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 & dom (((- (id Z)) (#) cos ) + sin ) = Z & ( for x being Real st x in Z holds
f . x = x * (sin . x) ) & Z = dom f & f | A is continuous holds
integral f,A = ((((- (id Z)) (#) cos ) + sin ) . (upper_bound A)) - ((((- (id Z)) (#) cos ) + sin ) . (lower_bound A))
let f be PartFunc of REAL ,REAL ; :: thesis: for Z being open Subset of REAL st A c= Z & dom (((- (id Z)) (#) cos ) + sin ) = Z & ( for x being Real st x in Z holds
f . x = x * (sin . x) ) & Z = dom f & f | A is continuous holds
integral f,A = ((((- (id Z)) (#) cos ) + sin ) . (upper_bound A)) - ((((- (id Z)) (#) cos ) + sin ) . (lower_bound A))
let Z be open Subset of REAL ; :: thesis: ( A c= Z & dom (((- (id Z)) (#) cos ) + sin ) = Z & ( for x being Real st x in Z holds
f . x = x * (sin . x) ) & Z = dom f & f | A is continuous implies integral f,A = ((((- (id Z)) (#) cos ) + sin ) . (upper_bound A)) - ((((- (id Z)) (#) cos ) + sin ) . (lower_bound A)) )
assume that
A1: A c= Z and
A2: dom (((- (id Z)) (#) cos ) + sin ) = Z and
A3: for x being Real st x in Z holds
f . x = x * (sin . x) and
A4: Z = dom f and
A5: f | A is continuous ; :: thesis: integral f,A = ((((- (id Z)) (#) cos ) + sin ) . (upper_bound A)) - ((((- (id Z)) (#) cos ) + sin ) . (lower_bound A))
A6: ((- (id Z)) (#) cos ) + sin is_differentiable_on Z by A2, FDIFF_4:46;
A7: for x being Real st x in dom ((((- (id Z)) (#) cos ) + sin ) `| Z) holds
((((- (id Z)) (#) cos ) + sin ) `| Z) . x = f . x

proof

let x be Real; :: thesis: ( x in dom ((((- (id Z)) (#) cos ) + sin ) `| Z) implies ((((- (id Z)) (#) cos ) + sin ) `| Z) . x = f . x )
assume x in dom ((((- (id Z)) (#) cos ) + sin ) `| Z) ; :: thesis: ((((- (id Z)) (#) cos ) + sin ) `| Z) . x = f . x
then A8: x in Z by A6, FDIFF_1:def 8;
then ((((- (id Z)) (#) cos ) + sin ) `| Z) . x = x * (sin . x) by A2, FDIFF_4:46
.= f . x by A3, A8 ;
hence ((((- (id Z)) (#) cos ) + sin ) `| Z) . x = f . x ; :: thesis: verum

end;

dom ((((- (id Z)) (#) cos ) + sin ) `| Z) = dom f by A4, A6, FDIFF_1:def 8;
then A9: (((- (id Z)) (#) cos ) + sin ) `| Z = f by A7, PARTFUN1:34;
( f is_integrable_on A & f | A is bounded ) by A1, A4, A5, INTEGRA5:10, INTEGRA5:11;
hence integral f,A = ((((- (id Z)) (#) cos ) + sin ) . (upper_bound A)) - ((((- (id Z)) (#) cos ) + sin ) . (lower_bound A)) by A1, A2, A9, FDIFF_4:46, INTEGRA5:13; :: thesis: verum