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 Th5:
for E being Element of sigma measurable_rectangles(S1,S2),
 V being Element of S2, f being E-measurable PartFunc of [:X1,X2:],ExtREAL
st M1 is sigma_finite & E = dom f
holds Integral1(M1,|.f.|) is V-measurable
proof
    let E be Element of sigma measurable_rectangles(S1,S2),
    V be Element of S2, f be E-measurable PartFunc of [:X1,X2:],ExtREAL;
    assume that
A1:  M1 is sigma_finite and
A2:  E = dom f;
A3: E = dom |.f.| by A2,MESFUNC1:def 10;
    |.f.| is E-measurable by A2,MESFUNC2:27;
    hence Integral1(M1,|.f.|) is V-measurable by A1,A3,MESFUN12:59;
end;
