let Z be open Subset of REAL ; :: thesis: for f being PartFunc of REAL , REAL st Z c= dom (cos * (f ^ )) & ( for x being Real st x in Z holds
( f . x = x & f . x <> 0 ) ) holds
( cos * (f ^ ) is_differentiable_on Z & ( for x being Real st x in Z holds
((cos * (f ^ )) `| Z) . x = (1 / (x ^2 )) * (sin . (1 / x)) ) )
let f be PartFunc of REAL , REAL ; :: thesis: ( Z c= dom (cos * (f ^ )) & ( for x being Real st x in Z holds
( f . x = x & f . x <> 0 ) ) implies ( cos * (f ^ ) is_differentiable_on Z & ( for x being Real st x in Z holds
((cos * (f ^ )) `| Z) . x = (1 / (x ^2 )) * (sin . (1 / x)) ) ) )
assume that
A1: Z c= dom (cos * (f ^ )) and
A2: for x being Real st x in Z holds
( f . x = x & f . x <> 0 ) ; :: thesis: ( cos * (f ^ ) is_differentiable_on Z & ( for x being Real st x in Z holds
((cos * (f ^ )) `| Z) . x = (1 / (x ^2 )) * (sin . (1 / x)) ) )
for y being set st y in Z holds
y in dom (f ^ ) by A1, FUNCT_1:21;
then A3: Z c= dom (f ^ ) by TARSKI:def 3;
dom (f ^ ) c= dom f by RFUNCT_1:11;
then A4: Z c= dom f by A3, XBOOLE_1:1;
then A5: ( f ^ is_differentiable_on Z & ( for x being Real st x in Z holds
((f ^ ) `| Z) . x = - (1 / (x ^2 )) ) ) by A2, Th4;
A6: for x being Real st x in Z holds
cos * (f ^ ) is_differentiable_in x

proof

let x be Real; :: thesis: ( x in Z implies cos * (f ^ ) is_differentiable_in x )
assume x in Z ; :: thesis: cos * (f ^ ) is_differentiable_in x
then A7: f ^ is_differentiable_in x by A5, FDIFF_1:16;
cos is_differentiable_in (f ^ ) . x by SIN_COS:68;
hence cos * (f ^ ) is_differentiable_in x by A7, FDIFF_2:13; :: thesis: verum

end;

then A8: cos * (f ^ ) is_differentiable_on Z by A1, FDIFF_1:16;
for x being Real st x in Z holds
((cos * (f ^ )) `| Z) . x = (1 / (x ^2 )) * (sin . (1 / x))

proof

let x be Real; :: thesis: ( x in Z implies ((cos * (f ^ )) `| Z) . x = (1 / (x ^2 )) * (sin . (1 / x)) )
assume A9: x in Z ; :: thesis: ((cos * (f ^ )) `| Z) . x = (1 / (x ^2 )) * (sin . (1 / x))
then A10: f ^ is_differentiable_in x by A5, FDIFF_1:16;
A11: diff cos ,((f ^ ) . x) = - (sin . ((f ^ ) . x)) by SIN_COS:68;
cos is_differentiable_in (f ^ ) . x by SIN_COS:68;
then diff (cos * (f ^ )),x = (diff cos ,((f ^ ) . x)) * (diff (f ^ ),x) by A10, FDIFF_2:13
.= - ((sin . ((f ^ ) . x)) * (diff (f ^ ),x)) by A11
.= - ((sin . ((f . x) " )) * (diff (f ^ ),x)) by A3, A9, RFUNCT_1:def 8
.= - ((sin . ((f . x) " )) * (((f ^ ) `| Z) . x)) by A5, A9, FDIFF_1:def 8
.= - ((sin . ((f . x) " )) * (- (1 / (x ^2 )))) by A2, A4, A9, Th4
.= - ((sin . (1 * (x " ))) * (- (1 / (x ^2 )))) by A2, A9
.= - ((sin . (1 / x)) * (- (1 / (x ^2 )))) by XCMPLX_0:def 9
.= (sin . (1 / x)) * (1 / (x ^2 )) ;
hence ((cos * (f ^ )) `| Z) . x = (1 / (x ^2 )) * (sin . (1 / x)) by A8, A9, FDIFF_1:def 8; :: thesis: verum

end;

hence ( cos * (f ^ ) is_differentiable_on Z & ( for x being Real st x in Z holds
((cos * (f ^ )) `| Z) . x = (1 / (x ^2 )) * (sin . (1 / x)) ) ) by A1, A6, FDIFF_1:16; :: thesis: verum