let Z be open Subset of REAL; :: thesis: ( Z c= dom (ln * cos) & ( for x being Real st x in Z holds

cos . x > 0 ) implies ( ln * cos is_differentiable_on Z & ( for x being Real st x in Z holds

((ln * cos) `| Z) . x = - (tan x) ) ) )

assume that

A1: Z c= dom (ln * cos) and

A2: for x being Real st x in Z holds

cos . x > 0 ; :: thesis: ( ln * cos is_differentiable_on Z & ( for x being Real st x in Z holds

((ln * cos) `| Z) . x = - (tan x) ) )

A3: cos is_differentiable_on Z by FDIFF_1:26, SIN_COS:67;

A4: for x being Real st x in Z holds

ln * cos is_differentiable_in x

for x being Real st x in Z holds

((ln * cos) `| Z) . x = - (tan x)

((ln * cos) `| Z) . x = - (tan x) ) ) by A1, A4, FDIFF_1:9; :: thesis: verum

cos . x > 0 ) implies ( ln * cos is_differentiable_on Z & ( for x being Real st x in Z holds

((ln * cos) `| Z) . x = - (tan x) ) ) )

assume that

A1: Z c= dom (ln * cos) and

A2: for x being Real st x in Z holds

cos . x > 0 ; :: thesis: ( ln * cos is_differentiable_on Z & ( for x being Real st x in Z holds

((ln * cos) `| Z) . x = - (tan x) ) )

A3: cos is_differentiable_on Z by FDIFF_1:26, SIN_COS:67;

A4: for x being Real st x in Z holds

ln * cos is_differentiable_in x

proof

then A5:
ln * cos is_differentiable_on Z
by A1, FDIFF_1:9;
let x be Real; :: thesis: ( x in Z implies ln * cos is_differentiable_in x )

assume x in Z ; :: thesis: ln * cos is_differentiable_in x

then ( cos is_differentiable_in x & cos . x > 0 ) by A2, A3, FDIFF_1:9;

hence ln * cos is_differentiable_in x by TAYLOR_1:20; :: thesis: verum

end;assume x in Z ; :: thesis: ln * cos is_differentiable_in x

then ( cos is_differentiable_in x & cos . x > 0 ) by A2, A3, FDIFF_1:9;

hence ln * cos is_differentiable_in x by TAYLOR_1:20; :: thesis: verum

for x being Real st x in Z holds

((ln * cos) `| Z) . x = - (tan x)

proof

hence
( ln * cos is_differentiable_on Z & ( for x being Real st x in Z holds
let x be Real; :: thesis: ( x in Z implies ((ln * cos) `| Z) . x = - (tan x) )

assume A6: x in Z ; :: thesis: ((ln * cos) `| Z) . x = - (tan x)

then ( cos is_differentiable_in x & cos . x > 0 ) by A2, A3, FDIFF_1:9;

then diff ((ln * cos),x) = (diff (cos,x)) / (cos . x) by TAYLOR_1:20

.= (- (sin . x)) / (cos . x) by SIN_COS:63

.= - ((sin . x) / (cos . x)) by XCMPLX_1:187

.= - ((sin x) / (cos . x)) by SIN_COS:def 17

.= - ((sin x) / (cos x)) by SIN_COS:def 19

.= - (tan x) by SIN_COS4:def 1 ;

hence ((ln * cos) `| Z) . x = - (tan x) by A5, A6, FDIFF_1:def 7; :: thesis: verum

end;assume A6: x in Z ; :: thesis: ((ln * cos) `| Z) . x = - (tan x)

then ( cos is_differentiable_in x & cos . x > 0 ) by A2, A3, FDIFF_1:9;

then diff ((ln * cos),x) = (diff (cos,x)) / (cos . x) by TAYLOR_1:20

.= (- (sin . x)) / (cos . x) by SIN_COS:63

.= - ((sin . x) / (cos . x)) by XCMPLX_1:187

.= - ((sin x) / (cos . x)) by SIN_COS:def 17

.= - ((sin x) / (cos x)) by SIN_COS:def 19

.= - (tan x) by SIN_COS4:def 1 ;

hence ((ln * cos) `| Z) . x = - (tan x) by A5, A6, FDIFF_1:def 7; :: thesis: verum

((ln * cos) `| Z) . x = - (tan x) ) ) by A1, A4, FDIFF_1:9; :: thesis: verum