
theorem Th77:
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)
st M1 is sigma_finite & M2 is sigma_finite
holds Integral(M1,(Y-vol(E,M2)))
        = Integral(Prod_Measure(M1,M2),chi(E,[:X1,X2:]))
    & Integral(M2,(X-vol(E,M1)))
        = Integral(Prod_Measure(M1,M2),chi(E,[:X1,X2:]))
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);
    assume that
A1:  M1 is sigma_finite and
A2:  M2 is sigma_finite;
    Integral(M2,(X-vol(E,M1))) = (product_sigma_Measure(M1,M2)).E
  & Integral(M1,(Y-vol(E,M2))) = (product_sigma_Measure(M1,M2)).E
      by A1,A2,MEASUR11:118,117;
    hence thesis by MESFUNC9:14;
end;
