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

proof

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

end;

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

proof

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

end;

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