let Z be open Subset of REAL ; :: thesis: ( Z c= dom (sin * ln ) implies ( sin * ln is_differentiable_on Z & ( for x being Real st x in Z holds
((sin * ln ) `| Z) . x = (cos . (log number_e ,x)) / x ) ) )
assume A1: Z c= dom (sin * ln ) ; :: thesis: ( sin * ln is_differentiable_on Z & ( for x being Real st x in Z holds
((sin * ln ) `| Z) . x = (cos . (log number_e ,x)) / x ) )
then for y being set st y in Z holds
y in dom ln by FUNCT_1:21;
then A2: Z c= dom ln by TARSKI:def 3;
then A3: ln is_differentiable_on Z by FDIFF_1:34, TAYLOR_1:18;
A4: 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 A5: ln is_differentiable_in x by A3, FDIFF_1:16;
sin is_differentiable_in ln . x by SIN_COS:69;
hence sin * ln is_differentiable_in x by A5, FDIFF_2:13; :: thesis: verum

end;

then A6: 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 . (log number_e ,x)) / x

proof

let x be Real; :: thesis: ( x in Z implies ((sin * ln ) `| Z) . x = (cos . (log number_e ,x)) / x )
assume A7: x in Z ; :: thesis: ((sin * ln ) `| Z) . x = (cos . (log number_e ,x)) / x
then A8: x in right_open_halfline 0 by A1, FUNCT_1:21, TAYLOR_1:18;
A9: ln is_differentiable_in x by A3, A7, FDIFF_1:16;
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 . (log number_e ,x)) * (diff ln ,x) by A8, TAYLOR_1:def 2
.= (cos . (log number_e ,x)) * (1 / x) by A2, A7, TAYLOR_1:18
.= (cos . (log number_e ,x)) / x by XCMPLX_1:100 ;
hence ((sin * ln ) `| Z) . x = (cos . (log number_e ,x)) / x by A6, A7, 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 . (log number_e ,x)) / x ) ) by A1, A4, FDIFF_1:16; :: thesis: verum