let Z be open Subset of REAL ; :: thesis: for f being PartFunc of REAL , REAL st Z c= dom ((id Z) (#) (sin * (f ^ ))) & ( for x being Real st x in Z holds
( f . x = x & f . x <> 0 ) ) holds
( (id Z) (#) (sin * (f ^ )) is_differentiable_on Z & ( for x being Real st x in Z holds
(((id Z) (#) (sin * (f ^ ))) `| Z) . x = (sin . (1 / x)) - ((1 / x) * (cos . (1 / x))) ) )
let f be PartFunc of REAL , REAL ; :: thesis: ( Z c= dom ((id Z) (#) (sin * (f ^ ))) & ( for x being Real st x in Z holds
( f . x = x & f . x <> 0 ) ) implies ( (id Z) (#) (sin * (f ^ )) is_differentiable_on Z & ( for x being Real st x in Z holds
(((id Z) (#) (sin * (f ^ ))) `| Z) . x = (sin . (1 / x)) - ((1 / x) * (cos . (1 / x))) ) ) )
assume A1: Z c= dom ((id Z) (#) (sin * (f ^ ))) ; :: thesis: ( ex x being Real st
( x in Z & not ( f . x = x & f . x <> 0 ) ) or ( (id Z) (#) (sin * (f ^ )) is_differentiable_on Z & ( for x being Real st x in Z holds
(((id Z) (#) (sin * (f ^ ))) `| Z) . x = (sin . (1 / x)) - ((1 / x) * (cos . (1 / x))) ) ) )
then Z c= (dom (id Z)) /\ (dom (sin * (f ^ ))) by VALUED_1:def 4;
then A2: ( Z c= dom (id Z) & Z c= dom (sin * (f ^ )) ) by XBOOLE_1:18;
assume A3: for x being Real st x in Z holds
( f . x = x & f . x <> 0 ) ; :: thesis: ( (id Z) (#) (sin * (f ^ )) is_differentiable_on Z & ( for x being Real st x in Z holds
(((id Z) (#) (sin * (f ^ ))) `| Z) . x = (sin . (1 / x)) - ((1 / x) * (cos . (1 / x))) ) )
for y being set st y in Z holds
y in dom (f ^ ) by A2, FUNCT_1:21;
then A4: Z c= dom (f ^ ) by TARSKI:def 3;
A5: ( sin * (f ^ ) is_differentiable_on Z & ( for x being Real st x in Z holds
((sin * (f ^ )) `| Z) . x = - ((1 / (x ^2 )) * (cos . (1 / x))) ) ) by A2, A3, Th5;
A6: for x being Real st x in Z holds
(id Z) . x = (1 * x) + 0 by FUNCT_1:35;
then A7: ( id Z is_differentiable_on Z & ( for x being Real st x in Z holds
((id Z) `| Z) . x = 1 ) ) by A2, FDIFF_1:31;

now

let x be Real; :: thesis: ( x in Z implies (((id Z) (#) (sin * (f ^ ))) `| Z) . x = (sin . (1 / x)) - ((1 / x) * (cos . (1 / x))) )
assume A8: x in Z ; :: thesis: (((id Z) (#) (sin * (f ^ ))) `| Z) . x = (sin . (1 / x)) - ((1 / x) * (cos . (1 / x)))
then f . x = x by A3;
then A9: x <> 0 by A3, A8;
(((id Z) (#) (sin * (f ^ ))) `| Z) . x = (((sin * (f ^ )) . x) * (diff (id Z),x)) + (((id Z) . x) * (diff (sin * (f ^ )),x)) by A1, A5, A7, A8, FDIFF_1:29
.= (((sin * (f ^ )) . x) * (((id Z) `| Z) . x)) + (((id Z) . x) * (diff (sin * (f ^ )),x)) by A7, A8, FDIFF_1:def 8
.= (((sin * (f ^ )) . x) * 1) + (((id Z) . x) * (diff (sin * (f ^ )),x)) by A2, A6, A8, FDIFF_1:31
.= ((sin * (f ^ )) . x) + (x * (diff (sin * (f ^ )),x)) by A8, FUNCT_1:35
.= ((sin * (f ^ )) . x) + (x * (((sin * (f ^ )) `| Z) . x)) by A5, A8, FDIFF_1:def 8
.= ((sin * (f ^ )) . x) + (x * (- ((1 / (x ^2 )) * (cos . (1 / x))))) by A2, A3, A8, Th5
.= ((sin * (f ^ )) . x) - ((x * (1 / (x * x))) * (cos . (1 / x)))
.= ((sin * (f ^ )) . x) - ((x * ((1 / x) * (1 / x))) * (cos . (1 / x))) by XCMPLX_1:103
.= ((sin * (f ^ )) . x) - (((x * (1 / x)) * (1 / x)) * (cos . (1 / x)))
.= ((sin * (f ^ )) . x) - ((1 * (1 / x)) * (cos . (1 / x))) by A9, XCMPLX_1:107
.= (sin . ((f ^ ) . x)) - ((1 / x) * (cos . (1 / x))) by A2, A8, FUNCT_1:22
.= (sin . ((f . x) " )) - ((1 / x) * (cos . (1 / x))) by A4, A8, RFUNCT_1:def 8
.= (sin . (1 * (x " ))) - ((1 / x) * (cos . (1 / x))) by A3, A8
.= (sin . (1 / x)) - ((1 / x) * (cos . (1 / x))) by XCMPLX_0:def 9 ;
hence (((id Z) (#) (sin * (f ^ ))) `| Z) . x = (sin . (1 / x)) - ((1 / x) * (cos . (1 / x))) ; :: thesis: verum

end;

hence ( (id Z) (#) (sin * (f ^ )) is_differentiable_on Z & ( for x being Real st x in Z holds
(((id Z) (#) (sin * (f ^ ))) `| Z) . x = (sin . (1 / x)) - ((1 / x) * (cos . (1 / x))) ) ) by A1, A5, A7, FDIFF_1:29; :: thesis: verum