Integer Multiplication has Zero

Theorem
Integer multiplication has a zero, which is $$0$$.

Proof
We need to show that:

$$\forall a, b, c \in \mathbb{N}: \left[\left[{a, b}\right]\right]_\boxminus \times \left[\left[{c, c}\right]\right]_\boxminus = \left[\left[{c, c}\right]\right]_\boxminus = \left[\left[{c, c}\right]\right]_\boxminus \times \left[\left[{a, b}\right]\right]_\boxminus$$.

From Natural Numbers form Semiring, we can take it for granted that addition and multiplication are commutative on the natural numbers $$\mathbb{N}$$.