Proportion is Transitive

Theorem
Proportion 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:


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

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


 * $x = \paren {j \times k} \times z$

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

The result follows from the definition of transitivity and proportion.