let Z be open Subset of REAL; for f, f1 being PartFunc of REAL,REAL st Z c= dom ((1 / 2) (#) (ln * f)) & f = f1 + (2 (#) sin) & ( for x being Real st x in Z holds
( f1 . x = 1 & f . x > 0 ) ) holds
( (1 / 2) (#) (ln * f) is_differentiable_on Z & ( for x being Real st x in Z holds
(((1 / 2) (#) (ln * f)) `| Z) . x = (cos . x) / (1 + (2 * (sin . x))) ) )
let f, f1 be PartFunc of REAL,REAL; ( Z c= dom ((1 / 2) (#) (ln * f)) & f = f1 + (2 (#) sin) & ( for x being Real st x in Z holds
( f1 . x = 1 & f . x > 0 ) ) implies ( (1 / 2) (#) (ln * f) is_differentiable_on Z & ( for x being Real st x in Z holds
(((1 / 2) (#) (ln * f)) `| Z) . x = (cos . x) / (1 + (2 * (sin . x))) ) ) )
assume that
A1:
Z c= dom ((1 / 2) (#) (ln * f))
and
A2:
f = f1 + (2 (#) sin)
and
A3:
for x being Real st x in Z holds
( f1 . x = 1 & f . x > 0 )
; ( (1 / 2) (#) (ln * f) is_differentiable_on Z & ( for x being Real st x in Z holds
(((1 / 2) (#) (ln * f)) `| Z) . x = (cos . x) / (1 + (2 * (sin . 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 (f1 + (2 (#) sin))
by A2, TARSKI:def 3;
A6:
for x being Real st x in Z holds
f1 . x = 1
by A3;
then A7:
f is_differentiable_on Z
by A2, A5, Lm6;
for x being Real st x in Z holds
ln * f is_differentiable_in x
then A8:
ln * f is_differentiable_on Z
by A4, FDIFF_1:9;
Z c= (dom f1) /\ (dom (2 (#) sin))
by A5, VALUED_1:def 1;
then A9:
Z c= dom (2 (#) sin)
by XBOOLE_1:18;
for x being Real st x in Z holds
(((1 / 2) (#) (ln * f)) `| Z) . x = (cos . x) / (1 + (2 * (sin . x)))
proof
let x be
Real;
( x in Z implies (((1 / 2) (#) (ln * f)) `| Z) . x = (cos . x) / (1 + (2 * (sin . x))) )
assume A10:
x in Z
;
(((1 / 2) (#) (ln * f)) `| Z) . x = (cos . x) / (1 + (2 * (sin . x)))
then A11:
f . x =
(f1 . x) + ((2 (#) sin) . x)
by A2, A5, VALUED_1:def 1
.=
1
+ ((2 (#) sin) . x)
by A3, A10
.=
1
+ (2 * (sin . x))
by A9, A10, VALUED_1:def 5
;
A12:
(
f is_differentiable_in x &
f . x > 0 )
by A3, A7, A10, FDIFF_1:9;
(((1 / 2) (#) (ln * f)) `| Z) . x =
(1 / 2) * (diff ((ln * f),x))
by A1, A8, A10, FDIFF_1:20
.=
(1 / 2) * ((diff (f,x)) / (f . x))
by A12, TAYLOR_1:20
.=
(1 / 2) * (((f `| Z) . x) / (f . x))
by A7, A10, FDIFF_1:def 7
.=
(1 / 2) * ((2 * (cos . x)) / (f . x))
by A2, A6, A5, A10, Lm6
.=
((1 / 2) * (2 * (cos . x))) / (f . x)
by XCMPLX_1:74
.=
(cos . x) / (1 + (2 * (sin . x)))
by A11
;
hence
(((1 / 2) (#) (ln * f)) `| Z) . x = (cos . x) / (1 + (2 * (sin . 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 = (cos . x) / (1 + (2 * (sin . x))) ) )
by A1, A8, FDIFF_1:20; verum