theorem
  y in Ball(x,r) & z in Ball(x,r) implies dist(y,z) < 2 * r
proof
  assume that
A1: y in Ball(x,r) and
A2: z in Ball(x,r);
  dist(x,z) < r by A2,Th41;
  then
A3: r + dist(x,z) < r + r by XREAL_1:6;
A4: dist(y,z) <= dist(y,x) + dist(x,z) by BHSP_1:35;
  dist(x,y) < r by A1,Th41;
  then dist(x,y) + dist(x,z) < r + dist(x,z) by XREAL_1:6;
  then dist(x,y) + dist(x,z) < 2 * r by A3,XXREAL_0:2;
  hence thesis by A4,XXREAL_0:2;
end;
