Joe N says: "In fact it emphasizes the fact the rule is based on the signs alone."
Paul Tanner III says: This is not correct.
Who's talking about ring axioms other than you Paul? I could explain exactly what I mean above but I don't think you could follow because the things your own mind inserts into the proceedings keep getting in the way. That's not something I can correct. Read the Martinez book, maybe, show that you can follow it, then maybe we can discuss.