let Z be open Subset of REAL ; :: thesis: ( Z c= dom (sin * ln ) & ( for x being Real st x in Z holds
x > 0 ) implies ( sin * ln is_differentiable_on Z & ( for x being Real st x in Z holds
((sin * ln ) `| Z) . x = (cos . (ln . x)) / x ) ) )
assume that
A1: Z c= dom (sin * ln ) and
A2: for x being Real st x in Z holds
x > 0 ; :: thesis: ( sin * ln is_differentiable_on Z & ( for x being Real st x in Z holds
((sin * ln ) `| Z) . x = (cos . (ln . x)) / x ) )
A3: for x being Real st x in Z holds
sin * ln is_differentiable_in x

proof

let x be Real; :: thesis: ( x in Z implies sin * ln is_differentiable_in x )
assume x in Z ; :: thesis: sin * ln is_differentiable_in x
then x > 0 by A2;
then A4: ln is_differentiable_in x by TAYLOR_1:18;
sin is_differentiable_in ln . x by SIN_COS:69;
hence sin * ln is_differentiable_in x by A4, FDIFF_2:13; :: thesis: verum

end;

then A5: sin * ln is_differentiable_on Z by A1, FDIFF_1:16;
for x being Real st x in Z holds
((sin * ln ) `| Z) . x = (cos . (ln . x)) / x

proof

let x be Real; :: thesis: ( x in Z implies ((sin * ln ) `| Z) . x = (cos . (ln . x)) / x )
assume A6: x in Z ; :: thesis: ((sin * ln ) `| Z) . x = (cos . (ln . x)) / x
then A7: x > 0 by A2;
then A8: x in right_open_halfline 0 by Lm3;
A9: ln is_differentiable_in x by A7, TAYLOR_1:18;
sin is_differentiable_in ln . x by SIN_COS:69;
then diff (sin * ln ),x = (diff sin ,(ln . x)) * (diff ln ,x) by A9, FDIFF_2:13
.= (cos . (ln . x)) * (diff ln ,x) by SIN_COS:69
.= (cos . (ln . x)) * (1 / x) by A8, TAYLOR_1:18
.= (cos . (ln . x)) / x by XCMPLX_1:100 ;
hence ((sin * ln ) `| Z) . x = (cos . (ln . x)) / x by A5, A6, FDIFF_1:def 8; :: thesis: verum

end;

hence ( sin * ln is_differentiable_on Z & ( for x being Real st x in Z holds
((sin * ln ) `| Z) . x = (cos . (ln . x)) / x ) ) by A1, A3, FDIFF_1:16; :: thesis: verum