Wreath product is associative

Statement

Suppose $G,H,K$ are groups. Suppose $H$ comes equipped with an action on a set $S$ and $K$ comes equipped with an action on a set $T$. Note that, by the action of wreath product on Cartesian product, we can use this to define an action of the external wreath product $H\wr K$ on $S\times T$. With these interpretations, we have that the external wreath product is associative up to isomorphism of groups:

$(G\wr H)\wr K\cong G\wr (H\wr K)$

Note that, to formulate this as an isomorphism of groups, we do not need to specify a group action of $G$ on any set. However, if we do equip $G$ with an action on a set $A$, then the two groups above have equivalent actions on the triple Cartesian product $A\times S\times T$ using the action of wreath product on Cartesian product.