let Z be open Subset of REAL ; :: thesis: ( 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 = (- (cos . x)) + (x * (sin . x)) ) ) )
A1: for x being Real st x in Z holds
(- (id Z)) . x = ((- 1) * x) + 0

proof

let x be Real; :: thesis: ( x in Z implies (- (id Z)) . x = ((- 1) * x) + 0 )
assume A2: x in Z ; :: thesis: (- (id Z)) . x = ((- 1) * x) + 0
(- (id Z)) . x = - ((id Z) . x) by VALUED_1:8
.= - x by A2, FUNCT_1:35
.= ((- 1) * x) + 0 ;
hence (- (id Z)) . x = ((- 1) * x) + 0 ; :: thesis: verum

end;

assume A3: Z c= dom ((- (id Z)) (#) cos ) ; :: thesis: ( (- (id Z)) (#) cos is_differentiable_on Z & ( for x being Real st x in Z holds
(((- (id Z)) (#) cos ) `| Z) . x = (- (cos . x)) + (x * (sin . x)) ) )
then Z c= (dom (- (id Z))) /\ (dom cos ) by VALUED_1:def 4;
then A4: Z c= dom (- (id Z)) by XBOOLE_1:18;
then A5: - (id Z) is_differentiable_on Z by A1, FDIFF_1:31;
A6: cos is_differentiable_on Z by FDIFF_1:34, SIN_COS:72;

now

let x be Real; :: thesis: ( x in Z implies (((- (id Z)) (#) cos ) `| Z) . x = (- (cos . x)) + (x * (sin . x)) )
assume A7: x in Z ; :: thesis: (((- (id Z)) (#) cos ) `| Z) . x = (- (cos . x)) + (x * (sin . x))
hence (((- (id Z)) (#) cos ) `| Z) . x = ((cos . x) * (diff (- (id Z)),x)) + (((- (id Z)) . x) * (diff cos ,x)) by A3, A5, A6, FDIFF_1:29
.= ((cos . x) * (((- (id Z)) `| Z) . x)) + (((- (id Z)) . x) * (diff cos ,x)) by A5, A7, FDIFF_1:def 8
.= ((cos . x) * (- 1)) + (((- (id Z)) . x) * (diff cos ,x)) by A4, A1, A7, FDIFF_1:31
.= ((cos . x) * (- 1)) + (((- (id Z)) . x) * (- (sin . x))) by SIN_COS:68
.= (- (cos . x)) + ((((- 1) * x) + 0 ) * (- (sin . x))) by A1, A7
.= (- (cos . x)) + (x * (sin . x)) ;
:: thesis: verum

end;

hence ( (- (id Z)) (#) cos is_differentiable_on Z & ( for x being Real st x in Z holds
(((- (id Z)) (#) cos ) `| Z) . x = (- (cos . x)) + (x * (sin . x)) ) ) by A3, A5, A6, FDIFF_1:29; :: thesis: verum