let Z be open Subset of REAL; :: thesis: ( Z c= dom (((id Z) ^) (#) cosec) implies ( - (((id Z) ^) (#) cosec) is_differentiable_on Z & ( for x being Real st x in Z holds
((- (((id Z) ^) (#) cosec)) `| Z) . x = ((1 / (sin . x)) / (x ^2)) + (((cos . x) / x) / ((sin . x) ^2)) ) ) )
assume A1: Z c= dom (((id Z) ^) (#) cosec) ; :: thesis: ( - (((id Z) ^) (#) cosec) is_differentiable_on Z & ( for x being Real st x in Z holds
((- (((id Z) ^) (#) cosec)) `| Z) . x = ((1 / (sin . x)) / (x ^2)) + (((cos . x) / x) / ((sin . x) ^2)) ) )
then A2: Z c= dom (- (((id Z) ^) (#) cosec)) by VALUED_1:8;
Z c= (dom ((id Z) ^)) /\ (dom cosec) by A1, VALUED_1:def 4;
then A3: Z c= dom ((id Z) ^) by XBOOLE_1:18;
A4: not 0 in Z

proof

assume K: 0 in Z ; :: thesis: contradiction
dom ((id Z) ^) = (dom (id Z)) \ ((id Z) " {0}) by RFUNCT_1:def 8
.= (dom (id Z)) \ {0} by Lm0, K ;
then not 0 in {0} by XBOOLE_0:def 5, K, A3;
hence contradiction by TARSKI:def 1; :: thesis: verum

end;

then A5: ((id Z) ^) (#) cosec is_differentiable_on Z by A1, FDIFF_9:33;
then A6: (- 1) (#) (((id Z) ^) (#) cosec) is_differentiable_on Z by A, A2, FDIFF_1:28;
for x being Real st x in Z holds
((- (((id Z) ^) (#) cosec)) `| Z) . x = ((1 / (sin . x)) / (x ^2)) + (((cos . x) / x) / ((sin . x) ^2))

proof

let x be Real; :: thesis: ( x in Z implies ((- (((id Z) ^) (#) cosec)) `| Z) . x = ((1 / (sin . x)) / (x ^2)) + (((cos . x) / x) / ((sin . x) ^2)) )
assume A7: x in Z ; :: thesis: ((- (((id Z) ^) (#) cosec)) `| Z) . x = ((1 / (sin . x)) / (x ^2)) + (((cos . x) / x) / ((sin . x) ^2))
((- (((id Z) ^) (#) cosec)) `| Z) . x = ((- 1) (#) ((((id Z) ^) (#) cosec) `| Z)) . x by A5, FDIFF_2:19, A
.= (- 1) * (((((id Z) ^) (#) cosec) `| Z) . x) by VALUED_1:6
.= (- 1) * ((- ((1 / (sin . x)) / (x ^2))) - (((cos . x) / x) / ((sin . x) ^2))) by A1, A4, A7, FDIFF_9:33
.= ((1 / (sin . x)) / (x ^2)) + (((cos . x) / x) / ((sin . x) ^2)) ;
hence ((- (((id Z) ^) (#) cosec)) `| Z) . x = ((1 / (sin . x)) / (x ^2)) + (((cos . x) / x) / ((sin . x) ^2)) ; :: thesis: verum

end;

hence ( - (((id Z) ^) (#) cosec) is_differentiable_on Z & ( for x being Real st x in Z holds
((- (((id Z) ^) (#) cosec)) `| Z) . x = ((1 / (sin . x)) / (x ^2)) + (((cos . x) / x) / ((sin . x) ^2)) ) ) by A6; :: thesis: verum