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 that
A1: Z c= dom ((- 2) (#) ((#R (1 / 2)) * f)) and
A2: 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)) ) )
A3: Z c= dom ((#R (1 / 2)) * f) by A1, VALUED_1:def 5;
then for y being set st y in Z holds
y in dom f by FUNCT_1:21;
then A4: Z c= dom f by TARSKI:def 3;
A5: for x being Real st x in Z holds
f . x = ((- 1) * x) + a

proof

let x be Real; :: thesis: ( x in Z implies f . x = ((- 1) * x) + a )
assume x in Z ; :: thesis: f . x = ((- 1) * x) + a
then f . x = a - x by A2;
hence f . x = ((- 1) * x) + a ; :: thesis: verum

end;

then A6: f is_differentiable_on Z by A4, FDIFF_1:31;

now

let x be Real; :: thesis: ( x in Z implies (#R (1 / 2)) * f is_differentiable_in x )
assume x in Z ; :: thesis: (#R (1 / 2)) * f is_differentiable_in x
then ( f is_differentiable_in x & f . x > 0 ) by A2, A6, FDIFF_1:16;
hence (#R (1 / 2)) * f is_differentiable_in x by TAYLOR_1:22; :: thesis: verum

end;

then A7: (#R (1 / 2)) * f is_differentiable_on Z by A3, 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 A8: x in Z ; :: thesis: (((- 2) (#) ((#R (1 / 2)) * f)) `| Z) . x = (a - x) #R (- (1 / 2))
then A9: f . x = a - x by A2;
A10: ( f is_differentiable_in x & f . x > 0 ) by A2, A6, A8, FDIFF_1:16;
(((- 2) (#) ((#R (1 / 2)) * f)) `| Z) . x = (- 2) * (diff ((#R (1 / 2)) * f),x) by A1, A7, A8, FDIFF_1:28
.= (- 2) * (((1 / 2) * ((f . x) #R ((1 / 2) - 1))) * (diff f,x)) by A10, TAYLOR_1:22
.= (- 2) * (((1 / 2) * ((f . x) #R ((1 / 2) - 1))) * ((f `| Z) . x)) by A6, A8, FDIFF_1:def 8
.= (- 2) * (((1 / 2) * ((a - x) #R ((1 / 2) - 1))) * (- 1)) by A4, A5, A8, A9, 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, A7, FDIFF_1:28; :: thesis: verum