If a natural number \(x\) is smaller than \(y\), then the inequality is preserved, if and only if we add an arbitrary natural number \(z\) to both sides of the inequality, formally:
\[x < y\Longleftrightarrow \begin{cases} z + x < z + y&\text{or}\\ x + z < y + z. \end{cases}\]
The same can be stated about the order relations smaller or greater "\( \le \)", greater or equal "\( \ge \)", and greater "\( > \)":
\[x \le y\Longleftrightarrow \begin{cases} z + x\le z + y&\text{or}\\ x + z\le y + z. \end{cases}\]
\[x \ge y\Longleftrightarrow \begin{cases} z + x \ge z + y&\text{or}\\ x + z \ge y + z. \end{cases}\]
\[x > y\Longleftrightarrow \begin{cases} z + x > z + y&\text{or}\\ x + z > y + z. \end{cases}\]
Proofs: 1
