(related to Proposition: Contraposition of Cancellative Law for Multiplying Natural Numbers)

According to the cancellation law for multiplying natural numbers, we have for all natural numbers \(x,y,z\in\mathbb N\) with \(z\neq 0\)

\[\begin{array}{rcl}z \cdot x=z \cdot y&\Longleftrightarrow &x=y,\\ x \cdot z=y \cdot z&\Longleftrightarrow& x=y. \end{array}\] By virtue of the proving principle by contraposition, it follows \[x \neq y\Longleftrightarrow \begin{cases} z \cdot x\neq z \cdot y,&\text{or}\\ x \cdot z\neq y \cdot z, \end{cases}\]

i.e. if any two natural numbers \(x\) and \(y\) are unequal, then the inequality is preserved, if we multiply both sides of the inequality by an arbitrary natural number \(z\neq 0\). Conversely, if we have an inequality, in which both sides are multiplied by the same natural number \(z\neq 0\), then we can "cancel it out", and still preserve the inequality.

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