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), 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) ) holds 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), G being FinSequence of REAL, s being FinSequence of
  Omega;
  assume that
A1: len G = 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 G holds G.n = X.(s.n);
  set P= Trivial-Probability (Omega);
  deffunc GF(Nat) = In(X.(s.$1) * P.{s.$1},REAL);
  consider F being FinSequence of REAL such that
A5: len F = len G & for j being Nat st j in dom F holds F.j = GF(j) from
  FINSEQ_2:sch 1;
A6: dom F = dom G by A5,FINSEQ_3:29;
  reconsider cG = (1/card (Omega))(#)G as FinSequence of REAL
      by RVSUM_1:145;
A7:  dom F = dom cG by A6,VALUED_1:def 5;
A8: for n being Nat st n in dom cG holds cG.n = F.n
  proof
    let n be Nat;
    assume
A9: n in dom cG;
    dom s = Seg len s by FINSEQ_1:def 3
      .= dom F by A1,A3,A5,FINSEQ_1:def 3;
    then s.n in Omega by A9,PARTFUN1:4,A7;
    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 cG.n = (1/card (Omega))*G.n by VALUED_1:6
      .= (1/card (Omega))*(X.(s.n)) by A4,A6,A9,A7
      .= GF(n) by A10
      .= F.n by A5,A9,A7;
  end;
A11: for n being Nat st n in dom cG holds cG.n = X.(s.n) * P.{s.n}
  proof
    let n be Nat;
    assume
A12: n in dom cG;
    dom s = Seg len s by FINSEQ_1:def 3
      .= dom F by A1,A3,A5,FINSEQ_1:def 3;
    then s.n in Omega by A12,PARTFUN1:4,A7;
    then reconsider A={s.n} as Singleton of Omega by RPR_1:4;
A13: P.{s.n} = prob(A) by Def1
      .=1/card (Omega) by RPR_1:14;
    thus cG.n = (1/card (Omega))*G.n by VALUED_1:6
      .= (1/card (Omega))*(X.(s.n)) by A4,A6,A12,A7
      .= GF(n) by A13
      .= X.(s.n) * P.{s.n};
  end;
  dom F = dom ((1/card (Omega))(#)G) by A6,VALUED_1:def 5;
  then
A14: (1/card (Omega))(#)G = F by FINSEQ_1:13,A8;
A15: len cG = card (Omega) by A1,A5,A14;
A16: s is one-to-one by A2;
 rng s = Omega & len s = card (Omega) by A2,A3;
 then expect(X,P) = Sum cG by Th31,A15,A16,A11;
  then expect(X,P) = Sum cG
  .= Sum((1/card (Omega))(#)G)
  .= (1/card Omega) * Sum(G) by RVSUM_1:87
  .= Sum(G) /card Omega by XCMPLX_1:99;
  hence thesis;
end;
