theorem Th109:
  z in Ball(x,r) iff |.x - z.| < r
proof
A1: z in { z2 : |.z2 - x.| < r } iff ex z1 st z = z1 & |.z1 - x.| < r;
  |.z - x.| = |.x - z.| by Th103;
  hence thesis by A1;
end;
