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 object st y in Z holds
y in dom ln by FUNCT_1:11;
then A2: Z c= dom ln by TARSKI:def 3;
then A3: ln is_differentiable_on Z by FDIFF_1:26, 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_1:9;
sin is_differentiable_in ln . x by SIN_COS:64;
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_1:9;
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 )
A7: sin is_differentiable_in ln . x by SIN_COS:64;
assume A8: x in Z ; :: thesis: ((sin * ln) `| Z) . x = (cos . (log (number_e,x))) / x
then A9: x in right_open_halfline 0 by A1, FUNCT_1:11, TAYLOR_1:18;
ln is_differentiable_in x by A3, A8, FDIFF_1:9;
then diff ((sin * ln),x) = (diff (sin,(ln . x))) * (diff (ln,x)) by A7, FDIFF_2:13
.= (cos . (ln . x)) * (diff (ln,x)) by SIN_COS:64
.= (cos . (log (number_e,x))) * (diff (ln,x)) by A9, TAYLOR_1:def 2
.= (cos . (log (number_e,x))) * (1 / x) by A2, A8, TAYLOR_1:18
.= (cos . (log (number_e,x))) / x by XCMPLX_1:99 ;
hence ((sin * ln) `| Z) . x = (cos . (log (number_e,x))) / x by A6, A8, FDIFF_1:def 7; :: 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_1:9; :: thesis: verum