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 $$y \circ z$$.


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

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


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


 * If $$x$$ and $$y$$ both commute with $$z$$, then $$x \circ y$$ commutes with $$z$$: