If $X$ is an affine variety then the affine restriction of the projective closure of $X $ is $X$ Proposition $9.50$ of Clader and Ross Book

I am self studying Algebraic geometry from Clader and Ross Beginning in Algebraic Geometry and I am struck on Proposition $9.50$ on page $270$.

Background: For each $i \in ${$0,1...,n$}, the $i$ th affine patch of $\mathbb{P}^n$ is the set $\mathbb{A}_i^n=${$[a_0,a_1,...,a_n] \in \mathbb{P^n}|a_i\neq 0$}, and for any set $X\subseteq \mathbb{P^n}$ , the intersection $X \cap \mathbb{A}_i^n$ is called the $i$ th affine restriction of $X$.

The projective closure of $X$ denoted by $\overline{X} \mathbb{P^n}$, is the intersection of all projective varieties that contain $j_0(X)$ where $j_0(X)$ is the image of the $X$ in the $1st $ affine patch $\mathbb{A}_0^n \subseteq \mathbb{P}^n$.

Statement of Proposition $9.50:$ If $ X\subseteq \mathbb{A}^n $ is an affine variety , then the affine restriction of the projective closure of $X$ is $X$.

The proof depends on the following statement which I am unable to prove and need help with: The $i$ th affine restriction of $X = V_{\mathbb{P}}(S)$ is given by $V_{\mathbb{A}}(S_i)\subseteq \mathbb{A}^n$ where $S_i =${$f(x_0,....,x_{i-1},1,x_{i+1},...,x_n)|f\in S$}

I have proved that there exists a natuaral bijection from $\mathbb{A}_i^n $ to $\mathbb{A}^n$, where $\mathbb{A}_i^n$ is as given in the definition of the affine patch but I am quite confused in rigorously showing that $i $ th affine restriction of $X=V_{\mathbb{P}}(S) $ is $V_{\mathbb{A}}(S_i)$.

Please help me with this.