Support of $x$ also belonging to the set of best responses when $x$ is a best reponse

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If $ \mathbf x $ belongs to the set of best responses to $\mathbf y$ i.e. to $BR(\mathbf y) = arg max \mathbf x \cdot A \mathbf y$, why do all of the pure strategies in the support of $\mathbf x $ also belong to $BR(\mathbf y)$?