let a be Real; 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; 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; ( 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 )
; ( (#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 A1, FUNCT_1:11;
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 A3, FDIFF_1:23;
A6:
for x being Real st x in Z holds
(#R (3 / 2)) * f is_differentiable_in x
then A7:
(#R (3 / 2)) * f is_differentiable_on Z
by A1, FDIFF_1:9;
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;
( x in Z implies (((#R (3 / 2)) * f) `| Z) . x = (3 / 2) * ((a + x) #R (1 / 2)) )
assume A8:
x in Z
;
(((#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 A2, A5, A8, FDIFF_1:9;
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 A5, A8, FDIFF_1:def 7
.=
((3 / 2) * ((a + x) #R ((3 / 2) - 1))) * 1
by A3, A4, A8, A9, FDIFF_1:23
.=
(3 / 2) * ((a + x) #R (1 / 2))
;
hence
(((#R (3 / 2)) * f) `| Z) . x = (3 / 2) * ((a + x) #R (1 / 2))
by A7, A8, FDIFF_1:def 7;
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 A1, A6, FDIFF_1:9; verum