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

assume A1: Z c= dom () ; :: thesis: ( cos * cot is_differentiable_on Z & ( for x being Real st x in Z holds
(() `| Z) . x = (sin (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
cos * cot is_differentiable_in x
proof end;
then A5: cos * cot is_differentiable_on Z by ;
for x being Real st x in Z holds
(() `| Z) . x = (sin (cot . x)) / ((sin . x) ^2)
proof
let x be Real; :: thesis: ( x in Z implies (() `| Z) . x = (sin (cot . x)) / ((sin . x) ^2) )
A6: cos is_differentiable_in cot . x by SIN_COS:63;
assume A7: x in Z ; :: thesis: (() `| Z) . x = (sin (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 (cos,(cot . x))) * (diff (cot,x)) by
.= (- (sin (cot . x))) * (diff (cot,x)) by SIN_COS:63
.= (- (sin (cot . x))) * (- (1 / ((sin . x) ^2))) by
.= (sin (cot . x)) / ((sin . x) ^2) ;
hence ((cos * cot) `| Z) . x = (sin (cot . x)) / ((sin . x) ^2) by ; :: thesis: verum
end;
hence ( cos * cot is_differentiable_on Z & ( for x being Real st x in Z holds
(() `| Z) . x = (sin (cot . x)) / ((sin . x) ^2) ) ) by ; :: thesis: verum