Proportion is Symmetric

Theorem
Proportionality is a symmetric relation.

That is:


 * $\forall x,y \in \R: x \propto y \implies y \propto x$

Proof
Let $x,y$ be arbitrary.

Let $x$ be proportional to $y$:


 * $x \propto y$.

Then by definition:

$\exists k \neq 0: x = k \times y$


 * $\implies y = k^{-1} \times x$

The result follows from the definition of symmetry and proportionality.