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 Th38:
  for S being non empty Subset-Family of X holds
  Product(n,S) = the set of all product f where f is Tuple of n,S
  proof
    let S be non empty Subset-Family of X;
    thus Product(n,S) c= the set of all product f where f is Tuple of n,S
    proof
      let x be object;
      assume x in Product(n,S);
      then consider g being Function such that
A1:   x = product g and
A2:   g in product (Seg n --> S) by Def2;
      g in n-tuples_on S by A2,Th8;
      then g is Tuple of n,S by FINSEQ_2:131;
      hence x in the set of all product f where f is Tuple of n,S by A1;
    end;
    thus the set of all product f where f is Tuple of n,S c= Product(n,S)
    proof
      let x be object;
      assume x in the set of all product f where f is Tuple of n,S;
      then consider f be Tuple of n,S such that
A3:   x = product f;
      f in n-tuples_on S by FINSEQ_2:131;
      then f in product(Seg n --> S) by Th8;
      hence thesis by A3,Def2;
    end;
  end;
