let Z be open Subset of REAL; :: thesis:
A1: for x being Real st x in Z holds
cos is_differentiable_in x by SIN_COS:63;
assume A2: Z c= dom (cos (#) (sin + cos)) ; :: thesis:
then A3: Z c= (dom (sin + cos)) /\ (dom cos) 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 cos by A3, XBOOLE_1:18;
then A6: cos is_differentiable_on Z by A1, FDIFF_1:9;
for x being Real st x in Z holds
((cos (#) (sin + cos)) `| Z) . x = (((cos . x) ^2) - ((2 * (sin . x)) * (cos . x))) - ((sin . x) ^2)

proof

let x be Real; :: thesis:
reconsider xx = x as Element of REAL by XREAL_0:def 1;
assume A7: x in Z ; :: thesis:
then ((cos (#) (sin + cos)) `| Z) . x = (((sin + cos) . x) * (diff (cos,x))) + ((cos . x) * (diff ((sin + cos),x))) by A2, A5, A6, FDIFF_1:21
.= (((sin . xx) + (cos . xx)) * (diff (cos,x))) + ((cos . x) * (diff ((sin + cos),x))) by VALUED_1:1
.= (((sin . x) + (cos . x)) * (- (sin . x))) + ((cos . x) * (diff ((sin + cos),x))) by SIN_COS:63
.= (((sin . x) + (cos . x)) * (- (sin . x))) + ((cos . x) * (((sin + cos) `| Z) . x)) by A5, A7, FDIFF_1:def 7
.= (((sin . x) + (cos . x)) * (- (sin . x))) + ((cos . x) * ((cos . x) - (sin . x))) by A4, A7, FDIFF_7:38 ;
hence ((cos (#) (sin + cos)) `| Z) . x = (((cos . x) ^2) - ((2 * (sin . x)) * (cos . x))) - ((sin . x) ^2) ; :: thesis:

end;

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