Integer Addition is Commutative/Proof 2

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


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

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

Then: