reserve Omega, Omega1, Omega2 for non empty set;
reserve Sigma for SigmaField of Omega;
reserve S1 for SigmaField of Omega1;
reserve S2 for SigmaField of Omega2;
reserve F for random_variable of S1,S2;

theorem Th19:
  for D be ManySortedSet of NAT holds
  (Product_dom(D)).0 = D.0 &
  (Product_dom(D)).1 = [:D.0,D.1:] &
  (Product_dom(D)).2 = [:D.0,D.1,D.2:] &
  (Product_dom(D)).3 = [:D.0,D.1,D.2,D.3:]
  proof
    let D be ManySortedSet of NAT;
    thus (Product_dom(D)).0 = D.0 by Def10;
    thus
    A1:(Product_dom(D)).1
    = (Product_dom(D)).( (0 qua Nat ) + 1)
    .= [:(Product_dom(D)).0,D.1:] by Def10
    .= [:D.0,D.1:] by Def10;
    thus
    A2:(Product_dom(D)).2 = [:(Product_dom(D)).1,D.(1+1):] by Def10
    .= [:D.0,D.1,D.2:] by ZFMISC_1:def 3,A1;
    thus (Product_dom(D)).3 = [:(Product_dom(D)).2,D.(2+1):] by Def10
    .= [:D.0,D.1,D.2,D.3 :] by ZFMISC_1:def 4,A2;
  end;
