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

assume A1: Z c= dom () ; :: thesis: ( sin * sec is_differentiable_on Z & ( for x being Real st x in Z holds
(() `| Z) . x = ((cos . (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
sin * sec is_differentiable_in x
proof end;
then A5: sin * sec is_differentiable_on Z by ;
for x being Real st x in Z holds
(() `| Z) . x = ((cos . (sec . x)) * (sin . x)) / ((cos . x) ^2)
proof
let x be Real; :: thesis: ( x in Z implies (() `| Z) . x = ((cos . (sec . x)) * (sin . x)) / ((cos . x) ^2) )
A6: sin is_differentiable_in sec . x by SIN_COS:64;
assume A7: x in Z ; :: thesis: (() `| Z) . x = ((cos . (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 (sin,(sec . x))) * (diff (sec,x)) by
.= (cos (sec . x)) * (diff (sec,x)) by SIN_COS:64
.= (cos (sec . x)) * ((sin . x) / ((cos . x) ^2)) by
.= ((cos . (sec . x)) * (sin . x)) / ((cos . x) ^2) ;
hence ((sin * sec) `| Z) . x = ((cos . (sec . x)) * (sin . x)) / ((cos . x) ^2) by ; :: thesis: verum
end;
hence ( sin * sec is_differentiable_on Z & ( for x being Real st x in Z holds
(() `| Z) . x = ((cos . (sec . x)) * (sin . x)) / ((cos . x) ^2) ) ) by ; :: thesis: verum