
theorem Th3:
for X1,X2 be non empty set, S1 be SigmaField of X1, S2 be SigmaField of X2,
F1 be Set_Sequence of S1, F2 be Set_Sequence of S2, n be Nat holds
   [:F1.n,F2.n:] is Element of sigma measurable_rectangles(S1,S2)
proof
   let X1,X2 be non empty set, S1 be SigmaField of X1, S2 be SigmaField of X2,
       F1 be Set_Sequence of S1, F2 be Set_Sequence of S2, n be Nat;
   set S = measurable_rectangles(S1,S2);
   F1.n in S1 & F2.n in S2 by MEASURE8:def 2; then
   [:F1.n,F2.n:] in the set of all [:A,B:]
      where A is Element of S1, B is Element of S2; then
A1: [:F1.n,F2.n:] in S by MEASUR10:def 5;
A2:S c= DisUnion S by SRINGS_3:12;
   DisUnion S c= sigma(DisUnion S) by PROB_1:def 9; then
   [:F1.n,F2.n:] is Element of sigma DisUnion (measurable_rectangles(S1,S2))
     by A1,A2;
   hence thesis by Th1;
end;
