
theorem Th21:
for m,n be non zero Nat, X be non-empty m-element FinSequence,
 S be sigmaFieldFamily of X st n < m holds
  Prod_Field SubFin(S,n+1)
  = sigma measurable_rectangles(Prod_Field SubFin(S,n),ElmFin(S,n+1))
proof
    let m,n be non zero Nat, X be non-empty m-element FinSequence,
    S be sigmaFieldFamily of X;
    assume
A1:  n < m; then
A2: n+1 <= m by NAT_1:13; then
A3: ElmFin(SubFin(X,n+1),n+1) = ElmFin(X,n+1) by Th8;
A4: ElmFin(SubFin(S,n+1),n+1) = ElmFin(S,n+1) by A2,Th12;

A5: n < n+1 by NAT_1:13; then
A6: CarProduct SubFin(SubFin(X,n+1),n) = CarProduct SubFin(X,n)
      by A2,Th7;
    consider Sn be SigmaField of CarProduct SubFin(SubFin(X,n+1),n)
     such that
A7:  Sn = (ProdSigmaFldFinSeq SubFin(S,n+1)).n
   & (ProdSigmaFldFinSeq SubFin(S,n+1)).(n+1)
      = sigma measurable_rectangles(Sn,ElmFin(SubFin(S,n+1),n+1))
        by A5,Def11;
    Sn = (ProdSigmaFldFinSeq S).n by A5,A7,A2,Th20;
    hence thesis by A7,A4,A3,A6,A1,Th20;
end;
