let Z be open Subset of REAL; :: thesis: ( Z c= dom ((1 / 2) (#) ((#Z 2) * sin)) implies ( (1 / 2) (#) ((#Z 2) * sin) is_differentiable_on Z & ( for x being Real st x in Z holds
(((1 / 2) (#) ((#Z 2) * sin)) `| Z) . x = (sin . x) * (cos . x) ) ) )

A1: now :: thesis: for x being Real st x in Z holds
(#Z 2) * sin is_differentiable_in x

let x be Real; :: thesis: ( x in Z implies (#Z 2) * sin is_differentiable_in x )
assume x in Z ; :: thesis: (#Z 2) * sin is_differentiable_in x
sin is_differentiable_in x by SIN_COS:64;
hence (#Z 2) * sin is_differentiable_in x by TAYLOR_1:3; :: thesis: verum

end;

assume A2: Z c= dom ((1 / 2) (#) ((#Z 2) * sin)) ; :: thesis: ( (1 / 2) (#) ((#Z 2) * sin) is_differentiable_on Z & ( for x being Real st x in Z holds
(((1 / 2) (#) ((#Z 2) * sin)) `| Z) . x = (sin . x) * (cos . x) ) )
then Z c= dom ((#Z 2) * sin) by VALUED_1:def 5;
then A3: (#Z 2) * sin is_differentiable_on Z by A1, FDIFF_1:9;
for x being Real st x in Z holds
(((1 / 2) (#) ((#Z 2) * sin)) `| Z) . x = (sin . x) * (cos . x)

proof

let x be Real; :: thesis: ( x in Z implies (((1 / 2) (#) ((#Z 2) * sin)) `| Z) . x = (sin . x) * (cos . x) )
A4: sin is_differentiable_in x by SIN_COS:64;
assume x in Z ; :: thesis: (((1 / 2) (#) ((#Z 2) * sin)) `| Z) . x = (sin . x) * (cos . x)
then (((1 / 2) (#) ((#Z 2) * sin)) `| Z) . x = (1 / 2) * (diff (((#Z 2) * sin),x)) by A2, A3, FDIFF_1:20
.= (1 / 2) * ((2 * ((sin . x) #Z (2 - 1))) * (diff (sin,x))) by A4, TAYLOR_1:3
.= (1 / 2) * ((2 * ((sin . x) #Z (2 - 1))) * (cos . x)) by SIN_COS:64
.= ((sin . x) #Z (2 - 1)) * (cos . x)
.= (sin . x) * (cos . x) by PREPOWER:35 ;
hence (((1 / 2) (#) ((#Z 2) * sin)) `| Z) . x = (sin . x) * (cos . x) ; :: thesis: verum

end;

hence ( (1 / 2) (#) ((#Z 2) * sin) is_differentiable_on Z & ( for x being Real st x in Z holds
(((1 / 2) (#) ((#Z 2) * sin)) `| Z) . x = (sin . x) * (cos . x) ) ) by A2, A3, FDIFF_1:20; :: thesis: verum