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Lemma 1.   Let \(X\) be a locally connected space. A sheaf \(F\) of \(R\)-modules on \(X\) is locally constant if and only if its étale space is a covering.

Proof. Suppose the étale space of \(F\) is a covering. Then every point \(x \in X\) has a neighborhood \(U\) such that the restriction is \(p \colon U \times M \to U\). Assuming \(U\) to be connected implies that the sections of this map are constant since the topology on \(M\) is discrete, and thus our sheaf is constant on \(U\). \(\blacksquare\)