let Z be open Subset of REAL; :: thesis: for f being PartFunc of REAL,REAL st Z c= dom ((1 / 2) (#) f) & f = #Z 2 holds
( (1 / 2) (#) f is_differentiable_on Z & ( for x being Real st x in Z holds
(((1 / 2) (#) f) `| Z) . x = x ) )
let f be PartFunc of REAL,REAL; :: thesis: ( Z c= dom ((1 / 2) (#) f) & f = #Z 2 implies ( (1 / 2) (#) f is_differentiable_on Z & ( for x being Real st x in Z holds
(((1 / 2) (#) f) `| Z) . x = x ) ) )
assume that
A1: Z c= dom ((1 / 2) (#) f) and
A2: f = #Z 2 ; :: thesis: ( (1 / 2) (#) f is_differentiable_on Z & ( for x being Real st x in Z holds
(((1 / 2) (#) f) `| Z) . x = x ) )
( Z c= dom f & ( for x being Real st x in Z holds
f is_differentiable_in x ) ) by A1, A2, TAYLOR_1:2, VALUED_1:def 5;
then A3: f is_differentiable_on Z by FDIFF_1:9;
A4: for x being Real st x in Z holds
(f `| Z) . x = 2 * x

proof

let x be Real; :: thesis: ( x in Z implies (f `| Z) . x = 2 * x )
2 * (x #Z (2 - 1)) = 2 * x by PREPOWER:35;
then A5: diff (f,x) = 2 * x by A2, TAYLOR_1:2;
assume x in Z ; :: thesis: (f `| Z) . x = 2 * x
hence (f `| Z) . x = 2 * x by A3, A5, FDIFF_1:def 7; :: thesis: verum

end;

for x being Real st x in Z holds
(((1 / 2) (#) f) `| Z) . x = x

proof

let x be Real; :: thesis: ( x in Z implies (((1 / 2) (#) f) `| Z) . x = x )
assume A6: x in Z ; :: thesis: (((1 / 2) (#) f) `| Z) . x = x
then (((1 / 2) (#) f) `| Z) . x = (1 / 2) * (diff (f,x)) by A1, A3, FDIFF_1:20
.= (1 / 2) * ((f `| Z) . x) by A3, A6, FDIFF_1:def 7
.= (1 / 2) * (2 * x) by A4, A6
.= x ;
hence (((1 / 2) (#) f) `| Z) . x = x ; :: thesis: verum

end;

hence ( (1 / 2) (#) f is_differentiable_on Z & ( for x being Real st x in Z holds
(((1 / 2) (#) f) `| Z) . x = x ) ) by A1, A3, FDIFF_1:20; :: thesis: verum