Open Neighborhood of Point in Topological Vector Space contains Sum of Open Neighborhoods/Corollary 2

Corollary
Let $K \in \set {\R, \C}$. Let $X$ be a topological vector space over $K$. Let $W$ be an open neighborhood of ${\mathbf 0}_X$.

Then there exists a balanced open neighborhood $U$ of ${\mathbf 0}_X$ such that:
 * $U + U \subseteq W$

Proof
From Open Neighborhood of Point in Topological Vector Space contains Sum of Open Neighborhoods: Corollary, there exists an open neighborhood $V$ of ${\mathbf 0}_X$ such that:
 * $V + V \subseteq W$

From Open Neighborhood of Origin in Topological Vector Space contains Balanced Open Neighborhood, there exists a balanced open neighborhood $U$ of ${\mathbf 0}_X$ such that $U \subseteq V$.

Then $U + U \subseteq V + V \subseteq W$.