
theorem Th22:
for m,n be non zero Nat, X be non-empty m-element FinSequence,
  S be sigmaFieldFamily of X, M be sigmaMeasureFamily of S st n <= m holds
  (ProdSigmaMesFinSeq M).n is sigma_Measure of Prod_Field SubFin(S,n)
proof
    let m,n be non zero Nat, X be non-empty m-element FinSequence,
    S be sigmaFieldFamily of X, M be sigmaMeasureFamily of S;
    assume
A1:  n <= m;

A2: 1 <= m by NAT_1:14;
A3: 1 <= n by NAT_1:14;
    set PM = ProdSigmaMesFinSeq M;

    defpred L[Nat] means 1 <= $1 & $1 <= m implies
     ex k be non zero Nat st k = $1
      & PM.$1 is sigma_Measure of Prod_Field SubFin(S,k);

A4: L[0];

A5: for i be Nat st L[i] holds L[i+1]
    proof
     let i be Nat;
     assume L[i];
     assume A6: 1 <= i+1 & i+1 <= m;

     per cases;
     suppose A7: i = 0;
A8:   1 in Seg m by A2;
A9:   1 in Seg 1;
      reconsider k = 1 as non zero Nat;
      take k;
      thus k = i+1 by A7;

      consider S1 be SigmaField of X.1 such that
A10:    S1 = S.1 & M.1 is sigma_Measure of S1 by A8,Def8;

A11:   ElmFin(X,k) = X.1 by Def1,NAT_1:14;
A12:   SubFin(X,k) = X|1 by Def5,NAT_1:14;

      CarProduct SubFin(X,k) = SubFin(X,k).1 by Def3; then
A13:   CarProduct SubFin(X,k) = ElmFin(X,k) by A12,A11,A9,FUNCT_1:49;

A14:   ElmFin(S,k) = S.1 by Def7,NAT_1:14;
A15:   SubFin(S,k) = S|1 by Def6,NAT_1:14;

      Prod_Field SubFin(S,k) = SubFin(S,k).1 by Def11; then
      ElmFin(S,k) = Prod_Field SubFin(S,k) by A14,A15,A9,FUNCT_1:49;
      hence PM.(i+1) is sigma_Measure of Prod_Field SubFin(S,k)
       by A11,A14,A10,A13,Def13,A7;
     end;
     suppose i <> 0; then
      reconsider k0 = i as non zero Nat;
A16:  1 <= k0 & k0 < m by A6,NAT_1:13,14; then
      consider PMk0 be sigma_Measure of Prod_Field SubFin(S,k0) such that
A17:   PMk0 = PM.k0
     & PM.(k0+1) = product_sigma_Measure(PMk0,ElmFin(M,k0+1))
        by Def13;
A18:   PM.(k0+1) is
       sigma_Measure of sigma
         measurable_rectangles(Prod_Field SubFin(S,k0),ElmFin(S,k0+1))
           by A17,MEASUR11:8;
A19:   [: CarProduct SubFin(X,k0),ElmFin(X,k0+1) :] = CarProduct SubFin(X,k0+1)
         by A16,Th9;
      Prod_Field SubFin(S,k0+1)
       = sigma measurable_rectangles(Prod_Field SubFin(S,k0),ElmFin(S,k0+1))
         by A16,Th21;
      hence ex k be non zero Nat st k = i+1 &
        PM.(i+1) is sigma_Measure of Prod_Field SubFin(S,k) by A18,A19;
     end;
    end;

    for n be Nat holds L[n] from NAT_1:sch 2(A4,A5); then
    ex n1 be non zero Nat st n1=n &
     PM.n is sigma_Measure of Prod_Field SubFin(S,n1) by A1,A3;
    hence (ProdSigmaMesFinSeq M).n is sigma_Measure of Prod_Field SubFin(S,n);
end;
