let Z be open Subset of REAL ; :: thesis: ( Z c= dom (cot (#) sec ) implies ( - (cot (#) sec ) is_differentiable_on Z & ( for x being Real st x in Z holds
((- (cot (#) sec )) `| Z) . x = ((1 / ((sin . x) ^2 )) / (cos . x)) - (((cot . x) * (sin . x)) / ((cos . x) ^2 )) ) ) )
assume A1: Z c= dom (cot (#) sec ) ; :: thesis: ( - (cot (#) sec ) is_differentiable_on Z & ( for x being Real st x in Z holds
((- (cot (#) sec )) `| Z) . x = ((1 / ((sin . x) ^2 )) / (cos . x)) - (((cot . x) * (sin . x)) / ((cos . x) ^2 )) ) )
then A2: Z c= dom (- (cot (#) sec )) by VALUED_1:8;
A3: cot (#) sec is_differentiable_on Z by A1, FDIFF_9:43;
then A4: (- 1) (#) (cot (#) sec ) is_differentiable_on Z by A, A2, FDIFF_1:28;
for x being Real st x in Z holds
((- (cot (#) sec )) `| Z) . x = ((1 / ((sin . x) ^2 )) / (cos . x)) - (((cot . x) * (sin . x)) / ((cos . x) ^2 ))

proof

let x be Real; :: thesis: ( x in Z implies ((- (cot (#) sec )) `| Z) . x = ((1 / ((sin . x) ^2 )) / (cos . x)) - (((cot . x) * (sin . x)) / ((cos . x) ^2 )) )
assume A5: x in Z ; :: thesis: ((- (cot (#) sec )) `| Z) . x = ((1 / ((sin . x) ^2 )) / (cos . x)) - (((cot . x) * (sin . x)) / ((cos . x) ^2 ))
((- (cot (#) sec )) `| Z) . x = ((- 1) (#) ((cot (#) sec ) `| Z)) . x by A, A3, FDIFF_2:19
.= (- 1) * (((cot (#) sec ) `| Z) . x) by VALUED_1:6
.= (- 1) * ((- ((1 / ((sin . x) ^2 )) / (cos . x))) + (((cot . x) * (sin . x)) / ((cos . x) ^2 ))) by A1, A5, FDIFF_9:43
.= ((1 / ((sin . x) ^2 )) / (cos . x)) - (((cot . x) * (sin . x)) / ((cos . x) ^2 )) ;
hence ((- (cot (#) sec )) `| Z) . x = ((1 / ((sin . x) ^2 )) / (cos . x)) - (((cot . x) * (sin . x)) / ((cos . x) ^2 )) ; :: thesis: verum

end;

hence ( - (cot (#) sec ) is_differentiable_on Z & ( for x being Real st x in Z holds
((- (cot (#) sec )) `| Z) . x = ((1 / ((sin . x) ^2 )) / (cos . x)) - (((cot . x) * (sin . x)) / ((cos . x) ^2 )) ) ) by A4; :: thesis: verum