let a be Real; :: thesis: for Z being open Subset of REAL
for f being PartFunc of REAL ,REAL st Z c= dom (2 (#) ((#R (1 / 2)) * f)) & ( for x being Real st x in Z holds
( f . x = a + x & f . x > 0 ) ) holds
( 2 (#) ((#R (1 / 2)) * f) is_differentiable_on Z & ( for x being Real st x in Z holds
((2 (#) ((#R (1 / 2)) * f)) `| Z) . x = (a + x) #R (- (1 / 2)) ) )
let Z be open Subset of REAL ; :: thesis: for f being PartFunc of REAL ,REAL st Z c= dom (2 (#) ((#R (1 / 2)) * f)) & ( for x being Real st x in Z holds
( f . x = a + x & f . x > 0 ) ) holds
( 2 (#) ((#R (1 / 2)) * f) is_differentiable_on Z & ( for x being Real st x in Z holds
((2 (#) ((#R (1 / 2)) * f)) `| Z) . x = (a + x) #R (- (1 / 2)) ) )
let f be PartFunc of REAL ,REAL ; :: thesis: ( Z c= dom (2 (#) ((#R (1 / 2)) * f)) & ( for x being Real st x in Z holds
( f . x = a + x & f . x > 0 ) ) implies ( 2 (#) ((#R (1 / 2)) * f) is_differentiable_on Z & ( for x being Real st x in Z holds
((2 (#) ((#R (1 / 2)) * f)) `| Z) . x = (a + x) #R (- (1 / 2)) ) ) )
assume A1: ( Z c= dom (2 (#) ((#R (1 / 2)) * f)) & ( for x being Real st x in Z holds
( f . x = a + x & f . x > 0 ) ) ) ; :: thesis: ( 2 (#) ((#R (1 / 2)) * f) is_differentiable_on Z & ( for x being Real st x in Z holds
((2 (#) ((#R (1 / 2)) * f)) `| Z) . x = (a + x) #R (- (1 / 2)) ) )
then A2: ( Z c= dom ((#R (1 / 2)) * f) & ( for x being Real st x in Z holds
( f . x = a + x & f . x > 0 ) ) ) by VALUED_1:def 5;
then for y being set st y in Z holds
y in dom f by FUNCT_1:21;
then A3: Z c= dom f by TARSKI:def 3;
A4: for x being Real st x in Z holds
f . x = (1 * x) + a by A1;
then A5: ( f is_differentiable_on Z & ( for x being Real st x in Z holds
(f `| Z) . x = 1 ) ) by A3, FDIFF_1:31;

now

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 A5, 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 A2, FDIFF_1:16;
for x being Real st x in Z holds
((2 (#) ((#R (1 / 2)) * f)) `| Z) . x = (a + x) #R (- (1 / 2))

proof

let x be Real; :: thesis: ( x in Z implies ((2 (#) ((#R (1 / 2)) * f)) `| Z) . x = (a + x) #R (- (1 / 2)) )
assume A9: x in Z ; :: thesis: ((2 (#) ((#R (1 / 2)) * f)) `| Z) . x = (a + x) #R (- (1 / 2))
then A10: f is_differentiable_in x by A5, FDIFF_1:16;
A11: ( f . x = a + x & f . x > 0 ) by A1, A9;
((2 (#) ((#R (1 / 2)) * f)) `| Z) . x = 2 * (diff ((#R (1 / 2)) * f),x) by A1, A8, A9, FDIFF_1:28
.= 2 * (((1 / 2) * ((f . x) #R ((1 / 2) - 1))) * (diff f,x)) by A10, A11, TAYLOR_1:22
.= 2 * (((1 / 2) * ((f . x) #R ((1 / 2) - 1))) * ((f `| Z) . x)) by A5, A9, FDIFF_1:def 8
.= 2 * (((1 / 2) * ((a + x) #R ((1 / 2) - 1))) * 1) by A3, A4, A9, A11, FDIFF_1:31
.= (a + x) #R (- (1 / 2)) ;
hence ((2 (#) ((#R (1 / 2)) * f)) `| Z) . x = (a + x) #R (- (1 / 2)) ; :: thesis: verum

end;

hence ( 2 (#) ((#R (1 / 2)) * f) is_differentiable_on Z & ( for x being Real st x in Z holds
((2 (#) ((#R (1 / 2)) * f)) `| Z) . x = (a + x) #R (- (1 / 2)) ) ) by A1, A8, FDIFF_1:28; :: thesis: verum