Translation of Closed Set in Topological Vector Space is Closed Set/Proof 1

Proof
We aim to show that $X \setminus \paren {F + x}$ is open.

Since $F$ is closed, $X \setminus F$ is open.

It follows from Translation of Open Set in Topological Vector Space is Open that $\paren {X \setminus F} + x$ is open.

From Translation of Complement of Set in Vector Space, we have:
 * $X \setminus \paren {F + x} = \paren {X \setminus F} + x$

Since we have established that $\paren {X \setminus F} + x$ is open, we have that $X \setminus \paren {F + x}$ is open.

So $F + x$ is closed.