let Z be open Subset of REAL; :: thesis: ( Z c= dom ((id Z) (#) sin) implies ( (id Z) (#) sin is_differentiable_on Z & ( for x being Real st x in Z holds
(((id Z) (#) sin) `| Z) . x = (sin . x) + (x * (cos . x)) ) ) )
A1: for x being Real st x in Z holds
(id Z) . x = (1 * x) + 0 by FUNCT_1:18;
assume A2: Z c= dom ((id Z) (#) sin) ; :: thesis: ( (id Z) (#) sin is_differentiable_on Z & ( for x being Real st x in Z holds
(((id Z) (#) sin) `| Z) . x = (sin . x) + (x * (cos . x)) ) )
then Z c= (dom (id Z)) /\ (dom sin) by VALUED_1:def 4;
then A3: Z c= dom (id Z) by XBOOLE_1:18;
then A4: id Z is_differentiable_on Z by A1, FDIFF_1:23;
A5: sin is_differentiable_on Z by FDIFF_1:26, SIN_COS:68;

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

let x be Real; :: thesis: ( x in Z implies (((id Z) (#) sin) `| Z) . x = (sin . x) + (x * (cos . x)) )
assume A6: x in Z ; :: thesis: (((id Z) (#) sin) `| Z) . x = (sin . x) + (x * (cos . x))
hence (((id Z) (#) sin) `| Z) . x = ((sin . x) * (diff ((id Z),x))) + (((id Z) . x) * (diff (sin,x))) by A2, A4, A5, FDIFF_1:21
.= ((sin . x) * (((id Z) `| Z) . x)) + (((id Z) . x) * (diff (sin,x))) by A4, A6, FDIFF_1:def 7
.= ((sin . x) * 1) + (((id Z) . x) * (diff (sin,x))) by A3, A1, A6, FDIFF_1:23
.= ((sin . x) * 1) + (((id Z) . x) * (cos . x)) by SIN_COS:64
.= (sin . x) + (x * (cos . x)) by A6, FUNCT_1:18 ;
:: thesis: verum

end;

hence ( (id Z) (#) sin is_differentiable_on Z & ( for x being Real st x in Z holds
(((id Z) (#) sin) `| Z) . x = (sin . x) + (x * (cos . x)) ) ) by A2, A4, A5, FDIFF_1:21; :: thesis: verum