Definition:Ordering on Integers/Definition 2

Definition
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.

Also see

 * Equivalence of Definitions of Ordering on Integers