
theorem Th39:
  for X1,X2 be non empty set,
      S1 be SigmaField of X1, S2 be SigmaField of X2,
      M1 be sigma_Measure of S1, M2 be sigma_Measure of S2 holds
  product-pre-Measure(M1,M2) is pre-Measure of measurable_rectangles(S1,S2)
proof
   let X1,X2 be non empty set, S1 be SigmaField of X1, S2 be SigmaField of X2,
   M1 be sigma_Measure of S1, M2 be sigma_Measure of S2;
   set P = measurable_rectangles(S1,S2);
   set M = product-pre-Measure(M1,M2);
A2:   for F be disjoint_valued FinSequence of P st Union F in P
    holds M.(Union F) = Sum(M*F) by Th38;
   for K be disjoint_valued Function of NAT,P st Union K in P
    holds M.(Union K) <= SUM(M*K) by Th35;
   hence thesis by A2,MEASURE9:def 7;
end;
