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 & f = #Z 2 & Z = dom ((1 / 2) (#) f) holds
integral ((id Z),A) = (((1 / 2) (#) f) . (upper_bound A)) - (((1 / 2) (#) f) . (lower_bound A))
let Z be open Subset of REAL; :: thesis: for f being PartFunc of REAL,REAL st A c= Z & f = #Z 2 & Z = dom ((1 / 2) (#) f) holds
integral ((id Z),A) = (((1 / 2) (#) f) . (upper_bound A)) - (((1 / 2) (#) f) . (lower_bound A))
let f be PartFunc of REAL,REAL; :: thesis: ( A c= Z & f = #Z 2 & Z = dom ((1 / 2) (#) f) implies integral ((id Z),A) = (((1 / 2) (#) f) . (upper_bound A)) - (((1 / 2) (#) f) . (lower_bound A)) )
assume that
A1: A c= Z and
A2: ( f = #Z 2 & Z = dom ((1 / 2) (#) f) ) ; :: thesis: integral ((id Z),A) = (((1 / 2) (#) f) . (upper_bound A)) - (((1 / 2) (#) f) . (lower_bound A))
A3: A c= dom (id Z) by A1;
then A4: (id Z) | A is bounded by INTEGRA5:10;
A5: (1 / 2) (#) f is_differentiable_on Z by A2, Th33;
A6: for x being Element of REAL st x in dom (((1 / 2) (#) f) `| Z) holds
(((1 / 2) (#) f) `| Z) . x = (id Z) . x

proof

let x be Element of REAL ; :: thesis: ( x in dom (((1 / 2) (#) f) `| Z) implies (((1 / 2) (#) f) `| Z) . x = (id Z) . x )
assume x in dom (((1 / 2) (#) f) `| Z) ; :: thesis: (((1 / 2) (#) f) `| Z) . x = (id Z) . x
then A7: x in Z by A5, FDIFF_1:def 7;
then (((1 / 2) (#) f) `| Z) . x = x by A2, Th33
.= (id Z) . x by A7, FUNCT_1:18 ;
hence (((1 / 2) (#) f) `| Z) . x = (id Z) . x ; :: thesis: verum

end;

dom (((1 / 2) (#) f) `| Z) = dom (id Z) by A5, FDIFF_1:def 7;
then A8: ((1 / 2) (#) f) `| Z = id Z by A6, PARTFUN1:5;
(id Z) | A is continuous ;
then id Z is_integrable_on A by A3, INTEGRA5:11;
hence integral ((id Z),A) = (((1 / 2) (#) f) . (upper_bound A)) - (((1 / 2) (#) f) . (lower_bound A)) by A1, A2, A4, A8, Th33, INTEGRA5:13; :: thesis: verum