Natural Number Less than or Equal to Successor of Another

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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 $m \le n^+$:

$m = n^+$

$\blacksquare$


Sources