Kanıt: $(X,\tau),$ Lindelöf; $A\in \mathcal{C}(X,\tau);$ $\mathcal{B}\subseteq \tau_A$ ve $A=\cup\mathcal{B}$ yani $\mathcal{B}$ ailesi, $A$'nın bir $\tau_A$-açık örtüsü olsun.
$\left.\begin{array}{r} (\mathcal{B}\subseteq \tau_A)(A=\cup\mathcal{B})\Rightarrow (\mathcal{A}:=\{T|(\forall B\in\mathcal{B})(\exists T\in \tau)(B=T\cap A)\}\subseteq \tau)(A\subseteq \cup\mathcal{A}) \\ \\ A\in\mathcal{C}(X,\tau)\Rightarrow \setminus A\in\tau\end{array}\right\} \Rightarrow$
$\left.\begin{array}{r} \Rightarrow (\mathcal{A}^*:= \mathcal{A}\cup\{\setminus A\}\subseteq\tau)(X=\cup \mathcal{A}^*) \\ \\ (X,\tau), \text{ Lindelöf uzayı}\end{array}\right\} \Rightarrow$
$\left.\begin{array}{r} \Rightarrow (\exists \mathcal{A}^{**}\subseteq \mathcal{A}^*)(|\mathcal{A}^{**}|\leq\aleph_0)(X=\cup \mathcal{A}^{**}) \\ \\ \mathcal{B}^*:=\{T\cap A|T\in\mathcal{A}^{**}\Rightarrow T\cap A\in \mathcal{B}\}\end{array}\right\} \Rightarrow (\exists \mathcal{B}^*\subseteq \mathcal{B})(|\mathcal{B}^*|\leq\aleph_0)(A=\cup \mathcal{B}^*).$