
theorem Th10:
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,
E be Element of sigma measurable_rectangles(S1,S2),
A be Element of S1, B be Element of S2
 st E = [:A,B:] holds product_sigma_Measure(M1,M2).E = M1.A * M2.B
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,
       E be Element of sigma measurable_rectangles(S1,S2),
       A be Element of S1, B be Element of S2;
   assume A1: E = [:A,B:]; then
A2:product_sigma_Measure(M1,M2).([:A,B:])
    = (sigma_Meas(C_Meas product_Measure(M1,M2))).([:A,B:]) by FUNCT_1:49;
A3:measurable_rectangles(S1,S2)
    c= Field_generated_by measurable_rectangles(S1,S2) by SRINGS_3:21;
   [:A,B:] in the set of all [:A,B:]
      where A is Element of S1, B is Element of S2; then
A4: [:A,B:] 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)).([:A,B:])
     = (C_Meas product_Measure(M1,M2)).([:A,B:]) 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).([:A,B:])
    = (product_Measure(M1,M2)).([:A,B:]) by A1,A2,A5,MEASURE4:def 3;
   hence thesis by A1,Th5;
end;
