reserve Omega for non empty set;
reserve r for Real;
reserve Sigma for SigmaField of Omega;
reserve P for Probability of Sigma;
reserve E for finite non empty set;

theorem
  for Omega be non empty finite set, M being sigma_Measure of
  Trivial-SigmaField (Omega), f be Function of Omega,REAL st M.Omega < +infty
holds ex x being FinSequence of ExtREAL, s being FinSequence of (Omega) st len
  x = card (Omega) & s is one-to-one & rng s = Omega & len s = card (Omega) & (
for n being Nat st n in dom x
  holds x.n = (f.(s.n) qua ExtReal) * M.{s.n}) & Integral
  (M,f) = Sum x
proof
  let Omega be non empty finite set, M be sigma_Measure of Trivial-SigmaField
  (Omega), f be Function of Omega,REAL;
  set s=canFS(Omega);
  assume M.Omega < +infty;
  then
A1: ex x being FinSequence of ExtREAL st len x = card (Omega) &( for n being
Nat st n in dom x
  holds x.n = (f.((canFS(Omega)). n) qua ExtReal) *
    M .{(canFS(Omega)
  ).n})& Integral(M,f) =Sum x by Lm10;
  rng s = Omega & len s = card (Omega) by FINSEQ_1:93,FUNCT_2:def 3;
  hence thesis by A1;
end;
