Morphisms into projective $n$-space defined by invertible sheaf and global sections

Question

In my algebraic geometry class we had a theorem that for a scheme $X$ over an affine scheme $\operatorname{Spec} A$ giving a morphism $X \to \mathbb{P}^n_A$ is the same as giving an invertible $\mathcal{O}_X$-sheaf $\mathcal{L}$ and $n + 1$ global sections $s_0, \dotsc , s_n \in \Gamma(X, \mathcal{L})$ which generate $\mathcal{L}$. In the proof we then defined $X_{s_i} = \{ x \in X \mid (s_i)_x \notin \mathfrak{m}_x \mathcal{L}_x\}$ and morphisms $$ f_i \colon X_{s_i} \to \mathbb{P}_A^n $$ by \begin{align*} A\Bigl[ \frac{x_0}{x_i}, \dotsc ,\frac{x_n}{x_i} \Bigr] &\to \Gamma(X_{s_i}, \mathcal{O}_{X_{s_i}}) \,, \\[0.5em] \frac{x_j}{x_i} &\mapsto s_j\vert_{X_{s_i}} (s_i\vert_{X_{s_i}})^{-1} \,. \end{align*} Here I am confused, the $s_j$ are global sections of $\mathcal{L}$ but why are then the $s_j\vert_{X_{s_i}} (s_i\vert_{X_{s_i}})^{-1} \in \Gamma(X_{s_i}, \mathcal{O}_{X_{s_i}})$? Im sorry if any of the notation is unclear, please ask and I will explain what is meant.