\begin{definition}[Definition:Ordering on Integers/Definition 2]
The integers are ordered on the relation $\le$ as follows:
Let $x$ and $y$ be defined as from the formal definition of integers:
:$x = \eqclass {x_1, x_2} {}$ and $y = \eqclass {y_1, y_2} {}$ where $x_1, x_2, y_1, y_2 \in \N$.
Then:
:$x < y \iff x_1 + y_2 \le x_2 + y_1$
where:
:$+$ denotes natural number addition
:$\le$ denotes natural number ordering.
\end{definition}