let Z be open Subset of REAL; :: thesis:
A1: for x being Real st x in Z holds
sin is_differentiable_in x by SIN_COS:64;
assume A2: Z c= dom (sin (#) (sin - cos)) ; :: thesis:
then A3: Z c= (dom (sin - cos)) /\ (dom sin) by VALUED_1:def 4;
then A4: Z c= dom (sin - cos) by XBOOLE_1:18;
then A5: sin - cos is_differentiable_on Z by FDIFF_7:39;
Z c= dom sin by A3, XBOOLE_1:18;
then A6: sin is_differentiable_on Z by A1, FDIFF_1:9;
for x being Real st x in Z holds
((sin (#) (sin - cos)) `| Z) . x = (((sin . x) ^2) + ((2 * (sin . x)) * (cos . x))) - ((cos . x) ^2)

proof

let x be Real; :: thesis:
assume A7: x in Z ; :: thesis:
then ((sin (#) (sin - cos)) `| Z) . x = (((sin - cos) . x) * (diff (sin,x))) + ((sin . x) * (diff ((sin - cos),x))) by A2, A5, A6, FDIFF_1:21
.= (((sin . x) - (cos . x)) * (diff (sin,x))) + ((sin . x) * (diff ((sin - cos),x))) by A4, A7, VALUED_1:13
.= (((sin . x) - (cos . x)) * (cos . x)) + ((sin . x) * (diff ((sin - cos),x))) by SIN_COS:64
.= (((sin . x) - (cos . x)) * (cos . x)) + ((sin . x) * (((sin - cos) `| Z) . x)) by A5, A7, FDIFF_1:def 7
.= (((sin . x) - (cos . x)) * (cos . x)) + ((sin . x) * ((cos . x) + (sin . x))) by A4, A7, FDIFF_7:39 ;
hence ((sin (#) (sin - cos)) `| Z) . x = (((sin . x) ^2) + ((2 * (sin . x)) * (cos . x))) - ((cos . x) ^2) ; :: thesis:

end;

hence ( sin (#) (sin - cos) is_differentiable_on Z & ( for x being Real st x in Z holds
((sin (#) (sin - cos)) `| Z) . x = (((sin . x) ^2) + ((2 * (sin . x)) * (cos . x))) - ((cos . x) ^2) ) ) by A2, A5, A6, FDIFF_1:21; :: thesis: