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_1:20;
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
.= (- 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