let Z be open Subset of REAL; :: thesis: ( Z c= dom (sec * cot) implies ( - (sec * cot) is_differentiable_on Z & ( for x being Real st x in Z holds
((- (sec * cot)) `| Z) . x = ((sin . (cot . x)) / ((sin . x) ^2)) / ((cos . (cot . x)) ^2) ) ) )
assume A1: Z c= dom (sec * cot) ; :: thesis: ( - (sec * cot) is_differentiable_on Z & ( for x being Real st x in Z holds
((- (sec * cot)) `| Z) . x = ((sin . (cot . x)) / ((sin . x) ^2)) / ((cos . (cot . x)) ^2) ) )
then A2: Z c= dom (- (sec * cot)) by VALUED_1:8;
A3: sec * cot is_differentiable_on Z by A1, FDIFF_9:39;
then A4: (- 1) (#) (sec * cot) is_differentiable_on Z by A2, FDIFF_1:20;
for x being Real st x in Z holds
((- (sec * cot)) `| Z) . x = ((sin . (cot . x)) / ((sin . x) ^2)) / ((cos . (cot . x)) ^2)

proof

let x be Real; :: thesis: ( x in Z implies ((- (sec * cot)) `| Z) . x = ((sin . (cot . x)) / ((sin . x) ^2)) / ((cos . (cot . x)) ^2) )
assume A5: x in Z ; :: thesis: ((- (sec * cot)) `| Z) . x = ((sin . (cot . x)) / ((sin . x) ^2)) / ((cos . (cot . x)) ^2)
((- (sec * cot)) `| Z) . x = ((- 1) (#) ((sec * cot) `| Z)) . x by A3, FDIFF_2:19
.= (- 1) * (((sec * cot) `| Z) . x) by VALUED_1:6
.= (- 1) * (- (((sin . (cot . x)) / ((sin . x) ^2)) / ((cos . (cot . x)) ^2))) by A1, A5, FDIFF_9:39
.= ((sin . (cot . x)) / ((sin . x) ^2)) / ((cos . (cot . x)) ^2) ;
hence ((- (sec * cot)) `| Z) . x = ((sin . (cot . x)) / ((sin . x) ^2)) / ((cos . (cot . x)) ^2) ; :: thesis: verum

end;

hence ( - (sec * cot) is_differentiable_on Z & ( for x being Real st x in Z holds
((- (sec * cot)) `| Z) . x = ((sin . (cot . x)) / ((sin . x) ^2)) / ((cos . (cot . x)) ^2) ) ) by A4; :: thesis: verum