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 = [.a,b.] 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 = [.a,b.] holds
integral f,A = integral f,a,b
let A be closed-interval Subset of REAL ; :: thesis: for a, b being Real st A = [.a,b.] holds
integral f,A = integral f,a,b
let a, b be Real; :: thesis: ( A = [.a,b.] implies integral f,A = integral f,a,b )
assume A0: A = [.a,b.] ; :: thesis: integral f,A = integral f,a,b
A1: dom (integral f,A) = Seg n by DefintN
.= dom (integral f,a,b) by Defintn ;

now

let i be Nat; :: thesis: ( i in dom (integral f,A) implies (integral f,A) . i = (integral f,a,b) . i )
assume B0: i in dom (integral f,A) ; :: thesis: (integral f,A) . i = (integral f,a,b) . i
then reconsider k = i as Element of NAT ;
B1: dom (integral f,A) = Seg n by DefintN;
then B2: (integral f,A) . k = integral ((proj k,n) * f),A by B0, DefintN;
(integral f,a,b) . k = integral ((proj k,n) * f),a,b by B0, B1, Defintn;
hence (integral f,A) . i = (integral f,a,b) . i by A0, INTEGRA5:19, B2; :: thesis: verum

end;

hence integral f,A = integral f,a,b by A1, FINSEQ_1:17; :: thesis: verum