reserve n for Nat;

theorem
  for x, y being Point of TOP-REAL n, x9 being Point of Euclid n st x9 =
  x & x <> y ex r being Real st not y in Ball (x9, r)
proof
  let x, y be Point of TOP-REAL n, x9 be Point of Euclid n;
  reconsider y9 = y as Point of Euclid n by TOPREAL3:8;
  reconsider r = dist (x9, y9)/2 as Real;
  assume x9 = x & x <> y;
  then
A1: dist (x9, y9) <> 0 by METRIC_1:2;
  take r;
  dist (x9, y9) >= 0 by METRIC_1:5;
  then dist (x9, y9) > r by A1,XREAL_1:216;
  hence thesis by METRIC_1:11;
end;
