
theorem Th6:
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,
F1 be Set_Sequence of S1, F2 be Set_Sequence of S2, n be Nat holds
  product_Measure(M1,M2).([:F1.n,F2.n:]) = M1.(F1.n) * M2.(F2.n)
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,
       F1 be Set_Sequence of S1, F2 be Set_Sequence of S2,
       n be Nat;
   F1.n in S1 & F2.n in S2 by MEASURE8:def 2;
   hence product_Measure(M1,M2).([:F1.n,F2.n:]) = M1.(F1.n) * M2.(F2.n)
     by Th5;
end;
