
theorem Th25:
for n be non zero Nat, X be non-empty (n+1)-element FinSequence,
    S be sigmaFieldFamily of X, M be sigmaMeasureFamily of S holds
 Prod_Measure M
 = product_sigma_Measure(Prod_Measure SubFin(M,n),ElmFin(M,n+1))
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;

A1: n < n+1 by NAT_1:13; then
    ex Mn be sigma_Measure of Prod_Field SubFin(S,n) st
     Mn = (ProdSigmaMesFinSeq M).n
   & Prod_Measure M = product_sigma_Measure(Mn,ElmFin(M,n+1))
       by Def13;
    hence thesis by A1,Th24;
end;
