# Product of Powers in B-Algebra

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## Theorem

Let $\struct {X, \circ}$ be a $B$-algebra.

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

Then:

- $x^m \circ x^n = \begin {cases} x^{m - n} & : m \ge n \\ 0 \circ x^{n - m} & : n > m \end {cases}$

## Proof

For $m \ge n$ the result follows from $B$-Algebra Power Law.

For $n > m$ the result follows from $B$-Algebra Power Law with Zero.

$\blacksquare$