let a be Real; :: thesis: for Z being open Subset of REAL
for f being PartFunc of REAL,REAL st Z c= dom ((#R (3 / 2)) * f) & ( for x being Real st x in Z holds
( f . x = a + x & f . x > 0 ) ) holds
( (#R (3 / 2)) * f is_differentiable_on Z & ( for x being Real st x in Z holds
(((#R (3 / 2)) * f) `| Z) . x = (3 / 2) * ((a + x) #R (1 / 2)) ) )

let Z be open Subset of REAL; :: thesis: for f being PartFunc of REAL,REAL st Z c= dom ((#R (3 / 2)) * f) & ( for x being Real st x in Z holds
( f . x = a + x & f . x > 0 ) ) holds
( (#R (3 / 2)) * f is_differentiable_on Z & ( for x being Real st x in Z holds
(((#R (3 / 2)) * f) `| Z) . x = (3 / 2) * ((a + x) #R (1 / 2)) ) )

let f be PartFunc of REAL,REAL; :: thesis: ( Z c= dom ((#R (3 / 2)) * f) & ( for x being Real st x in Z holds
( f . x = a + x & f . x > 0 ) ) implies ( (#R (3 / 2)) * f is_differentiable_on Z & ( for x being Real st x in Z holds
(((#R (3 / 2)) * f) `| Z) . x = (3 / 2) * ((a + x) #R (1 / 2)) ) ) )

assume that
A1: Z c= dom ((#R (3 / 2)) * f) and
A2: for x being Real st x in Z holds
( f . x = a + x & f . x > 0 ) ; :: thesis: ( (#R (3 / 2)) * f is_differentiable_on Z & ( for x being Real st x in Z holds
(((#R (3 / 2)) * f) `| Z) . x = (3 / 2) * ((a + x) #R (1 / 2)) ) )

for y being object st y in Z holds
y in dom f by ;
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 A2;
then A5: f is_differentiable_on Z by ;
A6: for x being Real st x in Z holds
(#R (3 / 2)) * f is_differentiable_in x
proof
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 ;
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 ;
for x being Real st x in Z holds
(((#R (3 / 2)) * f) `| Z) . x = (3 / 2) * ((a + x) #R (1 / 2))
proof
let x be Real; :: thesis: ( x in Z implies (((#R (3 / 2)) * f) `| Z) . x = (3 / 2) * ((a + x) #R (1 / 2)) )
assume A8: x in Z ; :: thesis: (((#R (3 / 2)) * f) `| Z) . x = (3 / 2) * ((a + x) #R (1 / 2))
then A9: f . x = a + x by A2;
( f is_differentiable_in x & f . x > 0 ) by ;
then diff (((#R (3 / 2)) * f),x) = ((3 / 2) * ((f . x) #R ((3 / 2) - 1))) * (diff (f,x)) by TAYLOR_1:22
.= ((3 / 2) * ((f . x) #R ((3 / 2) - 1))) * ((f `| Z) . x) by
.= ((3 / 2) * ((a + x) #R ((3 / 2) - 1))) * 1 by
.= (3 / 2) * ((a + x) #R (1 / 2)) ;
hence (((#R (3 / 2)) * f) `| Z) . x = (3 / 2) * ((a + x) #R (1 / 2)) by ; :: thesis: verum
end;
hence ( (#R (3 / 2)) * f is_differentiable_on Z & ( for x being Real st x in Z holds
(((#R (3 / 2)) * f) `| Z) . x = (3 / 2) * ((a + x) #R (1 / 2)) ) ) by ; :: thesis: verum