
theorem Th82:
for X1,X2 be non empty set, S1 be SigmaField of X1, S2 be SigmaField of X2,
  M1 be sigma_Measure of S1, M2 be sigma_Measure of S2,
  E be Element of sigma measurable_rectangles(S1,S2)
st M1 is sigma_finite & M2 is sigma_finite
holds
  Integral(Prod_Measure(M1,M2),chi(E,[:X1,X2:]))
   = Integral(M2,Integral1(M1,chi(E,[:X1,X2:])))
& Integral(Prod_Measure(M1,M2),chi(E,[:X1,X2:])|E)
   = Integral(M2,Integral1(M1,chi(E,[:X1,X2:])|E))
& Integral(Prod_Measure(M1,M2),chi(E,[:X1,X2:]))
   = Integral(M1,Integral2(M2,chi(E,[:X1,X2:])))
& Integral(Prod_Measure(M1,M2),chi(E,[:X1,X2:])|E)
   = Integral(M1,Integral2(M2,chi(E,[:X1,X2:])|E))
proof
    let X1,X2 be non empty set, S1 be SigmaField of X1, S2 be SigmaField of X2,
    M1 be sigma_Measure of S1, M2 be sigma_Measure of S2,
    E be Element of sigma measurable_rectangles(S1,S2);
    assume that
A1:  M1 is sigma_finite and
A2:  M2 is sigma_finite;

    X-vol(E,M1) = Integral1(M1,chi(E,[:X1,X2:])) by A1,Th64;
    hence
A4:  Integral(Prod_Measure(M1,M2),chi(E,[:X1,X2:]))
     = Integral(M2,Integral1(M1,chi(E,[:X1,X2:]))) by A1,A2,Th77;

A5: Integral(Prod_Measure(M1,M2),chi(E,[:X1,X2:])|E)
     = Prod_Measure(M1,M2).E by MESFUNC9:14
    .= Integral(Prod_Measure(M1,M2),chi(E,[:X1,X2:])) by MESFUNC9:14;
    hence
     Integral(Prod_Measure(M1,M2),chi(E,[:X1,X2:])|E)
      = Integral(M2,Integral1(M1,chi(E,[:X1,X2:])|E)) by A4,Th79;

    Y-vol(E,M2) = Integral2(M2,chi(E,[:X1,X2:])) by A2,Th65;
    hence
     Integral(Prod_Measure(M1,M2),chi(E,[:X1,X2:]))
     = Integral(M1,Integral2(M2,chi(E,[:X1,X2:]))) by A1,A2,Th77;
    hence
     Integral(Prod_Measure(M1,M2),chi(E,[:X1,X2:])|E)
      = Integral(M1,Integral2(M2,chi(E,[:X1,X2:])|E)) by A5,Th79;
end;
