Add a second proof in sheaves

Thanks to Samir Canning
https://stacks.math.columbia.edu/tag/00AL#comment-3423

sheaves.tex

@@ -5100,7 +5100,7 @@ \section{Glueing sheaves}
 \end{lemma}
 
 \begin{proof}
-Actually we can write a formula for the set of sections
+First proof. In this proof we give a formula for the set of sections
 of $\mathcal{F}$ over an open $W \subset X$. Namely, we define
 $$
 \mathcal{F}(W) =
@@ -5124,7 +5124,17 @@ \section{Glueing sheaves}
 of a glueing data assures that we may start with {\it any}
 section $s \in \mathcal{F}_i(W)$ and obtain a compatible
 collection by setting $s_i = s$ and $s_j = \varphi_{ij}(s_i|_{W \cap U_j})$.
-Thus the lemma follows.
+
+\medskip\noindent
+Second proof (sketch). Let $\mathcal{B}$ be the set of opens $U \subset X$
+such that $U \subset U_i$ for somje $i \in I$. Then $\mathcal{B}$
+is a base for the topology on $X$. For $U \in \mathcal{B}$ we pick
+$i \in I$ with $U \subset U_i$ and we set $\mathcal{F}(U) = \mathcal{F}_i(U)$.
+Using the isomorphisms $\varphi_{ij}$ we see that this prescription
+is ``independent of the choice of $i$''. Using the restriction mappings
+of $\mathcal{F}_i$ we find that $\mathcal{F}$ is a sheaf on $\mathcal{B}$.
+Finally, use Lemma \ref{lemma-extend-off-basis} to extend $\mathcal{F}$
+to a unique sheaf $\mathcal{F}$ on $X$.
 \end{proof}
 
 \begin{lemma}