Prove a set equidecomposable with a paradoxical set is paradoxical

Question

In Stan Wagon's book The Banach-Tarski Paradox, Proposition 3.4 is written as: "Suppose $G$ acts on $X$ and $E$, $E'$ are $G$-decomposable subsets of $X$. If $E$ is $G$-paradoxical, so is $E'$. I have not written the proof itself since it is omitted from Google Books, but what is a straightforward way to understand the intuition behind this relation? If the sets were isomorphic it would be fairly obvious, but I am missing the nuance of how a paradoxical decomposition 'lays in' with the relation under equidecomposability.