Natural Number Less than or Equal to Successor of Another

Theorem
Let $\N$ be the natural numbers.

Let $m, n \in \N$ such that $m \le n^+$.

Then either:


 * $(1): \quad m \le n$

or:
 * $(2): \quad m = n^+$

Proof
Let $m \le n^+$.

Suppose $m \le n$ is false.

Then:
 * $n^+ \le m$

and because $m \le n^+$:
 * $m = n^+$