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