let Z be open Subset of REAL; :: thesis: ( Z c= dom () implies ( cos * sec is_differentiable_on Z & ( for x being Real st x in Z holds
(() `| Z) . x = - (((sin . (sec . x)) * (sin . x)) / ((cos . x) ^2)) ) ) )

assume A1: Z c= dom () ; :: thesis: ( cos * sec is_differentiable_on Z & ( for x being Real st x in Z holds
(() `| Z) . x = - (((sin . (sec . x)) * (sin . x)) / ((cos . x) ^2)) ) )

dom () c= dom sec by RELAT_1:25;
then A2: Z c= dom sec by ;
A3: for x being Real st x in Z holds
cos * sec is_differentiable_in x
proof end;
then A5: cos * sec is_differentiable_on Z by ;
for x being Real st x in Z holds
(() `| Z) . x = - (((sin . (sec . x)) * (sin . x)) / ((cos . x) ^2))
proof
let x be Real; :: thesis: ( x in Z implies (() `| Z) . x = - (((sin . (sec . x)) * (sin . x)) / ((cos . x) ^2)) )
A6: cos is_differentiable_in sec . x by SIN_COS:63;
assume A7: x in Z ; :: thesis: (() `| Z) . x = - (((sin . (sec . x)) * (sin . x)) / ((cos . x) ^2))
then A8: cos . x <> 0 by ;
then sec is_differentiable_in x by FDIFF_9:1;
then diff ((),x) = (diff (cos,(sec . x))) * (diff (sec,x)) by
.= (- (sin (sec . x))) * (diff (sec,x)) by SIN_COS:63
.= (- (sin (sec . x))) * ((sin . x) / ((cos . x) ^2)) by
.= - (((sin . (sec . x)) * (sin . x)) / ((cos . x) ^2)) ;
hence ((cos * sec) `| Z) . x = - (((sin . (sec . x)) * (sin . x)) / ((cos . x) ^2)) by ; :: thesis: verum
end;
hence ( cos * sec is_differentiable_on Z & ( for x being Real st x in Z holds
(() `| Z) . x = - (((sin . (sec . x)) * (sin . x)) / ((cos . x) ^2)) ) ) by ; :: thesis: verum