let A be non empty 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) . (upper_bound A)) - (((- cot) + cosec) . (lower_bound 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) . (upper_bound A)) - (((- cot) + cosec) . (lower_bound 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) . (upper_bound A)) - (((- cot) + cosec) . (lower_bound A)) )
assume that
A1: A c= Z and
A2: for x being Real st x in Z holds
( 1 + (cos . x) <> 0 & 1 - (cos . x) <> 0 & f . x = 1 / (1 + (cos . x)) ) and
A3: dom ((- cot) + cosec) = Z and
A4: Z = dom f and
A5: f | A is continuous ; :: thesis: integral (f,A) = (((- cot) + cosec) . (upper_bound A)) - (((- cot) + cosec) . (lower_bound A))
A6: f is_integrable_on A by A1, A4, A5, INTEGRA5:11;
A7: for x being Real st x in Z holds
( 1 + (cos . x) <> 0 & 1 - (cos . x) <> 0 ) by A2;
then A8: (- cot) + cosec is_differentiable_on Z by A3, Th41;
A9: for x being Element of REAL st x in dom (((- cot) + cosec) `| Z) holds
(((- cot) + cosec) `| Z) . x = f . x

proof

let x be Element of 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 A10: x in Z by A8, FDIFF_1:def 7;
then (((- cot) + cosec) `| Z) . x = 1 / (1 + (cos . x)) by A3, A7, Th41
.= f . x by A2, A10 ;
hence (((- cot) + cosec) `| Z) . x = f . x ; :: thesis: verum

end;

dom (((- cot) + cosec) `| Z) = dom f by A4, A8, FDIFF_1:def 7;
then ((- cot) + cosec) `| Z = f by A9, PARTFUN1:5;
hence integral (f,A) = (((- cot) + cosec) . (upper_bound A)) - (((- cot) + cosec) . (lower_bound A)) by A1, A4, A5, A6, A8, INTEGRA5:10, INTEGRA5:13; :: thesis: verum