Integer Addition is Commutative

Theorem
The operation of addition on the set of integers $\Z$ is commutative:


 * $\forall x, y \in \Z: x + y = y + x$

Proof
From the formal definition of integers, $\left[\!\left[{a, b}\right]\!\right]$ is an equivalence class of ordered pairs of natural numbers.

It can be taken directly from Additive Group of Integers‎ that the integers under addition form an abelian group, from which commutativity follows a priori.

Alternatively, it can be proved explicitly as follows.

Let $x = \left[\!\left[{a, b}\right]\!\right]$ and $y = \left[\!\left[{c, d}\right]\!\right]$ for some $x, y \in \Z$.

Then: