A pair $(K, \rho_U)$ is called a solution in the text above. The uniqueness follows from Lemma \ref{lemma-uniqueness}. If $X$ has a finite covering by elements of $\mathcal{B}$ (for example if $X$ is quasi-compact), then the theorem is a consequence of Lemma \ref{lemma-solution-in-finite-case}. In the general case we argue in exactly the same manner, using transfinite induction and Lemma \ref{lemma-glueing-increasing-union}. \medskip\noindent First we use transfinite induction to choose opens $W_\alpha \subset X$ for any ordinal $\alpha$. Namely, we set $W_0 = \emptyset$. If $\alpha = \beta + 1$ is a successor, then either $W_\beta = X$ and we set $W_\alpha = X$ or $W_\beta \not = X$ and we set $W_\alpha = W_\beta \cup U_\alpha$ where $U_\alpha \in \mathcal{B}$ is not contained in $W_\beta$. If $\alpha$ is a limit ordinal we set $W_\alpha = \bigcup_{\beta < \alpha} W_\beta$. Then for large enough $\alpha$ we have $W_\alpha = X$. Observe that for every $\alpha$ the open $W_\alpha$ is a union of elements of $\mathcal{B}$. Hence if $\mathcal{B}_\alpha = \{U \in \mathcal{B}, U \subset W_\alpha\}$, then $$ S_\alpha = (\{K_U\}_{U \in \mathcal{B}_\alpha}, \{\rho_V^U\}_{V \subset U\text{ with }U, V \in \mathcal{B}_\alpha}) $$ is a system as in Lemma \ref{lemma-uniqueness} on the ringed space $W_\alpha$. \medskip\noindent We will show by transfinite induction that for every $\alpha$ the system $S_\alpha$ has a solution. This will prove the theorem as this system is the system given in the theorem for large $\alpha$. \medskip\noindent The case where $\alpha = \beta + 1$ is a successor ordinal. (This case was already treated in the proof of the lemma above but for clarity we repeat the argument.) Recall that $W_\alpha = W_\beta \cup U_\alpha$ for some $U_\alpha \in \mathcal{B}$ in this case. By induction hypothesis we have a solution $(K_{W_\beta}, \{\rho^{W_\beta}_U\}_{U \in \mathcal{B}_\beta})$ for the system $S_\beta$. Then we can consider the collection $\mathcal{B}_\alpha^* = \mathcal{B}_\alpha \cup \{W_\beta\}$ of opens of $W_\alpha$ and we see that we obtain a system $(\{K_U\}_{U \in \mathcal{B}_\alpha^*}, \{\rho_V^U\}_{V \subset U\text{ with }U, V \in \mathcal{B}_\alpha^*})$. Note that this new system also satisfies condition (3) by Lemma \ref{lemma-uniqueness} applied to the solution $K_{W_\beta}$. For this system we have $W_\alpha = W_\beta \cup U_\alpha$. This reduces us to the case handled in Lemma \ref{lemma-solution-in-finite-case}. \medskip\noindent The case where $\alpha$ is a limit ordinal. Recall that $W_\alpha = \bigcup_{\beta < \alpha} W_\beta$ in this case. For $\beta < \alpha$ let $(K_{W_\beta}, \{\rho^{W_\beta}_U\}_{U \in \mathcal{B}_\beta})$ be the solution for $S_\beta$. For $\gamma < \beta < \alpha$ the restriction $K_{W_\beta}|_{W_\gamma}$ endowed with the maps $\rho^{W_\beta}_U$, $U \in \mathcal{B}_\gamma$ is a solution for $S_\gamma$. By uniqueness we get unique isomorphisms $\rho_{W_\gamma}^{W_\beta} : K_{W_\beta}|_{W_\gamma} \to K_{W_\gamma}$ compatible with the maps $\rho^{W_\beta}_U$ and $\rho^{W_\gamma}_U$ for $U \in \mathcal{B}_\gamma$. These maps compose in the correct manner, i.e., $\rho_{W_\delta}^{W_\gamma} \circ \rho_{W_\gamma}^{W_\beta}|_{W_\delta} = \rho^{W_\delta}_{W_\beta}$ for $\delta < \gamma < \beta < \alpha$. Thus we may apply Lemma \ref{lemma-glueing-increasing-union} (note that the vanishing of negative exts is true for $K_{W_\beta}$ by Lemma \ref{lemma-uniqueness} applied to the solution $K_{W_\beta}$) to obtain $K_{W_\alpha}$ and isomorphisms $$ \rho_{W_\beta}^{W_\alpha} : K_{W_\alpha}|_{W_\beta} \longrightarrow K_{W_\beta} $$ compatible with the maps $\rho_{W_\gamma}^{W_\beta}$ for $\gamma < \beta < \alpha$. \medskip\noindent To show that $K_{W_\alpha}$ is a solution we still need to construct the isomorphisms $\rho_U^{W_\alpha} : K_{W_\alpha}|_U \to K_U$ for $U \in \mathcal{B}_\alpha$ satisfying certain compatibilities. We choose $\rho_U^{W_\alpha}$ to be the unique map such that for any $\beta < \alpha$ and any $V \in \mathcal{B}_\beta$ with $V \subset U$ the diagram $$ \xymatrix{ K_{W_\alpha}|_V \ar[r]_{\rho_U^{W_\alpha}|_V} \ar[d]_{\rho_{W_\beta}^{W_\alpha}|_V} & K_U|_V \ar[d]^{\rho_U^V} \\ K_{W_\beta} \ar[r]^{\rho_V^{W_\beta}} & K_V } $$ commutes. This makes sense because $$ (\{K_V\}_{V \subset U, V \in \mathcal{B}_\beta\text{ for some }\beta < \alpha}, \{\rho_V^{V'}\}_{V \subset V'\text{ with }V, V' \subset U \text{ and }V, V' \in \mathcal{B}_\beta\text{ for some }\beta < \alpha}) $$ is a system as in Lemma \ref{lemma-uniqueness} on the ringed space $U$ and because $(K_U, \rho^U_V)$ and $(K_{W_\alpha}|_U,  \rho_V^{W_\beta}\circ \rho_{W_\beta}^{W_\alpha}|_V)$ are both solutions for this system. This gives existence and uniqueness. We omit the proof that these maps satisfy the desired compatibilities (it is just bookkeeping).