Element Commutes with Product of Commuting Elements/General Theorem

Theorem
Let $\circ$ be a binary operation on a set $S$.

Let $\circ$ be associative.

Let $\left \langle {a_k} \right \rangle_{1 \le k \le n}$ be a sequence of terms of $S$.

Let $b \in S$.

If $b$ commutes with $a_k$ for each $k \in \left[{1 \,.\,.\, n}\right]$, then $b$ commutes with $a_1 \circ \cdots \circ a_n$.