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

for x being Real st x in Z holds

((sin * ln) `| Z) . x = (cos . (log (number_e,x))) / x

((sin * ln) `| Z) . x = (cos . (log (number_e,x))) / x ) ) by A1, A4, FDIFF_1:9; :: thesis: verum

((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

then A6:
sin * ln is_differentiable_on Z
by A1, FDIFF_1:9;
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;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

for x being Real st x in Z holds

((sin * ln) `| Z) . x = (cos . (log (number_e,x))) / x

proof

hence
( sin * ln is_differentiable_on Z & ( for x being Real st x in Z holds
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;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

((sin * ln) `| Z) . x = (cos . (log (number_e,x))) / x ) ) by A1, A4, FDIFF_1:9; :: thesis: verum