reserve n for Nat;

theorem Th9:
  for a, b being Real, x, y being Point of Euclid n st Ball
  (x, a) meets Ball (y, b) holds dist (x, y) < a + b
proof
  let a, b be Real, x, y be Point of Euclid n;
  assume Ball (x, a) meets Ball (y, b);
  then consider z being object such that
A1: z in Ball (x, a) and
A2: z in Ball (y, b) by XBOOLE_0:3;
  reconsider z as Point of Euclid n by A1;
  dist (y, z) < b by A2,METRIC_1:11;
  then
A3: dist (x, z) + dist (y, z) < dist (x, z) + b by XREAL_1:8;
A4: dist (x, z) + dist (y, z) >= dist (x, y) by METRIC_1:4;
  dist (x, z) < a by A1,METRIC_1:11;
  then dist (x, z) + b < a + b by XREAL_1:8;
  then dist (x, z) + dist (y, z) < a + b by A3,XXREAL_0:2;
  hence thesis by A4,XXREAL_0:2;
end;
