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

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

dom () c= dom cosec by RELAT_1:25;
then A2: Z c= dom cosec by ;
A3: for x being Real st x in Z holds
sin * cosec is_differentiable_in x
proof end;
then A5: sin * cosec is_differentiable_on Z by ;
for x being Real st x in Z holds
(() `| Z) . x = - (((cos . ()) * (cos . x)) / ((sin . x) ^2))
proof
let x be Real; :: thesis: ( x in Z implies (() `| Z) . x = - (((cos . ()) * (cos . x)) / ((sin . x) ^2)) )
A6: sin is_differentiable_in cosec . x by SIN_COS:64;
assume A7: x in Z ; :: thesis: (() `| Z) . x = - (((cos . ()) * (cos . x)) / ((sin . x) ^2))
then A8: sin . x <> 0 by ;
then cosec is_differentiable_in x by FDIFF_9:2;
then diff ((),x) = (diff (sin,())) * (diff (cosec,x)) by
.= (cos ()) * (diff (cosec,x)) by SIN_COS:64
.= (cos ()) * (- ((cos . x) / ((sin . x) ^2))) by
.= - (((cos . ()) * (cos . x)) / ((sin . x) ^2)) ;
hence ((sin * cosec) `| Z) . x = - (((cos . ()) * (cos . x)) / ((sin . x) ^2)) by ; :: thesis: verum
end;
hence ( sin * cosec is_differentiable_on Z & ( for x being Real st x in Z holds
(() `| Z) . x = - (((cos . ()) * (cos . x)) / ((sin . x) ^2)) ) ) by ; :: thesis: verum