let Z be open Subset of REAL ; :: thesis: ( sin * cos is_differentiable_on Z & ( for x being Real st x in Z holds
((sin * cos ) `| Z) . x = - ((cos . (cos . x)) * (sin . x)) ) )
A1: dom cos = REAL by SIN_COS:27;
( rng cos c= dom cos & dom sin = dom cos ) by SIN_COS:27;
then A2: dom (sin * cos ) = REAL by A1, RELAT_1:46;
A3: for x being Real st x in Z holds
sin * cos is_differentiable_in x

proof

let x be Real; :: thesis: ( x in Z implies sin * cos is_differentiable_in x )
assume x in Z ; :: thesis: sin * cos is_differentiable_in x
A4: cos is_differentiable_in x by SIN_COS:68;
sin is_differentiable_in cos . x by SIN_COS:69;
hence sin * cos is_differentiable_in x by A4, FDIFF_2:13; :: thesis: verum

end;

then A5: sin * cos is_differentiable_on Z by A2, FDIFF_1:16;
for x being Real st x in Z holds
((sin * cos ) `| Z) . x = - ((cos . (cos . x)) * (sin . x))

proof

let x be Real; :: thesis: ( x in Z implies ((sin * cos ) `| Z) . x = - ((cos . (cos . x)) * (sin . x)) )
assume A6: x in Z ; :: thesis: ((sin * cos ) `| Z) . x = - ((cos . (cos . x)) * (sin . x))
A7: cos is_differentiable_in x by SIN_COS:68;
sin is_differentiable_in cos . x by SIN_COS:69;
then diff (sin * cos ),x = (diff sin ,(cos . x)) * (diff cos ,x) by A7, FDIFF_2:13
.= (cos . (cos . x)) * (diff cos ,x) by SIN_COS:69
.= (cos . (cos . x)) * (- (sin . x)) by SIN_COS:68 ;
hence ((sin * cos ) `| Z) . x = - ((cos . (cos . x)) * (sin . x)) by A5, A6, FDIFF_1:def 8; :: thesis: verum

end;

hence ( sin * cos is_differentiable_on Z & ( for x being Real st x in Z holds
((sin * cos ) `| Z) . x = - ((cos . (cos . x)) * (sin . x)) ) ) by A2, A3, FDIFF_1:16; :: thesis: verum