(related to Proposition: Addition of Natural Numbers Is Cancellative)
If any two natural numbers are unequal \(x\neq y\), then the inequality is preserved, if and only if we add an arbitrary natural number \(z\) to both sides of the inequality, formally: \[x \neq y\Longleftrightarrow \begin{cases} z + x\neq z + y&\text{or}\\ x + z\neq y + z. \end{cases}\]
Proofs: 1