Definition:Power of Element

Theorem
Let $$\left({S, \circ}\right)$$ be an algebraic structure. Let $$a \in S$$.

Let $$\left({a_1, a_2, \ldots, a_n}\right)$$ be the ordered $n$-tuple defined by $$a_k = a$$ for each $$k \in \mathbb{N}_n$$.

Then:

$$\prod_{k=1}^n a_k = \circ^n a$$

Proof
Can be proved by induction.