theorem
  Ball(z1,r1) is open
proof
  let x;
  assume x in Ball(z1,r1); then
A1: |.z1 - x.| < r1 by Th109;
  take r = r1 - |.z1 - x.|;
  thus 0 < r by A1,XREAL_1:50;
  let z;
  assume |.z.| < r; then
A2: |.z.| + |.z1 - x.| < r + |.z1 - x.| by XREAL_1:6;
  z1 - x - z = z1 - (x + z) by Th74;
  then |.z1 - (x + z).| <= |.z.| + |.z1 - x.| by Th98;
  then |.z1 - (x + z).| < r + |.z1 - x.| by A2,XXREAL_0:2;
  hence thesis by Th109;
end;
