x in Z holds
((((id Z) ^) (#) cos) `| Z) . x = (- ((1 / x) * (sin . x))) - ((1 / (x ^2)) * (cos . x)) ) ) )
set f = id Z;
assume that
A1:
not 0 in Z
and
A2:
Z c= dom (((id Z) ^) (#) cos)
; ( ((id Z) ^) (#) cos is_differentiable_on Z & ( for x being Real st x in Z holds
((((id Z) ^) (#) cos) `| Z) . x = (- ((1 / x) * (sin . x))) - ((1 / (x ^2)) * (cos . x)) ) )
A3:
(id Z) ^ is_differentiable_on Z
by A1, Th4;
A4:
cos is_differentiable_on Z
by FDIFF_1:26, SIN_COS:67;
Z c= (dom ((id Z) ^)) /\ (dom cos)
by A2, VALUED_1:def 4;
then A5:
Z c= dom ((id Z) ^)
by XBOOLE_1:18;
now for x being Real st x in Z holds
((((id Z) ^) (#) cos) `| Z) . x = (- ((1 / x) * (sin . x))) - ((1 / (x ^2)) * (cos . x))let x be
Real;
( x in Z implies ((((id Z) ^) (#) cos) `| Z) . x = (- ((1 / x) * (sin . x))) - ((1 / (x ^2)) * (cos . x)) )assume A6:
x in Z
;
((((id Z) ^) (#) cos) `| Z) . x = (- ((1 / x) * (sin . x))) - ((1 / (x ^2)) * (cos . x))hence ((((id Z) ^) (#) cos) `| Z) . x =
((cos . x) * (diff (((id Z) ^),x))) + ((((id Z) ^) . x) * (diff (cos,x)))
by A2, A3, A4, FDIFF_1:21
.=
((cos . x) * ((((id Z) ^) `| Z) . x)) + ((((id Z) ^) . x) * (diff (cos,x)))
by A3, A6, FDIFF_1:def 7
.=
((cos . x) * (- (1 / (x ^2)))) + ((((id Z) ^) . x) * (diff (cos,x)))
by A1, A6, Th4
.=
((cos . x) * (- (1 / (x ^2)))) + ((((id Z) ^) . x) * (- (sin . x)))
by SIN_COS:63
.=
((cos . x) * (- (1 / (x ^2)))) + ((((id Z) . x) ") * (- (sin . x)))
by A5, A6, RFUNCT_1:def 2
.=
((cos . x) * (- (1 / (x ^2)))) + ((1 * (x ")) * (- (sin . x)))
by A6, FUNCT_1:18
.=
(- ((1 / (x ^2)) * (cos . x))) + ((1 / x) * (- (sin . x)))
by XCMPLX_0:def 9
.=
(- ((1 / x) * (sin . x))) - ((1 / (x ^2)) * (cos . x))
;
verum end;
hence
( ((id Z) ^) (#) cos is_differentiable_on Z & ( for x being Real st x in Z holds
((((id Z) ^) (#) cos) `| Z) . x = (- ((1 / x) * (sin . x))) - ((1 / (x ^2)) * (cos . x)) ) )
by A2, A3, A4, FDIFF_1:21; verum