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 = (cos . x) / ((cos . x) - 1) ) ) & dom (((id Z) + cot ) + cosec ) = Z & Z = dom f & f | A is continuous holds
integral f,A = ((((id Z) + cot ) + cosec ) . (upper_bound A)) - ((((id Z) + 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 = (cos . x) / ((cos . x) - 1) ) ) & dom (((id Z) + cot ) + cosec ) = Z & Z = dom f & f | A is continuous holds
integral f,A = ((((id Z) + cot ) + cosec ) . (upper_bound A)) - ((((id Z) + 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 = (cos . x) / ((cos . x) - 1) ) ) & dom (((id Z) + cot ) + cosec ) = Z & Z = dom f & f | A is continuous implies integral f,A = ((((id Z) + cot ) + cosec ) . (upper_bound A)) - ((((id Z) + 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 = (cos . x) / ((cos . x) - 1) ) and
A3: dom (((id Z) + cot ) + cosec ) = Z and
A4: Z = dom f and
A5: f | A is continuous ; :: thesis: integral f,A = ((((id Z) + cot ) + cosec ) . (upper_bound A)) - ((((id Z) + 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: ((id Z) + cot ) + cosec is_differentiable_on Z by A3, Th51;
A9: for x being Real st x in dom ((((id Z) + cot ) + cosec ) `| Z) holds
((((id Z) + cot ) + cosec ) `| Z) . x = f . x

proof

let x be Real; :: thesis: ( x in dom ((((id Z) + cot ) + cosec ) `| Z) implies ((((id Z) + cot ) + cosec ) `| Z) . x = f . x )
assume x in dom ((((id Z) + cot ) + cosec ) `| Z) ; :: thesis: ((((id Z) + cot ) + cosec ) `| Z) . x = f . x
then A10: x in Z by A8, FDIFF_1:def 8;
then ((((id Z) + cot ) + cosec ) `| Z) . x = (cos . x) / ((cos . x) - 1) by A3, A7, Th51
.= f . x by A2, A10 ;
hence ((((id Z) + cot ) + cosec ) `| Z) . x = f . x ; :: thesis: verum

end;

dom ((((id Z) + cot ) + cosec ) `| Z) = dom f by A4, A8, FDIFF_1:def 8;
then (((id Z) + cot ) + cosec ) `| Z = f by A9, PARTFUN1:34;
hence integral f,A = ((((id Z) + cot ) + cosec ) . (upper_bound A)) - ((((id Z) + cot ) + cosec ) . (lower_bound A)) by A1, A4, A5, A6, A8, INTEGRA5:10, INTEGRA5:13; :: thesis: verum