**Proof.**
Recall that $K_0$ is a covering of $X$ by Definition 25.3.3 and that this is equivalent to saying that $K_0 \to \{ X \to X\} $ is a covering in the sense of Definition 25.3.1. Hence (1) follows from (2) because it will prove that the composition $K_ n \to K_{n - 1} \to \ldots \to K_0 \to \{ X \to X\} $ is a covering by Lemma 25.3.2.

Proof of (2). Observe that $\mathop{\mathrm{Mor}}\nolimits (\Delta [n], K)_0 = K_ n$ by Simplicial, Lemma 14.17.4. Therefore (2) follows from Lemma 25.8.2 applied to the $n + 1$ different inclusions $\Delta [n - 1] \to \Delta [n]$.
$\square$

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