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 Th1:
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 & dom f = E
holds Integral(M1,Integral2(M2,|.f.|)) = Integral(Prod_Measure(M1,M2),|.f.|)
    & Integral(M2,Integral1(M1,|.f.|)) = Integral(Prod_Measure(M1,M2),|.f.|)
proof
    let E be Element of sigma measurable_rectangles(S1,S2),
    f be E-measurable PartFunc of [:X1,X2:],ExtREAL;
    assume that
A1:  M1 is sigma_finite and
A2:  M2 is sigma_finite and
A3:  dom f = E;
    E = dom |.f.| & |.f.| is E-measurable by A3,MESFUNC1:def 10,MESFUNC2:27;
    hence thesis by A1,A2,MESFUN12:84;
end;
