Commensurability of Elements of Proportional Magnitudes

Proof
Let $A$, $B$, $C$ and $D$ be four magnitudes in proportion:
 * $A : B = C : D$

Let $A$ be commensurable with $B$.

Then from Ratio of Commensurable Magnitudes:
 * $A$ has to $B$ the ratio which a number has to a number.

As $A : B = C : D$ it follows that:
 * $C$ has to $D$ the ratio which a number has to a number.

From Magnitudes with Rational Ratio are Commensurable it follows that:
 * $C$ is commensurable with $D$.

Let $A$ be incommensurable with $B$.

Then from Incommensurable Magnitudes have Irrational Ratio:
 * $A$ does not have to $B$ the ratio which a number has to a number.

As $A : B = C : D$ it follows that:
 * $C$ does not have to $D$ the ratio which a number has to a number.

From Magnitudes with Irrational Ratio are Incommensurable it follows that:
 * $C$ is incommensurable with $D$.