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), y be Element of X2
holds
  ( M1.Measurable-Y-section(E,y) <> 0
     implies Integral1(M1,Xchi(E,[:X1,X2:])).y = +infty ) &
  ( M1.Measurable-Y-section(E,y) = 0
     implies Integral1(M1,Xchi(E,[:X1,X2:])).y = 0 )
proof
    let E be Element of sigma measurable_rectangles(S1,S2),
    y be Element of X2;
    ProjPMap2(Xchi(E,[:X1,X2:]),y) = Xchi(Y-section(E,y),X1)
      by MESFUN12:35; then
A1: Integral1(M1,Xchi(E,[:X1,X2:])).y
     = Integral(M1,Xchi(Y-section(E,y),X1)) by MESFUN12:def 7;
A2: Measurable-Y-section(E,y) = Y-section(E,y) by MEASUR11:def 7;
    hence M1.Measurable-Y-section(E,y) <> 0 implies
     Integral1(M1,Xchi(E,[:X1,X2:])).y = +infty by A1,MEASUR10:33;
    assume M1.Measurable-Y-section(E,y) = 0;
    hence thesis by A1,A2,MEASUR10:33;
end;
