let Z be open Subset of REAL ; :: thesis: for f being PartFunc of REAL , REAL st Z c= dom ((#R (1 / 2)) * f) & ( for x being Real st x in Z holds
( f . x = x & f . x > 0 ) ) holds
( (#R (1 / 2)) * f is_differentiable_on Z & ( for x being Real st x in Z holds
(((#R (1 / 2)) * f) `| Z) . x = (1 / 2) * (x #R (- (1 / 2))) ) )
let f be PartFunc of REAL , REAL ; :: thesis: ( Z c= dom ((#R (1 / 2)) * f) & ( for x being Real st x in Z holds
( f . x = x & f . x > 0 ) ) implies ( (#R (1 / 2)) * f is_differentiable_on Z & ( for x being Real st x in Z holds
(((#R (1 / 2)) * f) `| Z) . x = (1 / 2) * (x #R (- (1 / 2))) ) ) )
assume A1: ( Z c= dom ((#R (1 / 2)) * f) & ( for x being Real st x in Z holds
( f . x = x & f . x > 0 ) ) ) ; :: thesis: ( (#R (1 / 2)) * f is_differentiable_on Z & ( for x being Real st x in Z holds
(((#R (1 / 2)) * f) `| Z) . x = (1 / 2) * (x #R (- (1 / 2))) ) )
then for y being set st y in Z holds
y in dom f by FUNCT_1:21;
then A2: Z c= dom f by TARSKI:def 3;
A3: for x being Real st x in Z holds
f . x = (1 * x) + 0 by A1;
then A4: ( f is_differentiable_on Z & ( for x being Real st x in Z holds
(f `| Z) . x = 1 ) ) by A2, FDIFF_1:31;
A5: for x being Real st x in Z holds
(#R (1 / 2)) * f is_differentiable_in x

proof

let x be Real; :: thesis: ( x in Z implies (#R (1 / 2)) * f is_differentiable_in x )
assume A6: x in Z ; :: thesis: (#R (1 / 2)) * f is_differentiable_in x
then A7: f is_differentiable_in x by A4, FDIFF_1:16;
f . x > 0 by A1, A6;
hence (#R (1 / 2)) * f is_differentiable_in x by A7, TAYLOR_1:22; :: thesis: verum

end;

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

proof

let x be Real; :: thesis: ( x in Z implies (((#R (1 / 2)) * f) `| Z) . x = (1 / 2) * (x #R (- (1 / 2))) )
assume A9: x in Z ; :: thesis: (((#R (1 / 2)) * f) `| Z) . x = (1 / 2) * (x #R (- (1 / 2)))
then A10: f is_differentiable_in x by A4, FDIFF_1:16;
A11: ( f . x = x & f . x > 0 ) by A1, A9;
then diff ((#R (1 / 2)) * f),x = ((1 / 2) * ((f . x) #R ((1 / 2) - 1))) * (diff f,x) by A10, TAYLOR_1:22
.= ((1 / 2) * ((f . x) #R ((1 / 2) - 1))) * ((f `| Z) . x) by A4, A9, FDIFF_1:def 8
.= ((1 / 2) * (x #R ((1 / 2) - 1))) * 1 by A2, A3, A9, A11, FDIFF_1:31
.= (1 / 2) * (x #R (- (1 / 2))) ;
hence (((#R (1 / 2)) * f) `| Z) . x = (1 / 2) * (x #R (- (1 / 2))) by A8, A9, FDIFF_1:def 8; :: thesis: verum

end;

hence ( (#R (1 / 2)) * f is_differentiable_on Z & ( for x being Real st x in Z holds
(((#R (1 / 2)) * f) `| Z) . x = (1 / 2) * (x #R (- (1 / 2))) ) ) by A1, A5, FDIFF_1:16; :: thesis: verum