theorem Th41:
  w in Ball(x,r) iff dist(x,w) < r
proof
  thus w in Ball(x,r) implies dist(x,w) < r
  proof
    assume w in Ball(x,r);
    then ||.x - w.|| < r by Th40;
    hence thesis by CSSPACE:def 16;
  end;
  assume dist(x,w) < r;
  then ||.x - w.|| < r by CSSPACE:def 16;
  hence thesis;
end;
