let Z be open Subset of REAL ; :: thesis: ( Z c= dom (cosec * sin ) implies ( - (cosec * sin ) is_differentiable_on Z & ( for x being Real st x in Z holds
((- (cosec * sin )) `| Z) . x = ((cos . x) * (cos . (sin . x))) / ((sin . (sin . x)) ^2 ) ) ) )
assume A1: Z c= dom (cosec * sin ) ; :: thesis: ( - (cosec * sin ) is_differentiable_on Z & ( for x being Real st x in Z holds
((- (cosec * sin )) `| Z) . x = ((cos . x) * (cos . (sin . x))) / ((sin . (sin . x)) ^2 ) ) )
then A2: Z c= dom (- (cosec * sin )) by VALUED_1:8;
A3: cosec * sin is_differentiable_on Z by A1, FDIFF_9:36;
then A4: (- 1) (#) (cosec * sin ) is_differentiable_on Z by A2, FDIFF_1:28, A;
for x being Real st x in Z holds
((- (cosec * sin )) `| Z) . x = ((cos . x) * (cos . (sin . x))) / ((sin . (sin . x)) ^2 )

proof

let x be Real; :: thesis: ( x in Z implies ((- (cosec * sin )) `| Z) . x = ((cos . x) * (cos . (sin . x))) / ((sin . (sin . x)) ^2 ) )
assume A5: x in Z ; :: thesis: ((- (cosec * sin )) `| Z) . x = ((cos . x) * (cos . (sin . x))) / ((sin . (sin . x)) ^2 )
((- (cosec * sin )) `| Z) . x = ((- 1) (#) ((cosec * sin ) `| Z)) . x by A3, FDIFF_2:19, A
.= (- 1) * (((cosec * sin ) `| Z) . x) by VALUED_1:6
.= (- 1) * (- (((cos . x) * (cos . (sin . x))) / ((sin . (sin . x)) ^2 ))) by A1, A5, FDIFF_9:36
.= ((cos . x) * (cos . (sin . x))) / ((sin . (sin . x)) ^2 ) ;
hence ((- (cosec * sin )) `| Z) . x = ((cos . x) * (cos . (sin . x))) / ((sin . (sin . x)) ^2 ) ; :: thesis: verum

end;

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