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:69;
A2: for x being Real st x in Z holds
cos is_differentiable_in x by SIN_COS:68;
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_1:16;
Z c= dom sin by A4, 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 (#) 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_1:29
.= ((cos . x) * (cos . x)) + ((sin . x) * (diff cos ,x)) by SIN_COS:69
.= ((cos . x) * (cos . x)) + ((sin . x) * (- (sin . x))) by SIN_COS:68
.= ((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_1:29; :: thesis: verum