Added the Mean Value Theorem.

src/fa/tvs/definition.tex

@@ -202,3 +202,12 @@
 
     (Uniqueness): Let $\mathcal{S} \subset 2^E$ be a topology on $E$ satisfying (1) and (2), then for each $x \in E$, $\cn_{(E, \mathcal{S})}(x) = \cn_{(E, \mathcal{T})}(x)$. By \ref{proposition:neighbourhoodcharacteristic}, $\mathcal{S} = \mathcal{T}$.
 \end{proof}
+
+
+\begin{proposition}
+\label{proposition:tvs-locally-connected}
+    Let $E$ be a TVS over $K \in \RC$, then $E$ is locally connected.
+\end{proposition}
+\begin{proof}
+    Let $U \in \cn(0)$ be radial, then for any $y \in U$, the mapping $t \mapsto ty$ is a path from $0$ to $y$ contained in $U$. Thus $U$ is path-connected. By \ref{proposition:tvs-0-neighbourhood-base}, the radial neighbourhoods of $0$ forms a fundamental system of neighbourhoods, so the path-connected neighbourhoods of $0$ forms a fundamental system as well.
+\end{proof}