reserve X for set;
reserve X,X1,X2 for non empty set;
reserve S for SigmaField of X;
reserve S1 for SigmaField of X1;
reserve S2 for SigmaField of X2;
reserve M for sigma_Measure of S;
reserve M1 for sigma_Measure of S1;
reserve M2 for sigma_Measure of S2;

theorem
for E being Element of sigma measurable_rectangles(S1,S2),
 f being E-measurable PartFunc of [:X1,X2:],ExtREAL
st M1 is sigma_finite & M2 is sigma_finite & E = dom f
holds f is_integrable_on Prod_Measure(M1,M2)
        iff Integral(M1,Integral2(M2,|.f.|)) < +infty by Lm1,Lm2;
