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 Th12:
  for Omega be non empty finite set, P be Probability of
  Trivial-SigmaField (Omega), f be Function of Omega,REAL, x be FinSequence of
REAL, s be 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) * P.{s.n}) holds Integral(P2M(P),f) =Sum x
proof
  let Omega be non empty finite set, P be Probability of Trivial-SigmaField (
Omega), f be Function of Omega,REAL, x be FinSequence of REAL, s be FinSequence
  of Omega;
  assume that
A1: len x = card (Omega) & s is one-to-one & rng s = Omega & len s =
  card (Omega) and
A2: for n being Nat st n in dom x holds x.n = f.(s.n) * P.{s.n};
  set M = P2M(P);
  rng x c= ExtREAL by NUMBERS:31;
  then reconsider x1= x as FinSequence of ExtREAL by FINSEQ_1:def 4;
A3: for n being Nat st n in dom x1 holds x1.n = (f.(s.n) ) * (P2M(P)).{s.n}
    proof
      let n be Nat;
      assume n in dom x1;
      then x1.n = f.(s.n) * P.{s.n} by A2;
      hence thesis by EXTREAL1:1;
    end;
  P.Omega in REAL by XREAL_0:def 1;
  then Integral(M,f) =Sum x1 by A1,A3,Th10,XXREAL_0:9
    .=Sum x by MESFUNC3:2;
  hence thesis;
end;
