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 A4: (#R (3 / 2)) * f is_differentiable_on Z by A2, Th27;
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 A5: x in Z ; :: thesis: (((2 / 3) (#) ((#R (3 / 2)) * f)) `| Z) . x = (a + x) #R (1 / 2)
hence (((2 / 3) (#) ((#R (3 / 2)) * f)) `| Z) . x = (2 / 3) * (diff (((#R (3 / 2)) * f),x)) by A1, A4, FDIFF_1:20
.= (2 / 3) * ((((#R (3 / 2)) * f) `| Z) . x) by A4, A5, FDIFF_1:def 7
.= (2 / 3) * ((3 / 2) * ((a + x) #R (1 / 2))) by A2, A3, A5, Th27
.= (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, A4, FDIFF_1:20; :: thesis: verum