let Z be open Subset of REAL ; :: thesis:
assume A1: Z c= dom (cos (#) (sin - cos )) ; :: thesis:
then A2: Z c= (dom (sin - cos )) /\ (dom cos ) by VALUED_1:def 4;
then A3: Z c= dom cos by XBOOLE_1:18;
A4: Z c= dom (sin - cos ) by A2, XBOOLE_1:18;
then A5: ( sin - cos is_differentiable_on Z & ( for x being Real st x in Z holds
((sin - cos ) `| Z) . x = (cos . x) + (sin . x) ) ) by FDIFF_7:39;
for x being Real st x in Z holds
cos is_differentiable_in x by SIN_COS:68;
then A6: cos is_differentiable_on Z by A3, FDIFF_1:16;
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 )

let x be Real; :: thesis:
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 A1, A5, A6, FDIFF_1:29
.= (((sin . x) - (cos . x)) * (diff cos ,x)) + ((cos . x) * (diff (sin - cos ),x)) by A4, A7, VALUED_1:13
.= (((sin . x) - (cos . x)) * (- (sin . x))) + ((cos . x) * (diff (sin - cos ),x)) by SIN_COS:68
.= (((sin . x) - (cos . x)) * (- (sin . x))) + ((cos . x) * (((sin - cos ) `| Z) . x)) by A5, A7, FDIFF_1:def 8
.= (((sin . x) - (cos . x)) * (- (sin . x))) + ((cos . x) * ((cos . x) + (sin . x))) by A4, A7, FDIFF_7:39 ;
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 A1, A5, A6, FDIFF_1:29; :: thesis: