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: 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
A2: cos is_differentiable_in sin . x by SIN_COS:63;
sin is_differentiable_in x by SIN_COS:64;
hence cos * sin is_differentiable_in x by A2, FDIFF_2:13; :: thesis: verum

end;

rng sin c= dom sin by SIN_COS:24;
then A3: dom (cos * sin) = REAL by RELAT_1:27, SIN_COS:24;
then A4: cos * sin is_differentiable_on Z by A1, FDIFF_1:9;
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 A5: x in Z ; :: thesis: ((cos * sin) `| Z) . x = - ((sin . (sin . x)) * (cos . x))
A6: cos is_differentiable_in sin . x by SIN_COS:63;
sin is_differentiable_in x by SIN_COS:64;
then diff ((cos * sin),x) = (diff (cos,(sin . x))) * (diff (sin,x)) by A6, FDIFF_2:13
.= (- (sin . (sin . x))) * (diff (sin,x)) by SIN_COS:63
.= (- (sin . (sin . x))) * (cos . x) by SIN_COS:64 ;
hence ((cos * sin) `| Z) . x = - ((sin . (sin . x)) * (cos . x)) by A4, A5, FDIFF_1:def 7; :: 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 A3, A1, FDIFF_1:9; :: thesis: verum