
theorem Th32:
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 st
  f is_integrable_on Prod_Measure M holds
ex g be PartFunc of [: CarProduct SubFin(X,n),ElmFin(X,n+1) :],ExtREAL st
   f = g
 & g is_integrable_on Prod_Measure(Prod_Measure SubFin(M,n),ElmFin(M,n+1))
 & Integral(Prod_Measure M,f)
    = Integral(Prod_Measure(Prod_Measure SubFin(M,n),ElmFin(M,n+1)),g)
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;
    assume
A1:  f is_integrable_on Prod_Measure M;

A2: Prod_Measure M = Prod_Measure(Prod_Measure SubFin(M,n),ElmFin(M,n+1))
     by Th28;

    reconsider g = f as
     PartFunc of [:CarProduct SubFin(X,n),ElmFin(X,n+1):],ExtREAL by Th6;
    take g;
    thus f = g;

A3: n < n+1 by NAT_1:13;

A4: len X = n+1 by CARD_1:def 7;
    SubFin(X,n+1) = X|(n+1) by Def5; then
A5: X = SubFin(X,n+1) by A4,FINSEQ_1:58;

A6: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 A6,FINSEQ_1:58; then
A7: Prod_Field S
     = sigma measurable_rectangles(Prod_Field SubFin(S,n),ElmFin(S,n+1))
        by A5,A3,Th21;
    hence g is_integrable_on
     Prod_Measure(Prod_Measure SubFin(M,n),ElmFin(M,n+1)) by A1,A2,Th6;

    thus
    Integral(Prod_Measure M,f)
     = Integral(Prod_Measure(Prod_Measure SubFin(M,n),ElmFin(M,n+1)),g)
       by A7,A2,Th6;
end;
