let n be Element of NAT ; :: thesis: for f being PartFunc of REAL ,(REAL n)
for A being closed-interval Subset of REAL
for a, b being Real st A = [.b,a.] holds
- (integral f,A) = integral f,a,b
let f be PartFunc of REAL ,(REAL n); :: thesis: for A being closed-interval Subset of REAL
for a, b being Real st A = [.b,a.] holds
- (integral f,A) = integral f,a,b
let A be closed-interval Subset of REAL ; :: thesis: for a, b being Real st A = [.b,a.] holds
- (integral f,A) = integral f,a,b
let a, b be Real; :: thesis: ( A = [.b,a.] implies - (integral f,A) = integral f,a,b )
assume A1: A = [.b,a.] ; :: thesis: - (integral f,A) = integral f,a,b

A2: now

let i be Nat; :: thesis: ( i in dom (- (integral f,A)) implies (- (integral f,A)) . i = (integral f,a,b) . i )
assume A3: i in dom (- (integral f,A)) ; :: thesis: (- (integral f,A)) . i = (integral f,a,b) . i
then reconsider k = i as Element of NAT ;
A4: dom (integral f,A) = Seg n by Def17;
A5: k in dom (integral f,A) by A3, VALUED_1:8;
then A6: (integral f,a,b) . k = integral ((proj k,n) * f),a,b by A4, Def18;
(- (integral f,A)) . k = - ((integral f,A) . k) by VALUED_1:8
.= - (integral ((proj k,n) * f),A) by A5, A4, Def17 ;
hence (- (integral f,A)) . i = (integral f,a,b) . i by A1, A6, INTEGRA5:20; :: thesis: verum

end;

dom (- (integral f,A)) = dom ((- 1) (#) (integral f,A)) by VALUED_1:def 6
.= dom (integral f,A) by VALUED_1:def 5
.= Seg n by Def17
.= dom (integral f,a,b) by Def18 ;
hence - (integral f,A) = integral f,a,b by A2, FINSEQ_1:17; :: thesis: verum