let a, b be Real; :: thesis: for Z being open Subset of REAL
for f being PartFunc of REAL ,REAL st Z c= dom (f (#) cosec ) & ( for x being Real st x in Z holds
f . x = (a * x) + b ) holds
( f (#) cosec is_differentiable_on Z & ( for x being Real st x in Z holds
((f (#) cosec ) `| Z) . x = (a / (sin . x)) - ((((a * x) + b) * (cos . x)) / ((sin . x) ^2 )) ) )
let Z be open Subset of REAL ; :: thesis: for f being PartFunc of REAL ,REAL st Z c= dom (f (#) cosec ) & ( for x being Real st x in Z holds
f . x = (a * x) + b ) holds
( f (#) cosec is_differentiable_on Z & ( for x being Real st x in Z holds
((f (#) cosec ) `| Z) . x = (a / (sin . x)) - ((((a * x) + b) * (cos . x)) / ((sin . x) ^2 )) ) )
let f be PartFunc of REAL ,REAL ; :: thesis: ( Z c= dom (f (#) cosec ) & ( for x being Real st x in Z holds
f . x = (a * x) + b ) implies ( f (#) cosec is_differentiable_on Z & ( for x being Real st x in Z holds
((f (#) cosec ) `| Z) . x = (a / (sin . x)) - ((((a * x) + b) * (cos . x)) / ((sin . x) ^2 )) ) ) )
assume that
A1: Z c= dom (f (#) cosec ) and
A2: for x being Real st x in Z holds
f . x = (a * x) + b ; :: thesis: ( f (#) cosec is_differentiable_on Z & ( for x being Real st x in Z holds
((f (#) cosec ) `| Z) . x = (a / (sin . x)) - ((((a * x) + b) * (cos . x)) / ((sin . x) ^2 )) ) )
A3: Z c= (dom f) /\ (dom cosec ) by A1, VALUED_1:def 4;
then A4: Z c= dom f by XBOOLE_1:18;
A5: Z c= dom cosec by A3, XBOOLE_1:18;
A6: ( f is_differentiable_on Z & ( for x being Real st x in Z holds
(f `| Z) . x = a ) ) by A2, A4, FDIFF_1:31;
A7: 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 A8: cosec is_differentiable_on Z by A5, FDIFF_1:16;
for x being Real st x in Z holds
((f (#) cosec ) `| Z) . x = (a / (sin . x)) - ((((a * x) + b) * (cos . x)) / ((sin . x) ^2 ))

proof

let x be Real; :: thesis: ( x in Z implies ((f (#) cosec ) `| Z) . x = (a / (sin . x)) - ((((a * x) + b) * (cos . x)) / ((sin . x) ^2 )) )
assume A9: x in Z ; :: thesis: ((f (#) cosec ) `| Z) . x = (a / (sin . x)) - ((((a * x) + b) * (cos . x)) / ((sin . x) ^2 ))
then ((f (#) cosec ) `| Z) . x = ((cosec . x) * (diff f,x)) + ((f . x) * (diff cosec ,x)) by A1, A6, A8, FDIFF_1:29
.= ((cosec . x) * ((f `| Z) . x)) + ((f . x) * (diff cosec ,x)) by A6, A9, FDIFF_1:def 8
.= ((cosec . x) * a) + ((f . x) * (diff cosec ,x)) by A2, A4, A9, FDIFF_1:31
.= ((cosec . x) * a) + (((a * x) + b) * (diff cosec ,x)) by A2, A9
.= ((cosec . x) * a) + (((a * x) + b) * (- ((cos . x) / ((sin . x) ^2 )))) by A7, A9
.= (a / (sin . x)) - ((((a * x) + b) * (cos . x)) / ((sin . x) ^2 )) by A5, A9, RFUNCT_1:def 8 ;
hence ((f (#) cosec ) `| Z) . x = (a / (sin . x)) - ((((a * x) + b) * (cos . x)) / ((sin . x) ^2 )) ; :: thesis: verum

end;

hence ( f (#) cosec is_differentiable_on Z & ( for x being Real st x in Z holds
((f (#) cosec ) `| Z) . x = (a / (sin . x)) - ((((a * x) + b) * (cos . x)) / ((sin . x) ^2 )) ) ) by A1, A6, A8, FDIFF_1:29; :: thesis: verum