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)) ) ) )
assume A1: 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 A2: ( Z c= dom (id Z) & Z c= dom sin ) by XBOOLE_1:18;
A3: for x being Real st x in Z holds
(id Z) . x = (1 * x) + 0 by FUNCT_1:35;
then A4: ( id Z is_differentiable_on Z & ( for x being Real st x in Z holds
((id Z) `| Z) . x = 1 ) ) by A2, FDIFF_1:31;
A5: sin is_differentiable_on Z by FDIFF_1:34, SIN_COS:73;

now

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 A1, A4, A5, FDIFF_1:29
.= ((sin . x) * (((id Z) `| Z) . x)) + (((id Z) . x) * (diff sin ,x)) by A4, A6, FDIFF_1:def 8
.= ((sin . x) * 1) + (((id Z) . x) * (diff sin ,x)) by A2, A3, A6, FDIFF_1:31
.= ((sin . x) * 1) + (((id Z) . x) * (cos . x)) by SIN_COS:69
.= (sin . x) + (x * (cos . x)) by A6, FUNCT_1:35 ;
:: 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 A1, A4, A5, FDIFF_1:29; :: thesis: verum