Proportion is Transitive

Theorem
Proportionality is a transitive relation.

That is:


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

Proof
Let $x,y,z$ be arbitrary.

Let $x$ be proportional to $y$ and $y$ to $z$:


 * $x \propto y \land y \propto z$.

Then by definition:

$\exists j,k \neq 0: x = j \times y \land y = k \times z$

Substituting $k \times z$ for $y$:


 * $x = \left({j \times k}\right) \times z$

So $j \times k$ is the desired constant of proportion.

The result follows from the definition of transitivity and proportionality.