
theorem Th35:
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,
 fn1 be PartFunc of [: CarProduct SubFin(X,n), ElmFin(X,n+1) :],ExtREAL
  st M is sigma_finite & f = fn1 & f is_integrable_on Prod_Measure M
   & for x be Element of CarProduct(SubFin(X,n)) holds
        (Integral2(ElmFin(M,n+1),|.fn1.|)).x < +infty
  holds
   Integral(Prod_Measure M,f)
     = Integral(Prod_Measure(Prod_Measure(SubFin(M,n)),ElmFin(M,n+1)),fn1)
 & ( for x being Element of CarProduct(SubFin(X,n)) holds
       ProjPMap1(fn1,x) is_integrable_on ElmFin(M,n+1) )
 & ( for U being Element of Prod_Field(SubFin(S,n)) holds
       Integral2(ElmFin(M,n+1),fn1) is U-measurable )
 & Integral2(ElmFin(M,n+1),fn1) is_integrable_on Prod_Measure(SubFin(M,n))
 & Integral(Prod_Measure(Prod_Measure(SubFin(M,n)),ElmFin(M,n+1)),fn1)
     = Integral(Prod_Measure(SubFin(M,n)),(Integral2(ElmFin(M,n+1),fn1)))
 & Integral2(ElmFin(M,n+1),fn1) in L1_Functions Prod_Measure(SubFin(M,n))
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,
    fn1 be PartFunc of [: CarProduct SubFin(X,n), ElmFin(X,n+1) :],ExtREAL;
    assume that
A1:  M is sigma_finite and
A2:  f = fn1 and
A3:  f is_integrable_on Prod_Measure M and
A4:  for x be Element of CarProduct(SubFin(X,n)) holds
        (Integral2(ElmFin(M,n+1),|.fn1.|)).x < +infty;

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

A6:Prod_Measure M = Prod_Measure(Prod_Measure SubFin(M,n),ElmFin(M,n+1))
      by Th28;

A7: n < n+1 by NAT_1:13; then
A8:Prod_Measure SubFin(M,n) is sigma_finite by A1,Th29,Th30;

A9:ElmFin(M,n+1) is sigma_finite by Th31,A1;

A10: len X = n+1 by CARD_1:def 7;
    SubFin(X,n+1) = X|(n+1) by Def5; then
A11: X = SubFin(X,n+1) by A10,FINSEQ_1:58;

A12:len S = n+1 by CARD_1:def 7;
    SubFin(S,n+1) = S|(n+1) by Def6; then
    S = SubFin(S,n+1) by A12,FINSEQ_1:58; then
    Prod_Field S
     = sigma measurable_rectangles(Prod_Field SubFin(S,n),ElmFin(S,n+1))
        by A7,A11,Th21;
    hence thesis by A4,A2,A3,A5,A6,A8,A9,MESFUN13:32;
end;
