Element Commutes with Product of Commuting Elements

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

Let $\circ$ be associative.

Let $x, y, z \in S$.

If $x$ commutes with both $y$ and $z$, then $x$ commutes with $y \circ z$.

Proof
The following are demonstrated by associativity of $\circ$ and the defined commutativity relations.