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
  for Omega be non empty finite set, X be Real-Valued-Random-Variable of
  Trivial-SigmaField (Omega)
 ex G being FinSequence of REAL, s being FinSequence of Omega
  st len G = card (Omega) & s is one-to-one & rng s = Omega
& len s = card (Omega) &
(for n being Nat st n in dom G holds G.n = X.(s.n) ) &
  expect(X,Trivial-Probability (Omega)) = (Sum G) / card (Omega)
proof
  let Omega be non empty finite set, X be Real-Valued-Random-Variable of
  Trivial-SigmaField (Omega);
  set P= Trivial-Probability (Omega);
  consider F being FinSequence of REAL, s being FinSequence of Omega such that
A1: len F = card (Omega) and
A2: s is one-to-one & rng s = Omega and
A3: len s = card (Omega) and
A4: for n being Nat st n in dom F holds F.n = X.(s.n) * P.{s.n} and
A5: expect(X,P) = Sum F by Th33;
  deffunc GF(Nat) = In(X.(s.$1),REAL);
  consider G being FinSequence of REAL such that
A6: len G = len F & for j being Nat st j in dom G holds G.j = GF(j) from
  FINSEQ_2:sch 1;
A7: dom F = dom G by A6,FINSEQ_3:29;
A8: now
    let n be Nat;
    assume
A9: n in dom F;
    dom s= Seg len s by FINSEQ_1:def 3
      .= dom F by A1,A3,FINSEQ_1:def 3;
    then s.n in Omega by A9,PARTFUN1:4;
    then reconsider A={s.n} as Singleton of Omega by RPR_1:4;
A10: P.{s.n} = prob(A) by Def1
      .=1/card (Omega) by RPR_1:14;
    thus ((1/card (Omega))(#)G).n = (1/card (Omega))*G.n by VALUED_1:6
      .= (1/card (Omega))*GF(n) by A6,A7,A9
      .= (1/card (Omega))*X.(s.n)
      .= X.(s.n) * P.{s.n} by A10
      .= F.n by A4,A9;
  end;
  take G,s;
  dom F = dom ((1/card (Omega))(#)G) by A7,VALUED_1:def 5;
  then
A11:(1/card (Omega))(#)G = F by A8,FINSEQ_1:13;
  thus len G = card (Omega) & s is one-to-one & rng s = Omega
& len s = card (Omega) by A1,A2,A3,A6;
  thus for n being Nat st n in dom G holds G.n = X.(s.n)
   proof let n be Nat;
    assume n in dom G;
     then G.n = GF(n) by A6;
    hence thesis;
   end;
   Sum((1/card (Omega))(#)G) = 1/card Omega * Sum G by RVSUM_1:87;
   hence thesis by A5,A11,XCMPLX_1:99;
end;
