let A be non empty closed_interval Subset of REAL; :: thesis: for Z being open Subset of REAL st A c= Z holds
integral (((id Z) (#) cos),A) = ((((id Z) (#) sin) + cos) . (upper_bound A)) - ((((id Z) (#) sin) + cos) . (lower_bound A))
let Z be open Subset of REAL; :: thesis: ( A c= Z implies integral (((id Z) (#) cos),A) = ((((id Z) (#) sin) + cos) . (upper_bound A)) - ((((id Z) (#) sin) + cos) . (lower_bound A)) )
assume A1: A c= Z ; :: thesis: integral (((id Z) (#) cos),A) = ((((id Z) (#) sin) + cos) . (upper_bound A)) - ((((id Z) (#) sin) + cos) . (lower_bound A))
dom (((id Z) (#) sin) + cos) = (dom ((id Z) (#) sin)) /\ REAL by SIN_COS:24, VALUED_1:def 1
.= dom ((id Z) (#) sin) by XBOOLE_1:28
.= (dom (id Z)) /\ REAL by SIN_COS:24, VALUED_1:def 4
.= dom (id Z) by XBOOLE_1:28 ;
then A2: dom (((id Z) (#) sin) + cos) = Z by RELAT_1:45;
then A3: ((id Z) (#) sin) + cos is_differentiable_on Z by FDIFF_4:47;
A4: for x being Real st x in Z holds
((id Z) (#) cos) . x = x * (cos . x)

proof

let x be Real; :: thesis: ( x in Z implies ((id Z) (#) cos) . x = x * (cos . x) )
assume A5: x in Z ; :: thesis: ((id Z) (#) cos) . x = x * (cos . x)
((id Z) (#) cos) . x = ((id Z) . x) * (cos . x) by VALUED_1:5
.= x * (cos . x) by A5, FUNCT_1:18 ;
hence ((id Z) (#) cos) . x = x * (cos . x) ; :: thesis: verum

end;

A6: for x being Element of REAL st x in dom ((((id Z) (#) sin) + cos) `| Z) holds
((((id Z) (#) sin) + cos) `| Z) . x = ((id Z) (#) cos) . x

proof

let x be Element of REAL ; :: thesis: ( x in dom ((((id Z) (#) sin) + cos) `| Z) implies ((((id Z) (#) sin) + cos) `| Z) . x = ((id Z) (#) cos) . x )
assume x in dom ((((id Z) (#) sin) + cos) `| Z) ; :: thesis: ((((id Z) (#) sin) + cos) `| Z) . x = ((id Z) (#) cos) . x
then A7: x in Z by A3, FDIFF_1:def 7;
then ((((id Z) (#) sin) + cos) `| Z) . x = x * (cos . x) by A2, FDIFF_4:47
.= ((id Z) (#) cos) . x by A4, A7 ;
hence ((((id Z) (#) sin) + cos) `| Z) . x = ((id Z) (#) cos) . x ; :: thesis: verum

end;

A8: dom ((id Z) (#) cos) = (dom (id Z)) /\ REAL by SIN_COS:24, VALUED_1:def 4
.= dom (id Z) by XBOOLE_1:28
.= Z by RELAT_1:45 ;
then dom ((((id Z) (#) sin) + cos) `| Z) = dom ((id Z) (#) cos) by A3, FDIFF_1:def 7;
then A9: (((id Z) (#) sin) + cos) `| Z = (id Z) (#) cos by A6, PARTFUN1:5;
((id Z) (#) cos) | A is continuous ;
then A10: (id Z) (#) cos is_integrable_on A by A1, A8, INTEGRA5:11;
((id Z) (#) cos) | A is bounded by A1, A8, INTEGRA5:10;
hence integral (((id Z) (#) cos),A) = ((((id Z) (#) sin) + cos) . (upper_bound A)) - ((((id Z) (#) sin) + cos) . (lower_bound A)) by A1, A2, A10, A9, FDIFF_4:47, INTEGRA5:13; :: thesis: verum