automata – How to prove that concatenating a language A and A* is commutative?

Suppose we have a language $$A$$. I want to prove that $$AA^*$$ is commutative. I know that this expression equals $$A^+$$, but I’m not sure how to go about a proof yet. This is my attempt so far.

If $$A$$ is a language, then
begin{align*} AA^* & = A(A^0cup A^1cupcdots) tag{Definition of A^*}\ & = A^1cup A^2cdots tag{Distributive law} \ & = (A^0cup A^1cupcdots)A tag{Distributive law} \ & = A^*A. end{align*}

Is this enough? I’m thinking that in order for it to be a proper proof, I have to show that $$AA^*$$ and $$A^*A$$ are subsets of each other. That is, take an element $$s = xy$$ where $$xin A$$ and $$yin A^*$$ and show that $$xin A^*$$ and $$yin A$$ (and vice versa). However, I get stuck very early when doing that method.