let n be Element of NAT ; :: thesis: for f being PartFunc of REAL,(REAL n)
for A being non empty 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 non empty 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 non empty 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 :: thesis: for i being Nat st i in dom (- (integral (f,A))) holds
(- (integral (f,A))) . i = (integral (f,a,b)) . i

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;

reconsider jj = 1 as Real ;
dom (- (integral (f,A))) = dom ((- jj) (#) (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_1:13; :: thesis: verum