Integer Multiplication Identity is One

Theorem
The identity of integer multiplication is $$1$$.

Proof
We need to show that $$\forall a, b, c \in \mathbb{N}: \left[\!\left[{a, b}\right]\!\right]_\boxminus \times \left[\!\left[{c+1, c}\right]\!\right]_\boxminus = \left[\!\left[{a, b}\right]\!\right]_\boxminus = \left[\!\left[{c+1, c}\right]\!\right]_\boxminus \times \left[\!\left[{a, b}\right]\!\right]_\boxminus$$.

From Natural Numbers form Semiring, we take it for granted that:
 * addition and multiplication are commutative and associative on the natural numbers $$\mathbb{N}$$;
 * natural number multiplication is distributive over natural number addition.

So:

$$ $$ $$ $$

So $$\left[\!\left[{a, b}\right]\!\right]_\boxminus \times \left[\!\left[{c+1, c}\right]\!\right]_\boxminus = \left[\!\left[{a, b}\right]\!\right]_\boxminus$$

The identity $$\left[\!\left[{a, b}\right]\!\right]_\boxminus = \left[\!\left[{c+1, c}\right]\!\right]_\boxminus \times \left[\!\left[{a, b}\right]\!\right]_\boxminus$$ is demonstrated similarly.