let Z be open Subset of REAL; :: thesis: ( Z c= dom (cos (#) cot) implies ( - (cos (#) cot) is_differentiable_on Z & ( for x being Real st x in Z holds
((- (cos (#) cot)) `| Z) . x = (cos . x) + ((cos . x) / ((sin . x) ^2)) ) ) )
assume A1: Z c= dom (cos (#) cot) ; :: thesis: ( - (cos (#) cot) is_differentiable_on Z & ( for x being Real st x in Z holds
((- (cos (#) cot)) `| Z) . x = (cos . x) + ((cos . x) / ((sin . x) ^2)) ) )
then A2: Z c= dom (- (cos (#) cot)) by VALUED_1:8;
A3: cos (#) cot is_differentiable_on Z by A1, FDIFF_10:11;
then A4: (- 1) (#) (cos (#) cot) is_differentiable_on Z by A2, FDIFF_1:20;
for x being Real st x in Z holds
((- (cos (#) cot)) `| Z) . x = (cos . x) + ((cos . x) / ((sin . x) ^2))

proof

let x be Real; :: thesis: ( x in Z implies ((- (cos (#) cot)) `| Z) . x = (cos . x) + ((cos . x) / ((sin . x) ^2)) )
assume A5: x in Z ; :: thesis: ((- (cos (#) cot)) `| Z) . x = (cos . x) + ((cos . x) / ((sin . x) ^2))
((- (cos (#) cot)) `| Z) . x = ((- 1) (#) ((cos (#) cot) `| Z)) . x by A3, FDIFF_2:19
.= (- 1) * (((cos (#) cot) `| Z) . x) by VALUED_1:6
.= (- 1) * ((- (cos . x)) - ((cos . x) / ((sin . x) ^2))) by A1, A5, FDIFF_10:11
.= (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
((- (cos (#) cot)) `| Z) . x = (cos . x) + ((cos . x) / ((sin . x) ^2)) ) ) by A4; :: thesis: verum