Formal Proof of Banach-Tarski Paradox

Banach-Tarski Paradox states that a ball in 3D space is equidecomposable with twice itself, i.e. we can break a ball into a finite number of pieces, and with these pieces, build two balls having the same size as the initial ball. This strange result is actually a Theorem which was proven in 1924 by Stefan Banach and Alfred Tarski using the Axiom of Choice.We present here a formal proof in Coq of this theorem.