
theorem
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_sigma_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;
A1: [:F1.n,F2.n:] is Element of sigma measurable_rectangles(S1,S2) by Th3;
 then
A2:product_sigma_Measure(M1,M2).([:F1.n,F2.n:])
    = (sigma_Meas(C_Meas product_Measure(M1,M2))).([:F1.n,F2.n:])
      by FUNCT_1:49;
A3:measurable_rectangles(S1,S2)
    c= Field_generated_by measurable_rectangles(S1,S2) by SRINGS_3:21;
   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
A4: [:F1.n,F2.n:] in measurable_rectangles(S1,S2) by MEASUR10:def 5;
   product_Measure(M1,M2) is completely-additive by MEASURE9:60; then
A5:(product_Measure(M1,M2)).([:F1.n,F2.n:])
     = (C_Meas product_Measure(M1,M2)).([:F1.n,F2.n:]) by A3,A4,MEASURE8:18;
   sigma measurable_rectangles(S1,S2)
    c= sigma_Field(C_Meas product_Measure(M1,M2)) by Th9; then
   product_sigma_Measure(M1,M2).([:F1.n,F2.n:])
    = (product_Measure(M1,M2)).([:F1.n,F2.n:]) by A1,A2,A5,MEASURE4:def 3;
   hence thesis by Th6;
end;
