Cube Number multiplied by Cube Number is Cube

Theorem
Let $a, b \in \N$ be natural numbers.

Let $a$ and $b$ be cube numbers.

Then $a b$ is also a cube number.

Proof
By the definition of cube number:
 * $\exists r \in \N: r^3 = a$
 * $\exists s \in \N: s^3 = b$

Thus:

Thus:
 * $\exists k = r s \in \N: a = k^3$

Hence the result by definition of cube number.