theorem Th47:
  w in cl_Ball(x,r) iff ||.x - w.|| <= r
proof
  thus w in cl_Ball(x,r) implies ||.x - w.|| <= r
  proof
    assume w in cl_Ball(x,r);
    then ex y be Point of X st w = y & ||.x - y.|| <= r;
    hence thesis;
  end;
  assume ||.x - w.|| <= r;
  hence thesis;
end;
