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) ) ) )
assume A1: 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 A2: Z c= dom ((#Z 2) * sin ) by VALUED_1:def 5;

now

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:69;
hence (#Z 2) * sin is_differentiable_in x by TAYLOR_1:3; :: thesis: verum

end;

then A3: (#Z 2) * sin is_differentiable_on Z by A2, FDIFF_1:16;
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) )
assume A4: x in Z ; :: thesis: (((1 / 2) (#) ((#Z 2) * sin )) `| Z) . x = (sin . x) * (cos . x)
A5: sin is_differentiable_in x by SIN_COS:69;
(((1 / 2) (#) ((#Z 2) * sin )) `| Z) . x = (1 / 2) * (diff ((#Z 2) * sin ),x) by A1, A3, A4, FDIFF_1:28
.= (1 / 2) * ((2 * ((sin . x) #Z (2 - 1))) * (diff sin ,x)) by A5, TAYLOR_1:3
.= (1 / 2) * ((2 * ((sin . x) #Z (2 - 1))) * (cos . x)) by SIN_COS:69
.= ((sin . x) #Z (2 - 1)) * (cos . x)
.= (sin . x) * (cos . x) by PREPOWER:45 ;
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 A1, A3, FDIFF_1:28; :: thesis: verum