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 M1 is sigma_finite & M2 is sigma_finite
 & f is_integrable_on Prod_Measure(M1,M2)
holds
    Integral(Prod_Measure(M1,M2),f)
     = Integral(M2,Integral1(M1,max+f)) - Integral(M2,Integral1(M1,max-f))
  & Integral(Prod_Measure(M1,M2),f)
     = Integral(M1,Integral2(M2,max+f)) - Integral(M1,Integral2(M2,max-f))
proof
    let f be PartFunc of [:X1,X2:],ExtREAL;
    assume that
A1:  M1 is sigma_finite and
A2:  M2 is sigma_finite and
A3:  f is_integrable_on Prod_Measure(M1,M2);
    set M = Prod_Measure(M1,M2);
    consider E be Element of sigma measurable_rectangles(S1,S2) such that
A4:  E = dom f & f is E-measurable by A3,MESFUNC5:def 17;
A5: max+f is nonnegative & max-f is nonnegative by MESFUN11:5;
A6: dom(max+f) = E & dom(max-f) = E by A4,MESFUNC2:def 2,def 3;
    max+f is E-measurable & max-f is E-measurable by A4,MESFUNC2:25,26; then
A7: Integral(M,max+f) = Integral(M2,Integral1(M1,max+f))
  & Integral(M,max-f) = Integral(M2,Integral1(M1,max-f))
  & Integral(M,max+f) = Integral(M1,Integral2(M2,max+f))
  & Integral(M,max-f) = Integral(M1,Integral2(M2,max-f))
       by A1,A2,A5,A6,MESFUN12:84;
    integral+(M,max+f) = Integral(M,max+f)
  & integral+(M,max-f) = Integral(M,max-f)
      by A4,A5,A6,MESFUNC2:25,26,MESFUNC5:88;
    hence thesis by A7,MESFUNC5:def 16;
end;
