reserve n for Nat;

theorem Th2:
  for p being Point of Euclid n, x, p9 being Point of TOP-REAL n,
  r being Real st p = p9 & x in Ball (p, r) holds |. x - p9 .| < 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 that
A1: p = p9 and
A2: x in Ball (p, r);
  dist (x9, p) < r by A2,METRIC_1:11;
  hence thesis by A1,SPPOL_1:39;
end;
