let Z be open Subset of REAL; :: thesis: ( Z c= dom (sin (#) cos) implies ( sin (#) cos is_differentiable_on Z & ( for x being Real st x in Z holds
((sin (#) cos) `| Z) . x = ((cos . x) ^2) - ((sin . x) ^2) ) ) )
A1: for x being Real st x in Z holds
sin is_differentiable_in x by SIN_COS:64;
A2: for x being Real st x in Z holds
cos is_differentiable_in x by SIN_COS:63;
assume A3: Z c= dom (sin (#) cos) ; :: thesis: ( sin (#) cos is_differentiable_on Z & ( for x being Real st x in Z holds
((sin (#) cos) `| Z) . x = ((cos . x) ^2) - ((sin . x) ^2) ) )
then A4: Z c= (dom sin) /\ (dom cos) by VALUED_1:def 4;
then Z c= dom cos by XBOOLE_1:18;
then A5: cos is_differentiable_on Z by A2, FDIFF_1:9;
Z c= dom sin by A4, 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 (#) cos) `| Z) . x = ((cos . x) ^2) - ((sin . x) ^2)

proof

let x be Real; :: thesis: ( x in Z implies ((sin (#) cos) `| Z) . x = ((cos . x) ^2) - ((sin . x) ^2) )
assume x in Z ; :: thesis: ((sin (#) cos) `| Z) . x = ((cos . x) ^2) - ((sin . x) ^2)
then ((sin (#) cos) `| Z) . x = ((diff (sin,x)) * (cos . x)) + ((sin . x) * (diff (cos,x))) by A3, A6, A5, FDIFF_1:21
.= ((cos . x) * (cos . x)) + ((sin . x) * (diff (cos,x))) by SIN_COS:64
.= ((cos . x) * (cos . x)) + ((sin . x) * (- (sin . x))) by SIN_COS:63
.= ((cos . x) ^2) - ((sin . x) ^2) ;
hence ((sin (#) cos) `| Z) . x = ((cos . x) ^2) - ((sin . x) ^2) ; :: thesis: verum

end;

hence ( sin (#) cos is_differentiable_on Z & ( for x being Real st x in Z holds
((sin (#) cos) `| Z) . x = ((cos . x) ^2) - ((sin . x) ^2) ) ) by A3, A6, A5, FDIFF_1:21; :: thesis: verum