reserve n for Nat;

theorem Th3:
  for p being Point of Euclid n, x, p9 being Point of TOP-REAL n,
  r being Real st p = p9 & |. x - p9 .| < r holds x in Ball (p, r)
proof
  let p be Point of Euclid n, x, p9 be Point of TOP-REAL n, r be Real;
  reconsider x9 = x as Point of Euclid n by TOPREAL3:8;
  assume p = p9 & |. x - p9 .| < r;
  then dist (x9, p) < r by SPPOL_1:39;
  hence thesis by METRIC_1:11;
end;
