let Z be open Subset of REAL; :: thesis: ( cos * sin is_differentiable_on Z & ( for x being Real st x in Z holds
(() `| Z) . x = - ((sin . (sin . x)) * (cos . x)) ) )

A1: for x being Real st x in Z holds
cos * sin is_differentiable_in x
proof end;
rng sin c= dom sin by SIN_COS:24;
then A3: dom () = REAL by ;
then A4: cos * sin is_differentiable_on Z by ;
for x being Real st x in Z holds
(() `| Z) . x = - ((sin . (sin . x)) * (cos . x))
proof
let x be Real; :: thesis: ( x in Z implies (() `| Z) . x = - ((sin . (sin . x)) * (cos . x)) )
assume A5: x in Z ; :: thesis: (() `| 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 ((),x) = (diff (cos,(sin . x))) * (diff (sin,x)) by
.= (- (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 ; :: thesis: verum
end;
hence ( cos * sin is_differentiable_on Z & ( for x being Real st x in Z holds
(() `| Z) . x = - ((sin . (sin . x)) * (cos . x)) ) ) by ; :: thesis: verum