reserve
  x, y for object,
  i, n for Nat,
  r, s for Real,
  f1, f2 for n-element real-valued FinSequence;
reserve e, e1 for Point of Euclid n;

theorem
  TopSpaceMetr Euclid n = product(Seg n --> R^1)
  proof
    set J = Seg n --> R^1;
    per cases;
    suppose
A1:   n = 0;
      then J = {} --> R^1;
      hence thesis by A1,Lm1;
    end;
    suppose
A2:   n <> 0;
A3:   REAL n = Funcs(Seg n,REAL) by FINSEQ_2:93;
A4:   Funcs(Seg n,REAL) = product Carrier J by Th25;
      then reconsider PP = product_prebasis J as
      Subset-Family of TopSpaceMetr Euclid n by FINSEQ_2:93;
A5:   PP is open by Th27;
      PP is quasi_prebasis by A2,Th26;
      hence thesis by A4,A3,A5,WAYBEL18:def 3;
    end;
  end;
