let A be closed-interval Subset of REAL; :: thesis: for f being PartFunc of REAL,REAL
for Z being open Subset of REAL st A c= Z & ( for x being Real st x in Z holds
f . x = (cos . (1 / x)) / ((x ^2) * ((sin . (1 / x)) ^2)) ) & Z c= dom (cosec * ((id Z) ^)) & Z = dom f & f | A is continuous holds
integral (f,A) = ((cosec * ((id Z) ^)) . (upper_bound A)) - ((cosec * ((id Z) ^)) . (lower_bound A))
let f be PartFunc of REAL,REAL; :: thesis: for Z being open Subset of REAL st A c= Z & ( for x being Real st x in Z holds
f . x = (cos . (1 / x)) / ((x ^2) * ((sin . (1 / x)) ^2)) ) & Z c= dom (cosec * ((id Z) ^)) & Z = dom f & f | A is continuous holds
integral (f,A) = ((cosec * ((id Z) ^)) . (upper_bound A)) - ((cosec * ((id Z) ^)) . (lower_bound A))
let Z be open Subset of REAL; :: thesis: ( A c= Z & ( for x being Real st x in Z holds
f . x = (cos . (1 / x)) / ((x ^2) * ((sin . (1 / x)) ^2)) ) & Z c= dom (cosec * ((id Z) ^)) & Z = dom f & f | A is continuous implies integral (f,A) = ((cosec * ((id Z) ^)) . (upper_bound A)) - ((cosec * ((id Z) ^)) . (lower_bound A)) )
assume A1: ( A c= Z & ( for x being Real st x in Z holds
f . x = (cos . (1 / x)) / ((x ^2) * ((sin . (1 / x)) ^2)) ) & Z c= dom (cosec * ((id Z) ^)) & Z = dom f & f | A is continuous ) ; :: thesis: integral (f,A) = ((cosec * ((id Z) ^)) . (upper_bound A)) - ((cosec * ((id Z) ^)) . (lower_bound A))
then A2: ( f is_integrable_on A & f | A is bounded ) by INTEGRA5:10, INTEGRA5:11;
A3: Z c= dom ((id Z) ^) by A1, FUNCT_1:171;
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: cosec * ((id Z) ^) is_differentiable_on Z by A1, FDIFF_9:9;
A6: for x being Real st x in dom ((cosec * ((id Z) ^)) `| Z) holds
((cosec * ((id Z) ^)) `| Z) . x = f . x

proof

let x be Real; :: thesis: ( x in dom ((cosec * ((id Z) ^)) `| Z) implies ((cosec * ((id Z) ^)) `| Z) . x = f . x )
assume x in dom ((cosec * ((id Z) ^)) `| Z) ; :: thesis: ((cosec * ((id Z) ^)) `| Z) . x = f . x
then A7: x in Z by A5, FDIFF_1:def 8;
then ((cosec * ((id Z) ^)) `| Z) . x = (cos . (1 / x)) / ((x ^2) * ((sin . (1 / x)) ^2)) by A1, A4, FDIFF_9:9
.= f . x by A1, A7 ;
hence ((cosec * ((id Z) ^)) `| Z) . x = f . x ; :: thesis: verum

end;

dom ((cosec * ((id Z) ^)) `| Z) = dom f by A1, A5, FDIFF_1:def 8;
then (cosec * ((id Z) ^)) `| Z = f by A6, PARTFUN1:34;
hence integral (f,A) = ((cosec * ((id Z) ^)) . (upper_bound A)) - ((cosec * ((id Z) ^)) . (lower_bound A)) by A1, A2, A5, INTEGRA5:13; :: thesis: verum