let X be non empty set ; :: thesis:
let S be SigmaField of X; :: thesis:
let M be sigma_Measure of S; :: thesis:
let f be PartFunc of X,ExtREAL; :: thesis:
let r be Real; :: thesis:
assume that
A1: dom f in S and
A2: 0 <= r and
A3: for x being object st x in dom f holds
f . x = r ; :: thesis:

suppose A4: dom f = {} ; :: thesis:

then A5: M . (dom f) = 0 by VALUED_0:def 19;
integral' (M,f) = 0 by A4, Def14;
hence integral' (M,f) = r * (M . (dom f)) by A5; :: thesis:

end;

suppose A6: dom f <> {} ; :: thesis:

then integral' (M,f) = integral (M,f) by Def14;
hence integral' (M,f) = r * (M . (dom f)) by A1, A2, A3, A6, Th103; :: thesis:

end;

end;