Uniqueness of the adjacent cluster under mutation

Let $(\mathbf x, \mathbf p, B)$ denote a seed in the cluster algebra with coefficients, or let $(\mathbf x,B)$ denote a seed in the coefficient-free cluster algebra. A cluster is the set of cluster variables underlying the seed. Uniqueness of the adjacent cluster under mutation. Any seed $(\mathbf x, \mathbf p, B)$, or $(\mathbf x,B)$, is uniquely determined by its cluster $\mathbf x$. Thus, for any cluster $\mathbf x$ and any $x\in\mathbf x$, there is a unique cluster $\mathbf x'$ such that $\mathbf x'\cap\mathbf x=\mathbf x\setminus\{x\}$. This asserts that each cluster variable determines a unique neighboring cluster obtained by replacing it, so the exchange graph and cluster complex have the expected mutation structure; the supplied context does not indicate whether the claim is proved or remains open.