
theorem Th34:
for n be non zero Nat, X be non-empty (n+1)-element FinSequence,
  S be sigmaFieldFamily of X, M be sigmaMeasureFamily of S holds
   Prod_Field S
    = sigma measurable_rectangles(Prod_Field SubFin(S,n),ElmFin(S,n+1))
proof
    let n be non zero Nat, X be non-empty (n+1)-element FinSequence,
     S be sigmaFieldFamily of X, M be sigmaMeasureFamily of S;

A1: n < n+1 by NAT_1:13;
A2:  len X = n+1 by CARD_1:def 7;
     SubFin(X,n+1) = X|(n+1) by Def5; then
A3:  X = SubFin(X,n+1) by A2,FINSEQ_1:58;
A4: len S = n+1 by CARD_1:def 7;
     SubFin(S,n+1) = S|(n+1) by Def6; then
     S = SubFin(S,n+1) by A4,FINSEQ_1:58;
    hence Prod_Field S
     = sigma measurable_rectangles(Prod_Field SubFin(S,n),ElmFin(S,n+1))
        by A3,A1,Th21;
end;
