let a be Real; :: thesis: for Z being open Subset of REAL
for f being PartFunc of REAL,REAL st Z c= dom ((- (2 / 3)) (#) ((#R (3 / 2)) * f)) & ( for x being Real st x in Z holds
( f . x = a - x & f . x > 0 ) ) holds
( (- (2 / 3)) (#) ((#R (3 / 2)) * f) is_differentiable_on Z & ( for x being Real st x in Z holds
(((- (2 / 3)) (#) ((#R (3 / 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 / 3)) (#) ((#R (3 / 2)) * f)) & ( for x being Real st x in Z holds
( f . x = a - x & f . x > 0 ) ) holds
( (- (2 / 3)) (#) ((#R (3 / 2)) * f) is_differentiable_on Z & ( for x being Real st x in Z holds
(((- (2 / 3)) (#) ((#R (3 / 2)) * f)) `| Z) . x = (a - x) #R (1 / 2) ) )
let f be PartFunc of REAL,REAL; :: thesis: ( Z c= dom ((- (2 / 3)) (#) ((#R (3 / 2)) * f)) & ( for x being Real st x in Z holds
( f . x = a - x & f . x > 0 ) ) implies ( (- (2 / 3)) (#) ((#R (3 / 2)) * f) is_differentiable_on Z & ( for x being Real st x in Z holds
(((- (2 / 3)) (#) ((#R (3 / 2)) * f)) `| Z) . x = (a - x) #R (1 / 2) ) ) )
assume that
A1: Z c= dom ((- (2 / 3)) (#) ((#R (3 / 2)) * f)) and
A2: for x being Real st x in Z holds
( f . x = a - x & f . x > 0 ) ; :: thesis: ( (- (2 / 3)) (#) ((#R (3 / 2)) * f) is_differentiable_on Z & ( for x being Real st x in Z holds
(((- (2 / 3)) (#) ((#R (3 / 2)) * f)) `| Z) . x = (a - x) #R (1 / 2) ) )
A3: Z c= dom ((#R (3 / 2)) * f) by A1, VALUED_1:def 5;
then for y being object st y in Z holds
y in dom f by FUNCT_1:11;
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_1:23;

now :: thesis: for x being Real st x in Z holds
(#R (3 / 2)) * f is_differentiable_in x

let x be Real; :: thesis: ( x in Z implies (#R (3 / 2)) * f is_differentiable_in x )
assume x in Z ; :: thesis: (#R (3 / 2)) * f is_differentiable_in x
then ( f is_differentiable_in x & f . x > 0 ) by A2, A6, FDIFF_1:9;
hence (#R (3 / 2)) * f is_differentiable_in x by TAYLOR_1:22; :: thesis: verum

end;

then A7: (#R (3 / 2)) * f is_differentiable_on Z by A3, FDIFF_1:9;
for x being Real st x in Z holds
(((- (2 / 3)) (#) ((#R (3 / 2)) * f)) `| Z) . x = (a - x) #R (1 / 2)

proof

let x be Real; :: thesis: ( x in Z implies (((- (2 / 3)) (#) ((#R (3 / 2)) * f)) `| Z) . x = (a - x) #R (1 / 2) )
assume A8: x in Z ; :: thesis: (((- (2 / 3)) (#) ((#R (3 / 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_1:9;
(((- (2 / 3)) (#) ((#R (3 / 2)) * f)) `| Z) . x = (- (2 / 3)) * (diff (((#R (3 / 2)) * f),x)) by A1, A7, A8, FDIFF_1:20
.= (- (2 / 3)) * (((3 / 2) * ((f . x) #R ((3 / 2) - 1))) * (diff (f,x))) by A10, TAYLOR_1:22
.= (- (2 / 3)) * (((3 / 2) * ((f . x) #R ((3 / 2) - 1))) * ((f `| Z) . x)) by A6, A8, FDIFF_1:def 7
.= (- (2 / 3)) * (((3 / 2) * ((a - x) #R ((3 / 2) - 1))) * (- 1)) by A4, A5, A8, A9, FDIFF_1:23
.= (a - x) #R (1 / 2) ;
hence (((- (2 / 3)) (#) ((#R (3 / 2)) * f)) `| Z) . x = (a - x) #R (1 / 2) ; :: thesis: verum

end;

hence ( (- (2 / 3)) (#) ((#R (3 / 2)) * f) is_differentiable_on Z & ( for x being Real st x in Z holds
(((- (2 / 3)) (#) ((#R (3 / 2)) * f)) `| Z) . x = (a - x) #R (1 / 2) ) ) by A1, A7, FDIFF_1:20; :: thesis: verum