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

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

A2: for x being Real st x in Z holds
sin . x <> 0
proof
let x be Real; :: thesis: ( x in Z implies sin . x <> 0 )
assume x in Z ; :: thesis:
then x in dom () by ;
hence sin . x <> 0 by FDIFF_8:2; :: thesis: verum
end;
A3: for x being Real st x in Z holds
sin * cot is_differentiable_in x
proof end;
then A5: sin * cot is_differentiable_on Z by ;
for x being Real st x in Z holds
(() `| Z) . x = - ((cos (cot . x)) / ((sin . x) ^2))
proof
let x be Real; :: thesis: ( x in Z implies (() `| Z) . x = - ((cos (cot . x)) / ((sin . x) ^2)) )
A6: sin is_differentiable_in cot . x by SIN_COS:64;
assume A7: x in Z ; :: thesis: (() `| Z) . x = - ((cos (cot . x)) / ((sin . x) ^2))
then A8: sin . x <> 0 by A2;
then cot is_differentiable_in x by FDIFF_7:47;
then diff ((),x) = (diff (sin,(cot . x))) * (diff (cot,x)) by
.= (cos (cot . x)) * (diff (cot,x)) by SIN_COS:64
.= (cos (cot . x)) * (- (1 / ((sin . x) ^2))) by
.= - ((cos (cot . x)) / ((sin . x) ^2)) ;
hence ((sin * cot) `| Z) . x = - ((cos (cot . x)) / ((sin . x) ^2)) by ; :: thesis: verum
end;
hence ( sin * cot is_differentiable_on Z & ( for x being Real st x in Z holds
(() `| Z) . x = - ((cos (cot . x)) / ((sin . x) ^2)) ) ) by ; :: thesis: verum