Natural Numbers with Extension fulfil Naturally Ordered Semigroup Axioms 1, 3 and 4/Construction

Natural Numbers with Extension fulfil Naturally Ordered Semigroup Axioms 1, 3 and 4: Construction
Let $\N$ denote the set of natural numbers.

Let $\beta$ be an object such that $\beta \notin \N$

Let $M = \N \cup \set \beta$.

Let us extend the operation of natural number addition from $\N$ to $M$ by defining: