Injective burrows wheeler We know that the Burrows-Wheeler Transform is reversible, in the sense that, given
s=BWT(w) and an index I, it is possible to recover w by an algorithm. But how to prove that w is unique? In other way how to prove that Burrows-Wheeler is injective?
Let $\mathsf{BWT}^{-1}$ be the inverse Burrows-Wheeler transform, that is, $\mathsf{BWT}^{-1}(\mathsf{BWT}(w)) = w$.

If $\mathsf{BWT}(x) = \mathsf{BWT}(y)$ then
$$
x = \mathsf{BWT}^{-1}(\mathsf{BWT}(x)) = \mathsf{BWT}^{-1}(\mathsf{BWT}(y)) = y.
$$