let Z be open Subset of REAL; for f, f1 being PartFunc of REAL,REAL st Z c= dom ((1 / 2) (#) (ln * f)) & f = ((#Z 2) * exp_R) + f1 & ( for x being Real st x in Z holds
f1 . x = 1 ) holds
( (1 / 2) (#) (ln * f) is_differentiable_on Z & ( for x being Real st x in Z holds
(((1 / 2) (#) (ln * f)) `| Z) . x = (exp_R x) / ((exp_R x) + (exp_R (- x))) ) )
let f, f1 be PartFunc of REAL,REAL; ( Z c= dom ((1 / 2) (#) (ln * f)) & f = ((#Z 2) * exp_R) + f1 & ( for x being Real st x in Z holds
f1 . x = 1 ) implies ( (1 / 2) (#) (ln * f) is_differentiable_on Z & ( for x being Real st x in Z holds
(((1 / 2) (#) (ln * f)) `| Z) . x = (exp_R x) / ((exp_R x) + (exp_R (- x))) ) ) )
assume that
A1:
Z c= dom ((1 / 2) (#) (ln * f))
and
A2:
f = ((#Z 2) * exp_R) + f1
and
A3:
for x being Real st x in Z holds
f1 . x = 1
; ( (1 / 2) (#) (ln * f) is_differentiable_on Z & ( for x being Real st x in Z holds
(((1 / 2) (#) (ln * f)) `| Z) . x = (exp_R x) / ((exp_R x) + (exp_R (- x))) ) )
A4:
Z c= dom (ln * 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 A5:
Z c= dom (((#Z 2) * exp_R) + f1)
by A2, TARSKI:def 3;
then A6:
((#Z 2) * exp_R) + f1 is_differentiable_on Z
by A3, Th23;
Z c= (dom ((#Z 2) * exp_R)) /\ (dom f1)
by A5, VALUED_1:def 1;
then A7:
Z c= dom ((#Z 2) * exp_R)
by XBOOLE_1:18;
A8:
for x being Real st x in Z holds
f . x > 0
for x being Real st x in Z holds
ln * f is_differentiable_in x
then A11:
ln * f is_differentiable_on Z
by A4, FDIFF_1:9;
for x being Real st x in Z holds
(((1 / 2) (#) (ln * f)) `| Z) . x = (exp_R x) / ((exp_R x) + (exp_R (- x)))
proof
let x be
Real;
( x in Z implies (((1 / 2) (#) (ln * f)) `| Z) . x = (exp_R x) / ((exp_R x) + (exp_R (- x))) )
A12:
exp_R x > 0
by SIN_COS:55;
assume A13:
x in Z
;
(((1 / 2) (#) (ln * f)) `| Z) . x = (exp_R x) / ((exp_R x) + (exp_R (- x)))
then A14:
(
f is_differentiable_in x &
f . x > 0 )
by A2, A6, A8, FDIFF_1:9;
A15:
(((#Z 2) * exp_R) + f1) . x =
(((#Z 2) * exp_R) . x) + (f1 . x)
by A5, A13, VALUED_1:def 1
.=
((#Z 2) . (exp_R . x)) + (f1 . x)
by A7, A13, FUNCT_1:12
.=
((exp_R . x) #Z 2) + (f1 . x)
by TAYLOR_1:def 1
.=
((exp_R . x) #Z 2) + 1
by A3, A13
.=
((exp_R x) #Z (1 + 1)) + 1
by SIN_COS:def 23
.=
(((exp_R x) #Z 1) * ((exp_R x) #Z 1)) + 1
by A12, PREPOWER:44
.=
((exp_R x) * ((exp_R x) #Z 1)) + 1
by PREPOWER:35
.=
((exp_R x) * (exp_R x)) + 1
by PREPOWER:35
;
(((1 / 2) (#) (ln * f)) `| Z) . x =
(1 / 2) * (diff ((ln * f),x))
by A1, A11, A13, FDIFF_1:20
.=
(1 / 2) * ((diff (f,x)) / (f . x))
by A14, TAYLOR_1:20
.=
(1 / 2) * ((((((#Z 2) * exp_R) + f1) `| Z) . x) / ((((#Z 2) * exp_R) + f1) . x))
by A2, A6, A13, FDIFF_1:def 7
.=
(1 / 2) * ((2 * (exp_R (2 * x))) / (((exp_R x) * (exp_R x)) + 1))
by A3, A5, A13, A15, Th23
.=
((1 / 2) * (2 * (exp_R (2 * x)))) / (((exp_R x) * (exp_R x)) + 1)
by XCMPLX_1:74
.=
((exp_R (x + x)) / (exp_R x)) / ((((exp_R x) * (exp_R x)) + 1) / (exp_R x))
by A12, XCMPLX_1:55
.=
(((exp_R x) * (exp_R x)) / (exp_R x)) / ((((exp_R x) * (exp_R x)) + 1) / (exp_R x))
by SIN_COS:50
.=
(((exp_R x) * (exp_R x)) / (exp_R x)) / ((((exp_R x) * (exp_R x)) / (exp_R x)) + (1 / (exp_R x)))
by XCMPLX_1:62
.=
(exp_R x) / ((((exp_R x) * (exp_R x)) / (exp_R x)) + (1 / (exp_R x)))
by A12, XCMPLX_1:89
.=
(exp_R x) / ((exp_R x) + (1 / (exp_R x)))
by A12, XCMPLX_1:89
.=
(exp_R x) / ((exp_R x) + (exp_R (- x)))
by TAYLOR_1:4
;
hence
(((1 / 2) (#) (ln * f)) `| Z) . x = (exp_R x) / ((exp_R x) + (exp_R (- x)))
;
verum
end;
hence
( (1 / 2) (#) (ln * f) is_differentiable_on Z & ( for x being Real st x in Z holds
(((1 / 2) (#) (ln * f)) `| Z) . x = (exp_R x) / ((exp_R x) + (exp_R (- x))) ) )
by A1, A11, FDIFF_1:20; verum