(related to Corollary: Contraposition of Cancellative Law for Adding Natural Numbers)

According to the cancellation law for adding natural numbers, we have for all natural numbers \(x,y,z\in\mathbb N\): \[\begin{array}{rcl}z + x=z + y&\Longleftrightarrow &x=y,\\ x + z=y + z&\Longleftrightarrow& x=y. \end{array}\] 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 natural numbers \(x\) and \(y\) are unequal, then the inequality is preserved if we add an arbitrary natural number \(z\) to both sides of the inequality. Conversely, given an inequality, in which the same natural number \(z\) is added on both sides, we can "cancel it out" and preserve the inequality.

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  1. Landau, Edmund: "Grundlagen der Analysis", Heldermann Verlag, 2008