Actually, I did skip the remarks on purpose. Because I wanted to first give a clear logical explanation of the problem. And then to make my point that in order for this definition to work it should be proven that the element stated to exist is unique... I mean it should ring a bell that the "axioms" that are usually given depend on each other in a nontrivial way. I mean heaving to prove something that gives you the right to write the definition the way you did is an obvious math smell (the equivalent of the term code smell from programming). I also believe that definitions should be well thought out and easy to use the way we have given them.