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A well-known proof of Brouwer's fixed point theorem utilizes Sperner's Lemma. Can we go the other way round--find a simple proof of Sperner's lemma using the fixed point theorem?

Based on a search in google, there are some papers that demonstrate this; but they (as far as I've seen) all go through some intermediate lemma and the proofs are not very simple. I am looking for a simple proof.

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