let A be closed-interval Subset of REAL ; :: thesis: for Z being open Subset of REAL
for f being PartFunc of REAL ,REAL st A c= Z & ( for x being Real st x in Z holds
( 1 + (cos . x) <> 0 & 1 - (cos . x) <> 0 & f . x = 1 / (1 + (cos . x)) ) ) & dom ((- cot ) + cosec ) = Z & Z = dom f & f | A is continuous holds
integral f,A = (((- cot ) + cosec ) . (sup A)) - (((- cot ) + cosec ) . (inf A))
let Z be open Subset of REAL ; :: thesis: for f being PartFunc of REAL ,REAL st A c= Z & ( for x being Real st x in Z holds
( 1 + (cos . x) <> 0 & 1 - (cos . x) <> 0 & f . x = 1 / (1 + (cos . x)) ) ) & dom ((- cot ) + cosec ) = Z & Z = dom f & f | A is continuous holds
integral f,A = (((- cot ) + cosec ) . (sup A)) - (((- cot ) + cosec ) . (inf A))
let f be PartFunc of REAL ,REAL ; :: thesis: ( A c= Z & ( for x being Real st x in Z holds
( 1 + (cos . x) <> 0 & 1 - (cos . x) <> 0 & f . x = 1 / (1 + (cos . x)) ) ) & dom ((- cot ) + cosec ) = Z & Z = dom f & f | A is continuous implies integral f,A = (((- cot ) + cosec ) . (sup A)) - (((- cot ) + cosec ) . (inf A)) )
assume A1: ( A c= Z & ( for x being Real st x in Z holds
( 1 + (cos . x) <> 0 & 1 - (cos . x) <> 0 & f . x = 1 / (1 + (cos . x)) ) ) & dom ((- cot ) + cosec ) = Z & Z = dom f & f | A is continuous ) ; :: thesis: integral f,A = (((- cot ) + cosec ) . (sup A)) - (((- cot ) + cosec ) . (inf A))
then A2: ( f is_integrable_on A & f | A is bounded ) by INTEGRA5:10, INTEGRA5:11;
B: for x being Real st x in Z holds
( 1 + (cos . x) <> 0 & 1 - (cos . x) <> 0 ) by A1;
then A3: (- cot ) + cosec is_differentiable_on Z by A1, Th16;
A4: for x being Real st x in dom (((- cot ) + cosec ) `| Z) holds
(((- cot ) + cosec ) `| Z) . x = f . x

proof

let x be Real; :: thesis: ( x in dom (((- cot ) + cosec ) `| Z) implies (((- cot ) + cosec ) `| Z) . x = f . x )
assume x in dom (((- cot ) + cosec ) `| Z) ; :: thesis: (((- cot ) + cosec ) `| Z) . x = f . x
then A5: x in Z by A3, FDIFF_1:def 8;
then (((- cot ) + cosec ) `| Z) . x = 1 / (1 + (cos . x)) by A1, B, Th16
.= f . x by A1, A5 ;
hence (((- cot ) + cosec ) `| Z) . x = f . x ; :: thesis: verum

end;

dom (((- cot ) + cosec ) `| Z) = dom f by A1, A3, FDIFF_1:def 8;
then ((- cot ) + cosec ) `| Z = f by A4, PARTFUN1:34;
hence integral f,A = (((- cot ) + cosec ) . (sup A)) - (((- cot ) + cosec ) . (inf A)) by A1, A2, A3, INTEGRA5:13; :: thesis: verum