Integer Addition is Associative

Theorem
The operation of addition on the set of integers $$\Z$$ is associative:
 * $$\forall x, y, z \in \Z: x + \left({y + z}\right) = \left({x + y}\right) + z$$

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 a group, from which associativity follows a priori.

Alternatively, it can be proved explicitly as follows.

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

Then:

$$ $$ $$ $$ $$ $$ $$