reserve n for Nat;

theorem Th10:
  for a, b, c being Real, x, y, z being Point of Euclid n
st Ball (x, a) meets Ball (z, c) & Ball (z, c) meets Ball (y, b) holds dist (x,
  y) < a + b + 2*c
proof
  let a, b, c be Real, x, y, z be Point of Euclid n;
  assume Ball (x, a) meets Ball (z, c) & Ball (z, c) meets Ball (y, b);
  then
  dist (x, z) + dist (z, y) < (a + c) + dist (z, y) & (a + c) + dist (z, y
  ) < (a + c) + (c + b) by Th9,XREAL_1:8;
  then
A1: dist (x, z) + dist (z, y) < (a + c) + (c + b) by XXREAL_0:2;
  dist (x, y) <= dist (x, z) + dist (z, y) by METRIC_1:4;
  hence thesis by A1,XXREAL_0:2;
end;
