Quotients of 3 Unequal Numbers are Unequal

Theorem
Let $x, y, z \in \R_{\ne 0}$ be non-zero real numbers which are not all equal.

Then $\dfrac x y, \dfrac y z, \dfrac z x$ are also not all equal.

Proof
$\dfrac x y = \dfrac y z = \dfrac z x$.

This contradicts the assertion that $x, y, z$ are all unequal.

Hence the result by Proof by Contradiction.