# Proof

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 BY-SA 4.0!

Github:

### References

#### Bibliography

1. Kramer Jürg, von Pippich, Anna-Maria: "Von den natürlichen Zahlen zu den Quaternionen", Springer-Spektrum, 2013