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;
reserve f,g for Real-Valued-Random-Variable of Sigma;

theorem Th31:
  for Omega be non empty finite set, P be Probability of
  Trivial-SigmaField (Omega), X be Real-Valued-Random-Variable of
Trivial-SigmaField (Omega), F being FinSequence of REAL, s being FinSequence of
Omega st len F = card (Omega) & s is one-to-one & rng s = Omega & len s = card
  (Omega) & (for n being Nat st n in dom F holds F.n = X.(s.n) * P.{s.n}) holds
  expect(X,P) = Sum F
proof
  let Omega be non empty finite set, P be Probability of Trivial-SigmaField (
Omega), X be Real-Valued-Random-Variable of Trivial-SigmaField (Omega), F being
  FinSequence of REAL, s being FinSequence of Omega;
  assume len F = card (Omega) & s is one-to-one & rng s = Omega & len s =
  card ( Omega) & for n being Nat st n in dom F holds F.n = X.(s.n) * P.{s.n};
  then Integral(P2M(P),X) = Sum F by Th12;
  hence thesis by Def4,Th30;
end;
