
theorem Th12:
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,
E1,E2 be Element of sigma measurable_rectangles(S1,S2)
st E1 misses E2 holds
 product_sigma_Measure(M1,M2).(E1 \/ E2)
  = product_sigma_Measure(M1,M2).E1 + product_sigma_Measure(M1,M2).E2
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,
    E1,E2 be Element of sigma measurable_rectangles(S1,S2);
   assume A1: E1 misses E2;
   product_sigma_Measure(M1,M2) is sigma_Measure of
    sigma measurable_rectangles(S1,S2) by Th2;
   hence thesis by A1,MEASURE1:30;
end;
