In http://en.wikipedia.org/wiki/Banach%E2%80%93Tarski_paradox#Some_details.2C_fleshed_out paragraph 2, it seems to imply that we need all the $\rho^n(D)$ to be disjoint from each other but I can't understand why. It seems to me that it's only important to show that $D$ is disjoint from the rest of the $\rho^n(D)$. This is because I thought that the reason they need to be disjoint is to make sure that $\rho(\bar{D}) \cup (S^2/\bar{D}) = S^2/D$ which is only the case if D is disjoint from the rest of the $\rho^n(D)$.
Could someone explain why all the $\rho^n(D)$ need to be disjoint?