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:69;
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_7:38;
Z c= dom sin by A3, XBOOLE_1:18;
then A6: sin is_differentiable_on Z by A1, FDIFF_1:16;
for x being Real st x in Z holds
((sin (#) (sin + cos )) `| Z) . x = (((cos . x) ^2 ) + ((2 * (sin . x)) * (cos . x))) - ((sin . 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_1:29
.= (((sin . x) + (cos . x)) * (diff sin ,x)) + ((sin . x) * (diff (sin + cos ),x)) by VALUED_1:1
.= (((sin . x) + (cos . x)) * (cos . x)) + ((sin . x) * (diff (sin + cos ),x)) by SIN_COS:69
.= (((sin . x) + (cos . x)) * (cos . x)) + ((sin . x) * (((sin + cos ) `| Z) . x)) by A5, A7, FDIFF_1:def 8
.= (((sin . x) + (cos . x)) * (cos . x)) + ((sin . x) * ((cos . x) - (sin . x))) by A4, A7, FDIFF_7:38 ;
hence ((sin (#) (sin + cos )) `| Z) . x = (((cos . x) ^2 ) + ((2 * (sin . x)) * (cos . x))) - ((sin . 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 = (((cos . x) ^2 ) + ((2 * (sin . x)) * (cos . x))) - ((sin . x) ^2 ) ) ) by A2, A5, A6, FDIFF_1:29; :: thesis: