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 X being non empty set,
      S being non empty Subset-Family of X,
      x being Subset of Funcs(Seg n,X)
  holds
      x is Element of Product(n,S)
    iff
      (ex s being Tuple of n,S st for t being Element of Funcs(Seg n,X) holds
      (for i being Nat st i in Seg n holds t.i in s.i) iff t in x)
  by Lm1,Lm2;
