let Z be open Subset of REAL ; :: thesis: ( not 0 in Z implies ( sin * ((id Z) ^ ) is_differentiable_on Z & ( for x being Real st x in Z holds
((sin * ((id Z) ^ )) `| Z) . x = - ((1 / (x ^2 )) * (cos . (1 / x))) ) ) )
set f = id Z;
assume A1: not 0 in Z ; :: thesis: ( sin * ((id Z) ^ ) is_differentiable_on Z & ( for x being Real st x in Z holds
((sin * ((id Z) ^ )) `| Z) . x = - ((1 / (x ^2 )) * (cos . (1 / x))) ) )
then A2: Z c= dom (sin * ((id Z) ^ )) by Lm4;
then for y being set st y in Z holds
y in dom ((id Z) ^ ) by FUNCT_1:21;
then A3: Z c= dom ((id Z) ^ ) by TARSKI:def 3;
A4: (id Z) ^ is_differentiable_on Z by A1, Th4;
A5: for x being Real st x in Z holds
sin * ((id Z) ^ ) is_differentiable_in x

proof

let x be Real; :: thesis: ( x in Z implies sin * ((id Z) ^ ) is_differentiable_in x )
assume x in Z ; :: thesis: sin * ((id Z) ^ ) is_differentiable_in x
then A6: (id Z) ^ is_differentiable_in x by A4, FDIFF_1:16;
sin is_differentiable_in ((id Z) ^ ) . x by SIN_COS:69;
hence sin * ((id Z) ^ ) is_differentiable_in x by A6, FDIFF_2:13; :: thesis: verum

end;

then A7: sin * ((id Z) ^ ) is_differentiable_on Z by A2, FDIFF_1:16;
for x being Real st x in Z holds
((sin * ((id Z) ^ )) `| Z) . x = - ((1 / (x ^2 )) * (cos . (1 / x)))

proof

let x be Real; :: thesis: ( x in Z implies ((sin * ((id Z) ^ )) `| Z) . x = - ((1 / (x ^2 )) * (cos . (1 / x))) )
A8: sin is_differentiable_in ((id Z) ^ ) . x by SIN_COS:69;
assume A9: x in Z ; :: thesis: ((sin * ((id Z) ^ )) `| Z) . x = - ((1 / (x ^2 )) * (cos . (1 / x)))
then (id Z) ^ is_differentiable_in x by A4, FDIFF_1:16;
then diff (sin * ((id Z) ^ )),x = (diff sin ,(((id Z) ^ ) . x)) * (diff ((id Z) ^ ),x) by A8, FDIFF_2:13
.= (cos . (((id Z) ^ ) . x)) * (diff ((id Z) ^ ),x) by SIN_COS:69
.= (cos . (((id Z) . x) " )) * (diff ((id Z) ^ ),x) by A3, A9, RFUNCT_1:def 8
.= (cos . (((id Z) . x) " )) * ((((id Z) ^ ) `| Z) . x) by A4, A9, FDIFF_1:def 8
.= (cos . (((id Z) . x) " )) * (- (1 / (x ^2 ))) by A1, A9, Th4
.= (cos . (1 * (x " ))) * (- (1 / (x ^2 ))) by A9, FUNCT_1:35
.= (cos . (1 / x)) * (- (1 / (x ^2 ))) by XCMPLX_0:def 9 ;
hence ((sin * ((id Z) ^ )) `| Z) . x = - ((1 / (x ^2 )) * (cos . (1 / x))) by A7, A9, FDIFF_1:def 8; :: thesis: verum

end;

hence ( sin * ((id Z) ^ ) is_differentiable_on Z & ( for x being Real st x in Z holds
((sin * ((id Z) ^ )) `| Z) . x = - ((1 / (x ^2 )) * (cos . (1 / x))) ) ) by A2, A5, FDIFF_1:16; :: thesis: verum