Between two Similar Solid Numbers exist two Mean Proportionals

Proof
Let $m$ and $n$ be similar solid numbers.

Then for some $a, b, c, d, e, f \in \Z$ such that $a \le b \le c$ and $d \le e \le f$:


 * $m = a b c$
 * $n = d e f$

such that:
 * $\dfrac a d = \dfrac b e = \dfrac c f$

Let:
 * $r := a e = b d = b f = c e$

So:

and:

Thus by definition, $\tuple {a b, r, d e}$ is a geometric sequence.

By definition, $r$ is a mean proportional between $a b$ and $d e$.

Now consider the products $r c$ and $r f$.

We have:

and:

and:

That is:
 * $\tuple {a b c, b c d, b d f, d e f}$ is a geometric sequence.

Also from the above:
 * $\dfrac a d = \dfrac b e = \dfrac c f$

showing that $m$ is in triplicate ratio to $n$ as their sides.