Proportion is Symmetric

Theorem

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:

 $\ds \exists k \ne 0: \,$ $\ds x$ $=$ $\ds k \times y$ $\ds \implies \ \$ $\ds y$ $=$ $\ds k^{-1} \times x$

The result follows from the definition of symmetry and proportion.

$\blacksquare$