**Proof.**
Part (1) follows from the definition of a prime divisor (Spaces over Fields, Definition 72.6.2), Decent Spaces, Lemma 68.20.2, and the definition of a dimension function (Topology, Definition 5.20.1).

Let $D \subset X$ be an effective Cartier divisor. Let $Z \subset D$ be an irreducible component and let $\xi \in |Z|$ be the generic point. Choose an étale neighbourhood $(U, u) \to (X, \xi )$ where $U = \mathop{\mathrm{Spec}}(A)$ and $D \times _ X U$ is cut out by a nonzerodivisor $f \in A$, see Divisors on Spaces, Lemma 71.6.2. Then $u$ is a generic point of $V(f)$ by Decent Spaces, Lemma 68.20.1. Hence $\mathcal{O}_{U, u}$ has dimension $1$ by Krull's Hauptidealsatz (Algebra, Lemma 10.60.11). Thus $\xi $ is a codimension $1$ point on $X$ and $Z$ is a prime divisor as desired.
$\square$

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