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 cammutativity 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:

$$ $$ $$ $$ $$