Ratio of Products of Sides of Plane Numbers

Proof
Let $a$ and $b$ be plane numbers.

Let $a = c d$ and $b = e f$.

From we can find $g, h, k$ such that:
 * $c : e = g : h$
 * $d : f = h : k$

Let $d e = l$.

We also have that $a = c d$.

So from :
 * $c : e = a : l$

But we also have:
 * $c : e = g : h$

and so:
 * $g : h = a : l$

Since:
 * $e d = l$
 * $e f = b$

from :
 * $d : f = l : b$

But we also have:
 * $d : f = h : k$

and so:
 * $h : k = l : b$

But we have:
 * $g : h = a : l$

and so from :
 * $g : k = a : b$

But $g : k$ is the ratio compounded of the ratios of the sides of $a$ and $b$.

Therefore $a : b$ is the ratio compounded of the ratios of their sides.