
theorem Th20:
for m,n,k be non zero Nat, X be non-empty m-element FinSequence,
 S be sigmaFieldFamily of X st k <= n & n <= m holds
  (ProdSigmaFldFinSeq S).k = (ProdSigmaFldFinSeq SubFin(S,n)).k
proof
    let m,n,k be non zero Nat, X be non-empty m-element FinSequence,
    S be sigmaFieldFamily of X;
    assume that
A1:  k <= n and
A2:  n <= m;
A3: SubFin(S,n) = S|n by A2,Def6;

A4: 1 <= k by NAT_1:14;

    defpred P[Nat] means 1 <= $1 & $1 <= n implies
     (ProdSigmaFldFinSeq S).$1 = (ProdSigmaFldFinSeq SubFin(S,n)).$1;

A5: P[0];
A6: for i be Nat st P[i] holds P[i+1]
    proof
     let i be Nat;
     assume
A7:   P[i];
     assume
A8:   1 <= i+1 & i+1 <= n;
     per cases;
     suppose
A9:   i = 0; then
A10:   (ProdSigmaFldFinSeq S).(i+1) = S.1 by Def11;
A11:  (ProdSigmaFldFinSeq SubFin(S,n)).(i+1) = SubFin(S,n).1 by A9,Def11;
      1 <= n by A8,XXREAL_0:2; then
      1 in Seg n;
      hence (ProdSigmaFldFinSeq S).(i+1)
        = (ProdSigmaFldFinSeq SubFin(S,n)).(i+1) by A3,A10,A11,FUNCT_1:49;
     end;
     suppose i <> 0; then
      reconsider i1=i as non zero Nat;
A12:  i1 < n by A8,NAT_1:13; then
      i1 < m by A2,XXREAL_0:2; then
A13:  ex Si be SigmaField of CarProduct SubFin(X,i1) st
       Si = (ProdSigmaFldFinSeq S).i1
     & (ProdSigmaFldFinSeq S).(i1+1)
         = sigma measurable_rectangles(Si,ElmFin(S,i1+1)) by Def11;

A14:  ex Sj be SigmaField of CarProduct SubFin(SubFin(X,n),i1) st
       Sj = (ProdSigmaFldFinSeq SubFin(S,n)).i1
     & (ProdSigmaFldFinSeq SubFin(S,n)).(i1+1)
         = sigma measurable_rectangles(Sj,ElmFin(SubFin(S,n),i1+1))
           by A12,Def11;

A15:  ElmFin(S,i1+1) = ElmFin(SubFin(S,n),i1+1) by A2,A8,Th12;
      ElmFin(X,i1+1) = ElmFin(SubFin(X,n),i1+1) by A2,A8,Th8;
      hence (ProdSigmaFldFinSeq S).(i+1)
        = (ProdSigmaFldFinSeq SubFin(S,n)).(i+1)
         by A7,A12,A13,A15,A14,A2,Th7,NAT_1:14;
     end;
    end;
    for i be Nat holds P[i] from NAT_1:sch 2(A5,A6);
    hence thesis by A1,A4;
end;
