let Z be open Subset of REAL; :: thesis: ( Z c= dom () & ( 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
(() `| Z) . x = - (tan x) ) ) )

assume that
A1: Z c= dom () 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
(() `| Z) . x = - (tan x) ) )

A3: cos is_differentiable_on Z by ;
A4: for x being Real st x in Z holds
ln * cos is_differentiable_in x
proof end;
then A5: ln * cos is_differentiable_on Z by ;
for x being Real st x in Z holds
(() `| Z) . x = - (tan x)
proof
let x be Real; :: thesis: ( x in Z implies (() `| Z) . x = - (tan x) )
assume A6: x in Z ; :: thesis: (() `| Z) . x = - (tan x)
then ( cos is_differentiable_in x & cos . x > 0 ) by ;
then diff ((),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 ; :: thesis: verum
end;
hence ( ln * cos is_differentiable_on Z & ( for x being Real st x in Z holds
(() `| Z) . x = - (tan x) ) ) by ; :: thesis: verum