Banach-Tarski Paradox/Lemmata

Lemma for Banach-Tarski Paradox
Let $\approx$ denote the relation between sets in Euclidean space of $3$ dimensions defined as follows:


 * $X \approx Y$


 * there exists a partition of $X$ into disjoint sets:
 * $X = X_1 \cup X_2 \cup \cdots \cup X_m$
 * and a partition of $Y$ into the same number of disjoint sets:
 * $Y = Y_1 \cup Y_2 \cup \cdots \cup Y_m$
 * $Y = Y_1 \cup Y_2 \cup \cdots \cup Y_m$

such that $X_i$ is congruent to $Y_i$ for each $i \in \set {1, 2, \ldots, m}$.

Then:


 * $(1): \quad \approx$ is an equivalence relation
 * $(2): \quad$ If $X$ and $Y$ are disjoint unions of $X_1, X_2$ and $Y_1, Y_2$ respectively, and if $X_i \approx Y_i$ for each $i \in \set {1, 2, \ldots, m}$, then $X \approx Y$
 * $(3): \quad$ If $X_1 \subseteq Y \subseteq X$ and if $X \approx X_1$, then $X \approx Y$.