let Z be open Subset of REAL ; :: thesis: ( Z c= dom (- (cos * ln )) implies ( - (cos * ln ) is_differentiable_on Z & ( for x being Real st x in Z holds
((- (cos * ln )) `| Z) . x = (sin . (log number_e ,x)) / x ) ) )
assume A1: Z c= dom (- (cos * ln )) ; :: thesis: ( - (cos * ln ) is_differentiable_on Z & ( for x being Real st x in Z holds
((- (cos * ln )) `| Z) . x = (sin . (log number_e ,x)) / x ) )
then A2: Z c= dom (cos * ln ) by VALUED_1:8;
A3: Z c= dom ((- 1) (#) (cos * ln )) by A1;
for y being set st y in Z holds
y in dom ln by A2, FUNCT_1:21;
then A4: Z c= dom ln by TARSKI:def 3;
then A5: ln is_differentiable_on Z by FDIFF_1:34, TAYLOR_1:18;
for x being Real st x in Z holds
cos * ln is_differentiable_in x

proof

let x be Real; :: thesis: ( x in Z implies cos * ln is_differentiable_in x )
assume x in Z ; :: thesis: cos * ln is_differentiable_in x
then A6: ln is_differentiable_in x by A5, FDIFF_1:16;
cos is_differentiable_in ln . x by SIN_COS:68;
hence cos * ln is_differentiable_in x by A6, FDIFF_2:13; :: thesis: verum

end;

then A7: cos * ln is_differentiable_on Z by A2, FDIFF_1:16;
for x being Real st x in Z holds
((- (cos * ln )) `| Z) . x = (sin . (log number_e ,x)) / x

proof

let x be Real; :: thesis: ( x in Z implies ((- (cos * ln )) `| Z) . x = (sin . (log number_e ,x)) / x )
assume A8: x in Z ; :: thesis: ((- (cos * ln )) `| Z) . x = (sin . (log number_e ,x)) / x
then A9: x in right_open_halfline 0 by A2, FUNCT_1:21, TAYLOR_1:18;
A10: ln is_differentiable_in x by A5, A8, FDIFF_1:16;
A11: cos is_differentiable_in ln . x by SIN_COS:68;
((- (cos * ln )) `| Z) . x = (- 1) * (diff (cos * ln ),x) by A1, A7, A8, FDIFF_1:28
.= (- 1) * ((diff cos ,(ln . x)) * (diff ln ,x)) by A10, A11, FDIFF_2:13
.= (- 1) * ((- (sin . (ln . x))) * (diff ln ,x)) by SIN_COS:68
.= ((- 1) * (- (sin . (ln . x)))) * (diff ln ,x)
.= ((- 1) * (- (sin . (log number_e ,x)))) * (diff ln ,x) by A9, TAYLOR_1:def 2
.= ((- 1) * (- (sin . (log number_e ,x)))) * (1 / x) by A4, A8, TAYLOR_1:18
.= (sin . (log number_e ,x)) / x by XCMPLX_1:100 ;
hence ((- (cos * ln )) `| Z) . x = (sin . (log number_e ,x)) / x ; :: thesis: verum

end;

hence ( - (cos * ln ) is_differentiable_on Z & ( for x being Real st x in Z holds
((- (cos * ln )) `| Z) . x = (sin . (log number_e ,x)) / x ) ) by A3, A7, FDIFF_1:28; :: thesis: verum