let a be Real; for Z being open Subset of REAL
for f, f1 being PartFunc of REAL,REAL st Z c= dom ((2 / (3 * (log (number_e,a)))) (#) ((#R (3 / 2)) * (f + (exp_R * f1)))) & ( for x being Real st x in Z holds
( f . x = 1 & f1 . x = x * (log (number_e,a)) ) ) & a > 0 & a <> 1 holds
( (2 / (3 * (log (number_e,a)))) (#) ((#R (3 / 2)) * (f + (exp_R * f1))) is_differentiable_on Z & ( for x being Real st x in Z holds
(((2 / (3 * (log (number_e,a)))) (#) ((#R (3 / 2)) * (f + (exp_R * f1)))) `| Z) . x = (a #R x) * ((1 + (a #R x)) #R (1 / 2)) ) )
let Z be open Subset of REAL; for f, f1 being PartFunc of REAL,REAL st Z c= dom ((2 / (3 * (log (number_e,a)))) (#) ((#R (3 / 2)) * (f + (exp_R * f1)))) & ( for x being Real st x in Z holds
( f . x = 1 & f1 . x = x * (log (number_e,a)) ) ) & a > 0 & a <> 1 holds
( (2 / (3 * (log (number_e,a)))) (#) ((#R (3 / 2)) * (f + (exp_R * f1))) is_differentiable_on Z & ( for x being Real st x in Z holds
(((2 / (3 * (log (number_e,a)))) (#) ((#R (3 / 2)) * (f + (exp_R * f1)))) `| Z) . x = (a #R x) * ((1 + (a #R x)) #R (1 / 2)) ) )
let f, f1 be PartFunc of REAL,REAL; ( Z c= dom ((2 / (3 * (log (number_e,a)))) (#) ((#R (3 / 2)) * (f + (exp_R * f1)))) & ( for x being Real st x in Z holds
( f . x = 1 & f1 . x = x * (log (number_e,a)) ) ) & a > 0 & a <> 1 implies ( (2 / (3 * (log (number_e,a)))) (#) ((#R (3 / 2)) * (f + (exp_R * f1))) is_differentiable_on Z & ( for x being Real st x in Z holds
(((2 / (3 * (log (number_e,a)))) (#) ((#R (3 / 2)) * (f + (exp_R * f1)))) `| Z) . x = (a #R x) * ((1 + (a #R x)) #R (1 / 2)) ) ) )
assume that
A1:
Z c= dom ((2 / (3 * (log (number_e,a)))) (#) ((#R (3 / 2)) * (f + (exp_R * f1))))
and
A2:
for x being Real st x in Z holds
( f . x = 1 & f1 . x = x * (log (number_e,a)) )
and
A3:
a > 0
and
A4:
a <> 1
; ( (2 / (3 * (log (number_e,a)))) (#) ((#R (3 / 2)) * (f + (exp_R * f1))) is_differentiable_on Z & ( for x being Real st x in Z holds
(((2 / (3 * (log (number_e,a)))) (#) ((#R (3 / 2)) * (f + (exp_R * f1)))) `| Z) . x = (a #R x) * ((1 + (a #R x)) #R (1 / 2)) ) )
A5:
for x being Real st x in Z holds
f . x = (0 * x) + 1
by A2;
A6:
Z c= dom ((#R (3 / 2)) * (f + (exp_R * f1)))
by A1, VALUED_1:def 5;
then
for y being object st y in Z holds
y in dom (f + (exp_R * f1))
by FUNCT_1:11;
then A7:
Z c= dom (f + (exp_R * f1))
by TARSKI:def 3;
then A8:
Z c= (dom (exp_R * f1)) /\ (dom f)
by VALUED_1:def 1;
then A9:
Z c= dom (exp_R * f1)
by XBOOLE_1:18;
A10:
for x being Real st x in Z holds
f1 . x = x * (log (number_e,a))
by A2;
then A11:
exp_R * f1 is_differentiable_on Z
by A3, A9, Th11;
A12:
Z c= dom f
by A8, XBOOLE_1:18;
then A13:
f is_differentiable_on Z
by A5, FDIFF_1:23;
then A14:
f + (exp_R * f1) is_differentiable_on Z
by A7, A11, FDIFF_1:18;
A15:
for x being Real st x in Z holds
(f + (exp_R * f1)) . x > 0
then A17:
(#R (3 / 2)) * (f + (exp_R * f1)) is_differentiable_on Z
