Equidecomposable Nested Sets

Theorem

Let $A, B, C$ be sets such that $A$ and $C$ are equidecomposable and $A \subseteq B \subseteq C$.

Then $B$ and $C$ are equidecomposable.

Proof

Let $\left\{{X_k}\right\}_{k=1}^n$ be a decomposition of $A$ and $C$, so that there are isometries $\left\{{\phi_k}\right\}_{k=1}^n$ and $\left\{{\psi_k}\right\}_{k=1}^n$ such that:

$\displaystyle A = \bigcup_{k=1}^n \phi_k(X_k)$

$\displaystyle C = \bigcup_{k=1}^n \psi_k(X_k)$

Let $Y_k = \psi_k^{-1} \left({ B \cap \psi_k(X_k) }\right)$.

Then $\displaystyle \bigcup_{k=1}^n \psi_k (Y_k) = B \cap \bigcup_{k=1}^n \psi_k(X_k) = B \cap C = B$.

Let $Z_k = \phi^{-1} \left({ A \cap \phi_k(X_k) }\right)$.

Then $\displaystyle \bigcup_{k=1}^n \phi_k (Z_k) = A \cap \bigcup_{k=1}^n \phi_k(X_k) = A \cap C = A$.