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

According to the cancellation law for multiplying real numbers, we have for all real numbers \(x,y,a\in\mathbb R\) 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: \[x \neq y\Longleftrightarrow \begin{cases} z \cdot x\neq z \cdot y,\\ x \cdot z\neq y \cdot z, \end{cases}\]

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