Multiplication by 2 over 3 in Egyptian Fractions

Theorem
Let $\dfrac 1 n$ be an Egyptian fraction not equal to $\dfrac 2 3$.

In order to multiply $\dfrac 1 n$ by $\dfrac 2 3$ and have it that $\dfrac 1 n \times \dfrac 2 3$ is also expressed in Egyptian form, we have:


 * $\dfrac 1 n \times \dfrac 2 3 = \dfrac 1 {2 n} + \dfrac 1 {6 n}$

Proof
Note the case where we multiply $\dfrac 2 3$ by $\dfrac 2 3$ itself: