Proportion is Symmetric

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Theorem

Proportion 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:

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

The result follows from the definition of symmetry and proportion.

$\blacksquare$