theorem Th41:
  z in Ball(x,r) iff dist(x,z) < r
proof
  thus z in Ball(x,r) implies dist(x,z) < r
  proof
    assume z in Ball(x,r);
    then ||.x - z.|| < r by Th40;
    hence thesis by BHSP_1:def 5;
  end;
  assume dist(x,z) < r;
  then ||.x - z.|| < r by BHSP_1:def 5;
  hence thesis;
end;
