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_1:18
.= ((- 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_1:23;
A6: cos is_differentiable_on Z by FDIFF_1:26, SIN_COS:67;

now :: thesis: for x being Real st x in Z holds
(((- (id Z)) (#) cos) `| Z) . x = (- (cos . x)) + (x * (sin . x))

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_1:21
.= ((cos . x) * (((- (id Z)) `| Z) . x)) + (((- (id Z)) . x) * (diff (cos,x))) by A5, A7, FDIFF_1:def 7
.= ((cos . x) * (- 1)) + (((- (id Z)) . x) * (diff (cos,x))) by A4, A1, A7, FDIFF_1:23
.= ((cos . x) * (- 1)) + (((- (id Z)) . x) * (- (sin . x))) by SIN_COS:63
.= (- (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_1:21; :: thesis: verum