Quotient Topological Vector Space is Hausdorff iff Linear Subspace is Closed

Theorem
Let $\GF \in \set {\R, \C}$.

Let $\struct {X, \tau}$ be a topological vector space over $\GF$.

Let $N$ be a linear subspace of $X$.

Let $X/N$ be the quotient vector space of $X$ modulo $N$.

Let $\tau_N$ be the quotient topology on $X/N$.

Then $\struct {X/N, \tau_N}$ is Hausdorff :
 * $N$ is a closed linear subspace.