let Z be open Subset of REAL ; :: thesis: 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 ; :: thesis: ( 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 ) & ( for x being Real st x in Z holds
( f1 . x = 1 & f . x > 0 ) ) ) ; :: thesis: ( (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))) ) )
A3: ( f = f1 + (2 (#) sin ) & ( for x being Real st x in Z holds
f1 . x = 1 ) ) by A2;
A4: Z c= dom (ln * f) by A1, VALUED_1:def 5;
then for y being set st y in Z holds
y in dom f by FUNCT_1:21;
then A5: Z c= dom (f1 + (2 (#) sin )) by A2, TARSKI:def 3;
then A6: ( f is_differentiable_on Z & ( for x being Real st x in Z holds
(f `| Z) . x = 2 * (cos . x) ) ) by A3, Lm6;
Z c= (dom f1) /\ (dom (2 (#) sin )) by A5, VALUED_1:def 1;
then A7: ( Z c= dom f1 & Z c= dom (2 (#) sin ) ) by XBOOLE_1:18;
for x being Real st x in Z holds
ln * f is_differentiable_in x

proof

let x be Real; :: thesis: ( x in Z implies ln * f is_differentiable_in x )
assume A8: x in Z ; :: thesis: ln * f is_differentiable_in x
then A9: f is_differentiable_in x by A6, FDIFF_1:16;
f . x > 0 by A2, A8;
hence ln * f is_differentiable_in x by A9, TAYLOR_1:20; :: thesis: verum

end;

then A10: ln * f is_differentiable_on Z by A4, FDIFF_1:16;
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; :: thesis: ( x in Z implies (((1 / 2) (#) (ln * f)) `| Z) . x = (cos . x) / (1 + (2 * (sin . x))) )
assume A11: x in Z ; :: thesis: (((1 / 2) (#) (ln * f)) `| Z) . x = (cos . x) / (1 + (2 * (sin . x)))
then A12: f is_differentiable_in x by A6, FDIFF_1:16;
A13: f . x > 0 by A2, A11;
A14: f . x = (f1 . x) + ((2 (#) sin ) . x) by A2, A5, A11, VALUED_1:def 1
.= 1 + ((2 (#) sin ) . x) by A2, A11
.= 1 + (2 * (sin . x)) by A7, A11, VALUED_1:def 5 ;
(((1 / 2) (#) (ln * f)) `| Z) . x = (1 / 2) * (diff (ln * f),x) by A1, A10, A11, FDIFF_1:28
.= (1 / 2) * ((diff f,x) / (f . x)) by A12, A13, TAYLOR_1:20
.= (1 / 2) * (((f `| Z) . x) / (f . x)) by A6, A11, FDIFF_1:def 8
.= (1 / 2) * ((2 * (cos . x)) / (f . x)) by A3, A5, A11, Lm6
.= ((1 / 2) * (2 * (cos . x))) / (f . x) by XCMPLX_1:75
.= (cos . x) / (1 + (2 * (sin . x))) by A14 ;
hence (((1 / 2) (#) (ln * f)) `| Z) . x = (cos . x) / (1 + (2 * (sin . x))) ; :: thesis: 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, A10, FDIFF_1:28; :: thesis: verum