let Z be open Subset of REAL; :: thesis: ( Z c= dom () implies ( cot * cot is_differentiable_on Z & ( for x being Real st x in Z holds
(() `| Z) . x = (1 / ((sin . (cot . x)) ^2)) * (1 / ((sin . x) ^2)) ) ) )

assume A1: Z c= dom () ; :: thesis: ( cot * cot is_differentiable_on Z & ( for x being Real st x in Z holds
(() `| Z) . x = (1 / ((sin . (cot . x)) ^2)) * (1 / ((sin . x) ^2)) ) )

A2: for x being Real st x in Z holds
sin . (cot . x) <> 0
proof
let x be Real; :: thesis: ( x in Z implies sin . (cot . x) <> 0 )
assume x in Z ; :: thesis: sin . (cot . x) <> 0
then cot . x in dom cot by ;
hence sin . (cot . x) <> 0 by FDIFF_8:2; :: thesis: verum
end;
A3: 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;
A4: for x being Real st x in Z holds
cot * cot is_differentiable_in x
proof end;
then A7: cot * cot is_differentiable_on Z by ;
for x being Real st x in Z holds
(() `| Z) . x = (1 / ((sin . (cot . x)) ^2)) * (1 / ((sin . x) ^2))
proof
let x be Real; :: thesis: ( x in Z implies (() `| Z) . x = (1 / ((sin . (cot . x)) ^2)) * (1 / ((sin . x) ^2)) )
assume A8: x in Z ; :: thesis: (() `| Z) . x = (1 / ((sin . (cot . x)) ^2)) * (1 / ((sin . x) ^2))
then A9: sin . (cot . x) <> 0 by A2;
then A10: cot is_differentiable_in cot . x by FDIFF_7:47;
A11: sin . x <> 0 by A3, A8;
then cot is_differentiable_in x by FDIFF_7:47;
then diff ((),x) = (diff (cot,(cot . x))) * (diff (cot,x)) by
.= (- (1 / ((sin . (cot . x)) ^2))) * (diff (cot,x)) by
.= (- (1 / ((sin . (cot . x)) ^2))) * (- (1 / ((sin . x) ^2))) by ;
hence ((cot * cot) `| Z) . x = (1 / ((sin . (cot . x)) ^2)) * (1 / ((sin . x) ^2)) by ; :: thesis: verum
end;
hence ( cot * cot is_differentiable_on Z & ( for x being Real st x in Z holds
(() `| Z) . x = (1 / ((sin . (cot . x)) ^2)) * (1 / ((sin . x) ^2)) ) ) by ; :: thesis: verum