
theorem Th23:
for m,n,k be non zero Nat, X be non-empty m-element FinSequence,
  S be sigmaFieldFamily of X, M be sigmaMeasureFamily of S st
  k <= n & n <= m holds
  (ProdSigmaMesFinSeq SubFin(M,n)).k
   = (ProdSigmaMesFinSeq SubFin(M,k)).k
proof
    let m,n,k be non zero Nat, X be non-empty m-element FinSequence,
    S be sigmaFieldFamily of X, M be sigmaMeasureFamily of S;
    assume that
A1:  k <= n and
A2:  n <= m;

A3: k <= m by A1,A2,XXREAL_0:2;
A4: 1 <= n & 1 <= k by NAT_1:14;
    now assume k <> n;

     defpred Q[Nat] means 1 <= $1 & $1 <= k implies
      (ProdSigmaMesFinSeq SubFin(M,n)).$1
        = (ProdSigmaMesFinSeq SubFin(M,k)).$1;

A5:  Q[0];
A6:  for i be Nat st Q[i] holds Q[i+1]
     proof
      let i be Nat;
      assume
A7:    Q[i];
      assume
A8:    1 <= i+1 & i+1 <= k;
      per cases;
      suppose i = 0; then
A9:    (ProdSigmaMesFinSeq SubFin(M,n)).(i+1) = SubFin(M,n).1
     & (ProdSigmaMesFinSeq SubFin(M,k)).(i+1) = SubFin(M,k).1 by Def13;

       1 in Seg n & 1 in Seg k by A4; then
       (M|n).1 = M.1 & (M|k).1 = M.1 by FUNCT_1:49; then
       SubFin(M,n).1 = M.1 & SubFin(M,k).1 = M.1 by A1,A2,Def9,XXREAL_0:2;
       hence (ProdSigmaMesFinSeq SubFin(M,n)).(i+1)
        = (ProdSigmaMesFinSeq SubFin(M,k)).(i+1) by A9;
      end;
      suppose i <> 0; then
       reconsider i0 = i as non zero Nat;
A10:    i0 < k by A8,NAT_1:13; then
A11:    i0 < n by A1,XXREAL_0:2; then
A12:    ex M1 be sigma_Measure of Prod_Field SubFin(SubFin(S,n),i0) st
        M1 = (ProdSigmaMesFinSeq SubFin(M,n)).i0
      & (ProdSigmaMesFinSeq SubFin(M,n)).(i0+1)
          = product_sigma_Measure(M1,ElmFin(SubFin(M,n),i0+1)) by Def13;

A13:    ex M2 be sigma_Measure of Prod_Field SubFin(SubFin(S,k),i0) st
        M2 = (ProdSigmaMesFinSeq SubFin(M,k)).i0
      & (ProdSigmaMesFinSeq SubFin(M,k)).(i0+1)
          = product_sigma_Measure(M2,ElmFin(SubFin(M,k),i0+1)) by A10,Def13;

       CarProduct SubFin(SubFin(X,n),i0)
        = CarProduct SubFin(X,i0) by A2,A11,Th7; then
A14:   CarProduct SubFin(SubFin(X,n),i0)
        = CarProduct SubFin(SubFin(X,k),i0) by A3,A10,Th7;

       SubFin(SubFin(X,n),i0) = SubFin(X,i0) by A2,A11,Th7; then
A15:   SubFin(SubFin(X,n),i0) = SubFin(SubFin(X,k),i0) by A3,A10,Th7;

       SubFin(SubFin(S,n),i0) = SubFin(S,i0) by A2,A11,Th14; then
A16:    Prod_Field SubFin(SubFin(S,n),i0)
       = Prod_Field SubFin(SubFin(S,k),i0) by A15,A3,A10,Th14;

A17:    i0+1 <= n by A8,A1,XXREAL_0:2; then
       ElmFin(SubFin(X,n),i0+1) = ElmFin(X,i0+1) by A2,Th8; then
A18:    ElmFin(SubFin(X,n),i0+1) = ElmFin(SubFin(X,k),i0+1) by A3,A8,Th8;

       ElmFin(SubFin(S,n),i0+1) = ElmFin(S,i0+1) by A17,A2,Th12; then
A19:    ElmFin(SubFin(S,n),i0+1) = ElmFin(SubFin(S,k),i0+1) by A3,A8,Th12;

       ElmFin(SubFin(M,n),i0+1) = ElmFin(M,i0+1) by A17,A2,Th17;
       hence (ProdSigmaMesFinSeq SubFin(M,n)).(i+1)
        = (ProdSigmaMesFinSeq SubFin(M,k)).(i+1)
           by A7,A12,A13,A19,A18,A16,A14,A3,A8,Th17,NAT_1:13,14;
      end;
     end;

     for i be Nat holds Q[i] from NAT_1:sch 2(A5,A6);
     hence (ProdSigmaMesFinSeq SubFin(M,n)).k
      = (ProdSigmaMesFinSeq SubFin(M,k)).k by A4;
    end;
    hence thesis;
end;
