reserve n   for Nat,
        r,s for Real,
        x,y for Element of REAL n,
        p,q for Point of TOP-REAL n,
        e   for Point of Euclid n;
reserve n for non zero Nat;
reserve n for non zero Nat;
reserve n for Nat,
        X for set,
        S for Subset-Family of X;

theorem
  for n being non zero Nat,X being non empty set,
  S being non empty Subset-Family of X st S <> {{}} holds
  union Product(n,S) c= Funcs(Seg n,X)
  proof
    let n be non zero Nat,X be non empty set,S be non empty Subset-Family of X;
    assume S <> {{}};
    then union Product(n,S) c= union bool Funcs(Seg n,X) by ZFMISC_1:77,Th40;
    hence thesis by ZFMISC_1:81;
  end;
