
theorem
for n be non zero Nat, X be non-empty (n+1)-element FinSequence,
 S be sigmaFieldFamily of X, M be sigmaMeasureFamily of S,
 f be PartFunc of CarProduct X,ExtREAL, E be Element of Prod_Field S,
 g be PartFunc of [:CarProduct SubFin(X,n),ElmFin(X,n+1):],ExtREAL
 st M is sigma_finite & E = dom f & f is E-measurable & f = g
holds
   g is_integrable_on  Prod_Measure(Prod_Measure SubFin(M,n),ElmFin(M,n+1))
    iff
   Integral(Prod_Measure SubFin(M,n),Integral2(ElmFin(M,n+1),|.g.|)) < +infty
proof
    let n be non zero Nat, X be non-empty (n+1)-element FinSequence,
    S be sigmaFieldFamily of X, M be sigmaMeasureFamily of S,
    f be PartFunc of CarProduct X,ExtREAL, E be Element of Prod_Field S,
    g be PartFunc of [:CarProduct SubFin(X,n),ElmFin(X,n+1):],ExtREAL;
    assume that
A1:  M is sigma_finite and
A2:  E = dom f and
A3:  f is E-measurable and
A4:  f = g;

A5: CarProduct X = [:CarProduct(SubFin(X,n)),ElmFin(X,n+1):]
      by Th6;

A6: Prod_Field S
     = sigma measurable_rectangles(Prod_Field SubFin(S,n),ElmFin(S,n+1))
      by Th34;
    reconsider E1 = E as Element of
      sigma measurable_rectangles(Prod_Field SubFin(S,n),ElmFin(S,n+1))
        by Th34;

    n <= n+1 by NAT_1:12; then
A7: Prod_Measure SubFin(M,n) is sigma_finite by Th30,A1,Th29;

    ElmFin(M,n+1) is sigma_finite by Th31,A1;
    hence g is_integrable_on
     Prod_Measure(Prod_Measure SubFin(M,n),ElmFin(M,n+1))
        iff
    Integral(Prod_Measure SubFin(M,n),Integral2(ElmFin(M,n+1),|.g.|))
      < +infty by MESFUN13:11,A3,A5,A6,A4,A2,A7;
end;
