Schauder's Lemma in functional analysis states the following:

Let $E$ and $F$ be metrizable locally convex topological vector spaces, and let $E$ be Fréchet. Then if the linear continuous map $A:E\to F$ is nearly open, that is to say, for any neighborhood of the origin $U\subset E$ we have that $\emptyset\ne\operatorname{Int}(\overline{A(U)})\subset F$, then $A$ is surjective and open.

However, I've been unable to find a reference for a proof of this, and I haven't managed to figure out why it is true either. Either a proof or a reference to a proof would be much appreciated. The most I've been able to show is that $\overline{A(E)}=F$.