
theorem Th24:
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 = Prod_Measure SubFin(M,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;
    defpred P[Nat] means 1 <= $1 & $1 <= m implies
     ex k be non zero Nat st
      k = $1 & (ProdSigmaMesFinSeq M).$1 = Prod_Measure SubFin(M,k);

A2: P[0];
A3: for i be Nat st P[i] holds P[i+1]
    proof
     let i be Nat;
     assume
A4:   P[i];
     assume
A5:   1 <= i+1 & i+1 <= m;
     per cases;
     suppose
A6:   i = 0; then
      (ProdSigmaMesFinSeq M).(i+1) = M.1 by Def13; then
A7:  (ProdSigmaMesFinSeq M).(i+1) = ElmFin(M,1) by A6,A5,Def10;

      reconsider k = i+1 as non zero Nat;

      Prod_Measure SubFin(M,k) = SubFin(M,k).1 by A6,Def13; then
      Prod_Measure SubFin(M,k) = ElmFin(SubFin(M,k),1) by A6,Def10;
      hence ex k be non zero Nat st
       k = i+1 & (ProdSigmaMesFinSeq M).(i+1)
         = Prod_Measure SubFin(M,k) by A7,A5,Th17;
     end;
     suppose i <> 0; then
      reconsider k0 = i as non zero Nat;
      k0 < m by A5,NAT_1:13; then
A8:  ex Mk0 be sigma_Measure of Prod_Field SubFin(S,k0) st
       Mk0 = (ProdSigmaMesFinSeq M).k0
     & (ProdSigmaMesFinSeq M).(k0+1)
        = product_sigma_Measure(Mk0,ElmFin(M,k0+1)) by Def13;

      reconsider k = k0+1 as non zero Nat;
A9:  k0 < k by NAT_1:13; then
A10:  ex Mk02 be sigma_Measure of Prod_Field SubFin(SubFin(S,k),k0) st
        Mk02 = (ProdSigmaMesFinSeq SubFin(M,k)).k0
      & (ProdSigmaMesFinSeq SubFin(M,k)).(k0+1)
        = product_sigma_Measure(Mk02,ElmFin(SubFin(M,k),k0+1)) by Def13;

A11:   SubFin(X,k0) = SubFin(SubFin(X,k),k0) by A5,A9,Th7;
A12:   SubFin(S,k0) = SubFin(SubFin(S,k),k0) by A5,A9,Th14;

A13:  ElmFin(X,k0+1) = ElmFin(SubFin(X,k),k) by A5,Th8;
A14:  ElmFin(S,k0+1) = ElmFin(SubFin(S,k),k) by A5,Th12;
A15:  ElmFin(M,k0+1) = ElmFin(SubFin(M,k),k) by A5,Th17;

      (ProdSigmaMesFinSeq M).(k0+1) = Prod_Measure SubFin(M,k)
          by A8,A15,A14,A13,A10,A11,A12,A9,A4,A5,NAT_1:13,14,Th23;
      hence ex k be non zero Nat st k = i+1
       & (ProdSigmaMesFinSeq M).(i+1) = Prod_Measure SubFin(M,k);
     end;
    end;

    for i be Nat holds P[i] from NAT_1:sch 2(A2,A3); then
    P[n];
    hence (ProdSigmaMesFinSeq M).n = Prod_Measure SubFin(M,n)
      by A1,NAT_1:14;
end;
