How to prove that $\mathbb{P}^n_{\mathbb{Z}} \setminus D_+(x_i) \cong \mathbb{P}^{n-1}_{\mathbb{Z}}$?

I am currently studying algebraic geometry and trying to understand projective spaces.

let $S = \mathbb{Z}[x_0, \dots, x_n]$ so that $\mathbb{P}^n_{\mathbb{Z}} = \operatorname{Proj}(S)$.

I read that if we remove the standard open affine subset $D_+(x_i) = \{ \mathfrak{p} \in \operatorname{Proj}(S) \mid x_i \notin \mathfrak{p} \}$ from the projective space $\mathbb{P}^n_{\mathbb{Z}}$, the complement is isomorphic to $\mathbb{P}^{n-1}_{\mathbb{Z}}$.

However, I don't really know how to approach the proof. Could someone please explain it to me simply, step by step?

Thank you in advance!