(related to Proposition: Contraposition of Cancellative Law for Adding Integers)
According to the cancellation law for adding integers, we have for all integers \(x,y,z\in\mathbb Z\): \[\begin{array}{rcl}z + x=z + y&\Longleftrightarrow &x=y,\\ x + z=y + z&\Longleftrightarrow& x=y. \end{array}\] By virtue of the proving principle by contraposition, it follows \[x \neq y\Longleftrightarrow \begin{cases} z + x\neq z + y,&\text{or}\\ x + z\neq y + z, \end{cases}\]
i.e. if any two integers \(x\) and \(y\) are unequal, then the inequality is preserved if we add an arbitrary integer \(z\) to both sides of the inequality. Conversely, given an inequality, in which the same integer \(z\) is added on both sides, we can "cancel it out" and preserve the inequality.
