
theorem Th104:
  for X be non empty set, S be SigmaField of X, M be
sigma_Measure of S, f be PartFunc of X,ExtREAL, r be Real st dom f in S & 0 <=
r & (for x be object st x in dom f holds f.x = r) holds integral'(M,f) = r
  * M.(dom f)
proof
  let X be non empty set;
  let S be SigmaField of X;
  let M be sigma_Measure of S;
  let f be PartFunc of X,ExtREAL;
  let r be Real;
  assume that
A1: dom f in S and
A2: 0 <= r and
A3: for x be object st x in dom f holds f.x = r;
  per cases;
  suppose
A4: dom f = {};
    then
A5: M.(dom f) = 0 by VALUED_0:def 19;
    integral'(M,f) = 0 by A4,Def14;
    hence thesis by A5;
  end;
  suppose
A6: dom f <> {};
    then integral'(M,f) = integral(M,f) by Def14;
    hence thesis by A1,A2,A3,A6,Th103;
  end;
end;
