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 = (- (cos . x)) - ((cos . x) / ((sin . x) ^2)) ) ) )

A1: for x being Real st x in Z holds
cos is_differentiable_in x by SIN_COS:63;
assume A2: Z c= dom () ; :: thesis: ( cos (#) cot is_differentiable_on Z & ( for x being Real st x in Z holds
(() `| Z) . x = (- (cos . x)) - ((cos . x) / ((sin . x) ^2)) ) )

then A3: Z c= () /\ () by VALUED_1:def 4;
then A4: Z c= dom cot by XBOOLE_1:18;
for x being Real st x in Z holds
cot is_differentiable_in x
proof end;
then A5: cot is_differentiable_on Z by ;
Z c= dom cos by ;
then A6: cos is_differentiable_on Z by ;
A7: for x being Real st x in Z holds
diff (cot,x) = - (1 / ((sin . x) ^2))
proof
let x be Real; :: thesis: ( x in Z implies diff (cot,x) = - (1 / ((sin . x) ^2)) )
assume x in Z ; :: thesis: diff (cot,x) = - (1 / ((sin . x) ^2))
then sin . x <> 0 by ;
hence diff (cot,x) = - (1 / ((sin . x) ^2)) by FDIFF_7:47; :: thesis: verum
end;
for x being Real st x in Z holds
(() `| Z) . x = (- (cos . x)) - ((cos . x) / ((sin . x) ^2))
proof
let x be Real; :: thesis: ( x in Z implies (() `| Z) . x = (- (cos . x)) - ((cos . x) / ((sin . x) ^2)) )
assume A8: x in Z ; :: thesis: (() `| Z) . x = (- (cos . x)) - ((cos . x) / ((sin . x) ^2))
then (() `| Z) . x = ((diff (cos,x)) * (cot . x)) + ((cos . x) * (diff (cot,x))) by
.= ((cot . x) * (- (sin . x))) + ((cos . x) * (diff (cot,x))) by SIN_COS:63
.= ((cot . x) * (- (sin . x))) + ((cos . x) * (- (1 / ((sin . x) ^2)))) by A7, A8
.= (((cos . x) / (sin . x)) * (- ((sin . x) / 1))) - ((cos . x) / ((sin . x) ^2)) by
.= (- ((cos . x) * ((sin . x) / (sin . x)))) - ((cos . x) / ((sin . x) ^2))
.= (- ((cos . x) * 1)) - ((cos . x) / ((sin . x) ^2)) by
.= (- (cos . x)) - ((cos . x) / ((sin . x) ^2)) ;
hence ((cos (#) cot) `| Z) . x = (- (cos . x)) - ((cos . x) / ((sin . x) ^2)) ; :: thesis: verum
end;
hence ( cos (#) cot is_differentiable_on Z & ( for x being Real st x in Z holds
(() `| Z) . x = (- (cos . x)) - ((cos . x) / ((sin . x) ^2)) ) ) by ; :: thesis: verum