Product of Powers in B-Algebra

Theorem
Let $(X, \circ)$ be a B-Algebra.

Let $x \in X$ and $m, n \in \N$.

Then:


 * $x^m \circ x^n = \begin{cases}

x^{m-n} & \text{if $m \ge n$} \\ 0 \circ x^{n-m} & \text{if $n > m$} \end{cases}$

Proof
For $m \ge n$ the result follows from: B-Algebra Power Law of Differences

For $n > m$ the result follows from: B-Algebra Power Law of Differences 2