let Z be open Subset of REAL ; :: thesis: ( not 0 in Z & 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 )) ) ) )
set f = id Z;
assume A1: ( not 0 in Z & 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 )) ) )
A2: for x being Real st x in Z holds
(id Z) . x = x by FUNCT_1:35;
A3: Z c= (dom ((id Z) ^ )) /\ (dom cosec ) by A1, VALUED_1:def 4;
then A4: Z c= dom ((id Z) ^ ) by XBOOLE_1:18;
A5: Z c= dom cosec by A3, XBOOLE_1:18;
A8: ( (id Z) ^ is_differentiable_on Z & ( for x being Real st x in Z holds
(((id Z) ^ ) `| Z) . x = - (1 / (x ^2 )) ) ) by FDIFF_5:4, A1;
A9: for x being Real st x in Z holds
( cosec is_differentiable_in x & diff cosec ,x = - ((cos . x) / ((sin . x) ^2 )) )

proof

let x be Real; :: thesis: ( x in Z implies ( cosec is_differentiable_in x & diff cosec ,x = - ((cos . x) / ((sin . x) ^2 )) ) )
assume x in Z ; :: thesis: ( cosec is_differentiable_in x & diff cosec ,x = - ((cos . x) / ((sin . x) ^2 )) )
then sin . x <> 0 by A5, RFUNCT_1:13;
hence ( cosec is_differentiable_in x & diff cosec ,x = - ((cos . x) / ((sin . x) ^2 )) ) by Th2; :: thesis: verum

end;

then for x being Real st x in Z holds
cosec is_differentiable_in x ;
then A10: cosec is_differentiable_on Z by A5, FDIFF_1:16;
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 A11: x in Z ; :: thesis: ((((id Z) ^ ) (#) cosec ) `| Z) . x = (- ((1 / (sin . x)) / (x ^2 ))) - (((cos . x) / x) / ((sin . x) ^2 ))
then ((((id Z) ^ ) (#) cosec ) `| Z) . x = ((cosec . x) * (diff ((id Z) ^ ),x)) + ((((id Z) ^ ) . x) * (diff cosec ,x)) by A1, A8, A10, FDIFF_1:29
.= ((cosec . x) * ((((id Z) ^ ) `| Z) . x)) + ((((id Z) ^ ) . x) * (diff cosec ,x)) by A8, A11, FDIFF_1:def 8
.= ((cosec . x) * (- (1 / (x ^2 )))) + ((((id Z) ^ ) . x) * (diff cosec ,x)) by A11, FDIFF_5:4, A1
.= (- ((cosec . x) * (1 / (x ^2 )))) + ((((id Z) ^ ) . x) * (- ((cos . x) / ((sin . x) ^2 )))) by A9, A11
.= (- (((sin . x) " ) * (1 / (x ^2 )))) + ((((id Z) ^ ) . x) * ((- (cos . x)) / ((sin . x) ^2 ))) by A5, A11, RFUNCT_1:def 8
.= (- ((1 / (sin . x)) / (x ^2 ))) + ((((id Z) . x) " ) * ((- (cos . x)) / ((sin . x) ^2 ))) by A4, A11, RFUNCT_1:def 8
.= (- ((1 / (sin . x)) / (x ^2 ))) + ((1 / x) * ((- (cos . x)) / ((sin . x) ^2 ))) by A2, A11
.= (- ((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 A1, A8, A10, FDIFF_1:29; :: thesis: verum