let Z be open Subset of REAL ; :: thesis:
set f = id Z;
assume A1: ( not 0 in Z & Z c= dom (((id Z) ^ ) (#) cos ) ) ; :: thesis:
then Z c= (dom ((id Z) ^ )) /\ (dom cos ) by VALUED_1:def 4;
then A2: ( Z c= dom ((id Z) ^ ) & Z c= dom cos ) by XBOOLE_1:18;
A4: ( (id Z) ^ is_differentiable_on Z & ( for x being Real st x in Z holds
(((id Z) ^ ) `| Z) . x = - (1 / (x ^2 )) ) ) by A1, Th4;
A5: cos is_differentiable_on Z by FDIFF_1:34, SIN_COS:72;

let x be Real; :: thesis:
assume A6: x in Z ; :: thesis:
hence ((((id Z) ^ ) (#) cos ) `| Z) . x = ((cos . x) * (diff ((id Z) ^ ),x)) + ((((id Z) ^ ) . x) * (diff cos ,x)) by A1, A4, A5, FDIFF_1:29
.= ((cos . x) * ((((id Z) ^ ) `| Z) . x)) + ((((id Z) ^ ) . x) * (diff cos ,x)) by A4, A6, FDIFF_1:def 8
.= ((cos . x) * (- (1 / (x ^2 )))) + ((((id Z) ^ ) . x) * (diff cos ,x)) by A1, A6, Th4
.= ((cos . x) * (- (1 / (x ^2 )))) + ((((id Z) ^ ) . x) * (- (sin . x))) by SIN_COS:68
.= ((cos . x) * (- (1 / (x ^2 )))) + ((((id Z) . x) " ) * (- (sin . x))) by A2, A6, RFUNCT_1:def 8
.= ((cos . x) * (- (1 / (x ^2 )))) + ((1 * (x " )) * (- (sin . x))) by A6, FUNCT_1:35
.= (- ((1 / (x ^2 )) * (cos . x))) + ((1 / x) * (- (sin . x))) by XCMPLX_0:def 9
.= (- ((1 / x) * (sin . x))) - ((1 / (x ^2 )) * (cos . x)) ;
:: thesis:

end;

hence ( ((id Z) ^ ) (#) cos is_differentiable_on Z & ( for x being Real st x in Z holds
((((id Z) ^ ) (#) cos ) `| Z) . x = (- ((1 / x) * (sin . x))) - ((1 / (x ^2 )) * (cos . x)) ) ) by A1, A4, A5, FDIFF_1:29; :: thesis: