Proportion is Symmetric

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:

\(\ds \exists k \ne 0: \, \)

\(\ds x\)

\(=\)

\(\ds k \times y\)

\(\ds \leadsto \ \ \)

\(\ds y\)

\(=\)

\(\ds k^{-1} \times x\)

The result follows from the definition of symmetry and proportion.

$\blacksquare$