let Z be open Subset of REAL ; :: thesis: ( not 0 in Z & Z c= dom (((id Z) ^ ) (#) sin ) implies ( ((id Z) ^ ) (#) sin is_differentiable_on Z & ( for x being Real st x in Z holds
((((id Z) ^ ) (#) sin ) `| Z) . x = ((1 / x) * (cos . x)) - ((1 / (x ^2 )) * (sin . x)) ) ) )
set f = id Z;
assume A1: ( not 0 in Z & Z c= dom (((id Z) ^ ) (#) sin ) ) ; :: thesis: ( ((id Z) ^ ) (#) sin is_differentiable_on Z & ( for x being Real st x in Z holds
((((id Z) ^ ) (#) sin ) `| Z) . x = ((1 / x) * (cos . x)) - ((1 / (x ^2 )) * (sin . x)) ) )
A: for x being Real st x in Z holds
(id Z) . x = x by FUNCT_1:35;
Z c= (dom ((id Z) ^ )) /\ (dom sin ) by A1, VALUED_1:def 4;
then A2: ( Z c= dom ((id Z) ^ ) & Z c= dom sin ) 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: sin is_differentiable_on Z by FDIFF_1:34, SIN_COS:73;

now

let x be Real; :: thesis: ( x in Z implies ((((id Z) ^ ) (#) sin ) `| Z) . x = ((1 / x) * (cos . x)) - ((1 / (x ^2 )) * (sin . x)) )
assume A6: x in Z ; :: thesis: ((((id Z) ^ ) (#) sin ) `| Z) . x = ((1 / x) * (cos . x)) - ((1 / (x ^2 )) * (sin . x))
hence ((((id Z) ^ ) (#) sin ) `| Z) . x = ((sin . x) * (diff ((id Z) ^ ),x)) + ((((id Z) ^ ) . x) * (diff sin ,x)) by A1, A4, A5, FDIFF_1:29
.= ((sin . x) * ((((id Z) ^ ) `| Z) . x)) + ((((id Z) ^ ) . x) * (diff sin ,x)) by A4, A6, FDIFF_1:def 8
.= ((sin . x) * (- (1 / (x ^2 )))) + ((((id Z) ^ ) . x) * (diff sin ,x)) by A1, A6, Th4
.= ((sin . x) * (- (1 / (x ^2 )))) + ((((id Z) ^ ) . x) * (cos . x)) by SIN_COS:69
.= ((sin . x) * (- (1 / (x ^2 )))) + ((((id Z) . x) " ) * (cos . x)) by A2, A6, RFUNCT_1:def 8
.= ((sin . x) * (- (1 / (x ^2 )))) + ((1 * (x " )) * (cos . x)) by A6, A
.= (- ((1 / (x ^2 )) * (sin . x))) + ((1 / x) * (cos . x)) by XCMPLX_0:def 9
.= ((1 / x) * (cos . x)) - ((1 / (x ^2 )) * (sin . x)) ;
:: thesis: verum

end;

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