We use the real number axioms to prove two very important properties: that multiplication by 0 results in 0, and multiplication by -1 results in the additive inverse. These are NOT axioms, but are proven from the axioms. Often in mathematics, we wonder "why" a familiar property is the case--often, they arise not as assumptions but as trivial expansions upon an even more fundamental set of axioms. Of course, you could dispense with the "why's", but what's the fun in that?!
