reserve n for Nat;

theorem
  for M be symmetric triangle MetrStruct for r be Real
  for p,q,x be Element of M st p in Ball(x,r) & q in Ball(x,r) holds
  dist(p,q) < 2*r
proof
  let M be symmetric triangle MetrStruct;
  let r be Real;
  let p,q,x be Element of M;
  assume that
A1: p in Ball(x,r) and
A2: q in Ball(x,r);
A3: dist(p,x) < r by A1,METRIC_1:11;
A4: dist(x,q) < r by A2,METRIC_1:11;
A5: dist(p,q) <= dist(p,x) + dist(x,q) by METRIC_1:4;
  dist(p,x) + dist(x,q) < r+r by A3,A4,XREAL_1:8;
  hence thesis by A5,XXREAL_0:2;
end;
