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

Construction
The algebraic structure:
 * $\struct {M, +}$

is a commutative monoid such that $0$ is the identity.

Closure
It is directly apparent from the definition that $\struct {M, +}$ is closed.

Commutativity
We know from Natural Number Addition is Commutative that:
 * $\forall a, b \in \N: a + b = b + a$

It is also seen from the definition that:


 * $\forall a \in \N: a + \beta = \beta + a$

Thus $+$ is commutative on $M$.

Associativity
We know from Natural Number Addition is Associative that:
 * $\forall a, b, c \in \N: \paren {a + b} + c = a + \paren {b + c}$

It remains to check the other cases.

In the below, $a$ and $b$ are arbitrary non-zero elements of $\N$.

Because of commutativity it is sufficient to test the various combinations of $a$, $b$ and $0$ with $\beta$.

So:

All combinations of $a$, $b$ and $0$ with $\beta$ have been verified for associativity.

Hence $+$ is associative on $M$.

Identity Element
We have that Identity Element of Natural Number Addition is Zero.

That is:
 * $\forall a \in \N: a + 0 = a = 0 + a$

We also have that:
 * $\beta + 0 = \beta = 0 + \beta$

and it is seen that $0$ is the identity element for $\struct {M, +}$.

Hence the result.