by A6, FDIFF_1:9;
A18:
log (number_e,a) <> 0
proof
A19:
number_e <> 1
by TAYLOR_1:11;
assume
log (
number_e,
a)
= 0
;
contradiction
then
log (
number_e,
a)
= log (
number_e,1)
by SIN_COS2:13, TAYLOR_1:13;
then a =
number_e to_power (log (number_e,1))
by A3, A19, POWER:def 3, TAYLOR_1:11
.=
1
by A19, POWER:def 3, TAYLOR_1:11
;
hence
contradiction
by A4;
verum
end;
for x being Real st x in Z holds
(((2 / (3 * (log (number_e,a)))) (#) ((#R (3 / 2)) * (f + (exp_R * f1)))) `| Z) . x = (a #R x) * ((1 + (a #R x)) #R (1 / 2))
proof
let x be
Real;
( x in Z implies (((2 / (3 * (log (number_e,a)))) (#) ((#R (3 / 2)) * (f + (exp_R * f1)))) `| Z) . x = (a #R x) * ((1 + (a #R x)) #R (1 / 2)) )
A20:
3
* (log (number_e,a)) <> 0
by A18;
assume A21:
x in Z
;
(((2 / (3 * (log (number_e,a)))) (#) ((#R (3 / 2)) * (f + (exp_R * f1)))) `| Z) . x = (a #R x) * ((1 + (a #R x)) #R (1 / 2))
then A22:
((f + (exp_R * f1)) `| Z) . x =
(diff (f,x)) + (diff ((exp_R * f1),x))
by A7, A13, A11, FDIFF_1:18
.=
(diff (f,x)) + (((exp_R * f1) `| Z) . x)
by A11, A21, FDIFF_1:def 7
.=
((f `| Z) . x) + (((exp_R * f1) `| Z) . x)
by A13, A21, FDIFF_1:def 7
.=
0 + (((exp_R * f1) `| Z) . x)
by A12, A5, A21, FDIFF_1:23
.=
(a #R x) * (log (number_e,a))
by A3, A10, A9, A21, Th11
;
A23:
(f + (exp_R * f1)) . x =
(f . x) + ((exp_R * f1) . x)
by A7, A21, VALUED_1:def 1
.=
(f . x) + (exp_R . (f1 . x))
by A9, A21, FUNCT_1:12
.=
1
+ (exp_R . (f1 . x))
by A2, A21
.=
1
+ (exp_R . (x * (log (number_e,a))))
by A2, A21
.=
1
+ (a #R x)
by A3, Th1
;
(
f + (exp_R * f1) is_differentiable_in x &
(f + (exp_R * f1)) . x > 0 )
by A14, A15, A21, FDIFF_1:9;
then diff (
((#R (3 / 2)) * (f + (exp_R * f1))),
x) =
((3 / 2) * (((f + (exp_R * f1)) . x) #R ((3 / 2) - 1))) * (diff ((f + (exp_R * f1)),x))
by TAYLOR_1:22
.=
((3 / 2) * ((1 + (a #R x)) #R (1 / 2))) * ((a #R x) * (log (number_e,a)))
by A14, A21, A23, A22, FDIFF_1:def 7
.=
(((3 * (log (number_e,a))) / 2) * (a #R x)) * ((1 + (a #R x)) #R (1 / 2))
;
then (((2 / (3 * (log (number_e,a)))) (#) ((#R (3 / 2)) * (f + (exp_R * f1)))) `| Z) . x =
(2 / (3 * (log (number_e,a)))) * ((((3 * (log (number_e,a))) / 2) * (a #R x)) * ((1 + (a #R x)) #R (1 / 2)))
by A1, A17, A21, FDIFF_1:20
.=
(((2 / (3 * (log (number_e,a)))) * ((3 * (log (number_e,a))) / 2)) * (a #R x)) * ((1 + (a #R x)) #R (1 / 2))
.=
(1 * (a #R x)) * ((1 + (a #R x)) #R (1 / 2))
by A20, XCMPLX_1:112
.=
(a #R x) * ((1 + (a #R x)) #R (1 / 2))
;
hence
(((2 / (3 * (log (number_e,a)))) (#) ((#R (3 / 2)) * (f + (exp_R * f1)))) `| Z) . x = (a #R x) * ((1 + (a #R x)) #R (1 / 2))
;
verum
end;
hence
( (2 / (3 * (log (number_e,a)))) (#) ((#R (3 / 2)) * (f + (exp_R * f1))) is_differentiable_on Z & ( for x being Real st x in Z holds
(((2 / (3 * (log (number_e,a)))) (#) ((#R (3 / 2)) * (f + (exp_R * f1)))) `| Z) . x = (a #R x) * ((1 + (a #R x)) #R (1 / 2)) ) )
by A1, A17, FDIFF_1:20; verum