Banach-Tarski Paradox/Lemma 2

Relation Definition
Let $X$ and $Y$ be disjoint unions of $X_1, X_2$ and $Y_1, Y_2$ respectively.

Let $X_i \approx Y_i$ for each $i \in \set {1, 2, \ldots, m}$.

Then:
 * $X \approx Y$