B-Algebra Power Law with Zero

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

Let $n, m \in \N_{>0}$ such that $n > m$.

Then:
 * $\forall x \in X: n, m \in \N_{>0} \implies x^m \circ x^n = 0 \circ x^{n - m}$

where $x^k$ for $k \in \N_{>0}$ denotes the $k$th power of the element $x$.

Proof
First we show that:
 * $\forall x \in X: x \circ x^2 = 0 \circ x$

Now we show that:
 * $\forall x \in X: m \in \N_{>0}: x \circ x^m = 0 \circ x^{m - 1}$

Now we prove the original proposition using a proof by induction.

The base case for $n = 1, m = 2$ was established in the first lemma.

Let us assume $x^m \circ x^n = 0 \circ x^{n - m}$ for some $n, m \in \N_{>0}$.

Then:

Hence the result.