Subtraction on Numbers is Anticommutative/Natural Numbers

Theorem
The operation of subtraction on the natural numbers $\N$ is anticommutative, and defined only when $a = b$:

That is:
 * $a - b = b - a \iff a = b$

Proof
$a - b$ is defined on $\N$ only if $a \ge b$.

If $a > b$, then although $a - b$ is defined, $b - a$ is not.

So for $a - b = b - a$ it is necessary for both to be defined.

This happens only when $a = b$.

Hence the result.