Well, the problem lies in the fact that I failed to differentiate between definitions and theorems. And the difference is crucial. Absolutely crucial.
A definition is basically the naming of a certain property or group of properties. So for example, say we observe that pretty often the order of multiplication in the natural numbers does not change the value of the product (i.e. 3x4=4x3). So instead of saying that whole shpiel each time, we decide to define the term "commutative" (which in "real maths" loosely means a x b = b x a or a + b = b + a). You can't prove what commutative means. That's just what you've decided to call that particular property.
If you are alert, you will observe that this is pretty similar to how I explained what an axiom was. I promise that there is an important distinction. An axiom is something you assume to always be true. A definition is simply the name you give to something which is sometimes true in some situations.
So, back to theorems and definitions.
So we've defined commutativity. Now the more we look at the natural numbers, the more we get the feeling that all natural numbers will be commutative under multiplication. Now we're getting into theorem territory. The minute we make the claim "natural numbers are commutative under multiplication (i.e. it doesn't matter which order you multiply natural numbers in)", we are saying that our definition of commutativity applies to a very large group of numbers.
This has to be proved. And I'm not going to tackle that enormous topic now. Suffice it to say that it is important to distinguish clearly (in Maths notes or elsewhere) between theorems and definitions.
Briefly (and slightly poetically):
Theorem = Claim
Definition = Name