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 Th13:
  for Omega be non empty finite set, P be Probability of
  Trivial-SigmaField (Omega), f be Function of Omega,REAL holds ex 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 = f.(s.n) * P.{s.n}) & Integral(P2M(P),f) = Sum F
proof
  let Omega be non empty finite set, P be Probability of Trivial-SigmaField (
  Omega), f be Function of Omega,REAL;
  set s=canFS(Omega);
A1: len s = card Omega by FINSEQ_1:93;
  (ex F being FinSequence of REAL st len F = card (Omega) &( for n being
  Nat st n in dom F holds F.n = f.((canFS(Omega)).n) * P. {(canFS( Omega)).n})&
  Integral(P2M(P),f) =Sum F )& rng s = Omega by Lm11,FUNCT_2:def 3;
  hence thesis by A1;
end;
