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 $y$:

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

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

The result follows from the definition of transitivity and proportion.

$\blacksquare$