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 f being PartFunc of [:X1,X2:],ExtREAL
st M2 is sigma_finite & f is_integrable_on Prod_Measure(M1,M2)
holds
  Integral(M1,max+(Integral2(M2,|.f.|))) = Integral(M1,Integral2(M2,|.f.|))
& Integral(M1,max-(Integral2(M2,|.f.|))) = 0
proof
    let f be PartFunc of [:X1,X2:],ExtREAL;
    assume that
A1:  M2 is sigma_finite and
A2:  f is_integrable_on Prod_Measure(M1,M2);
    consider A be Element of sigma measurable_rectangles(S1,S2) such that
A3:  A = dom f & f is A-measurable by A2,MESFUNC5:def 17;
    reconsider SX1 = X1 as Element of S1 by MEASURE1:7;
A4: Integral2(M2,|.f.|) is SX1-measurable by A1,A3,Th4;
    A = dom |.f.| & |.f.| is A-measurable
      by A3,MESFUNC1:def 10,MESFUNC2:27; then
A5: Integral2(M2,|.f.|) is nonnegative by MESFUN12:66;
    hence Integral(M1,max+(Integral2(M2,|.f.|)))
             = Integral(M1,Integral2(M2,|.f.|)) by MESFUN11:31;
    SX1 = dom(Integral2(M2,|.f.|)) by FUNCT_2:def 1;
    hence Integral(M1,max-(Integral2(M2,|.f.|))) = 0 by A4,A5,MESFUN11:53;
end;
