Proof
(related to Proposition: Algebraic Structure Of Natural Numbers Together With Addition)
The proof will consist of four steps:
Step 1: Demonstrate that the addition of natural numbers "\( + \)" is associative.
This follows from the corresponding proposition.
Step 2: Demonstrate that the addition of natural numbers "\( + \)" is commutative.
This follows from the corresponding proposition.
Step 3: Demonstrate that \( ( \mathbb N, + )\) forms a commutative monoid.
 Step 1 shows that "\( + \)" is associative, so \( ( \mathbb N, + )\) forms a semigroup.
 Because adding natural numbers is associative and because the natural number \(0\) is neutral with respect to addition, we have \(n + 0=0 + n=n\), thus the element \(0\in\mathbb N\) is the identity element of the semigroup.
 Because \( ( \mathbb N, + )\) forms a semigroup with an identity element (which is \(0\)), it is a monoid.
 Together with step 2, this shows that \( ( \mathbb N, + )\) forms a communtative monoid.
Step 4: Show that \((\mathbb N, +)\) is cancellative.
This follows from the corresponding proposition.
These four steps show altogether that the set \((\mathbb N, + )\), i.e. the set of natural numbers, together with the addition
"\( + \)" as a binary operation, forms a cancellative commutative monoid.
∎
Thank you to the contributors under CC BYSA 4.0!
 Github:

References
Bibliography
 Kramer Jürg, von Pippich, AnnaMaria: "Von den natürlichen Zahlen zu den Quaternionen", SpringerSpektrum, 2013