Translation of Closed Set in Topological Vector Space is Closed Set/Proof 2

Proof
Define a mapping $T_{-x} : X \to X$ by:
 * $\map {c_\lambda} y = y + x$

for each $y \in X$.

From Translation Mapping on Topological Vector Space is Homeomorphism, $T_{-x}$ is a homeomorphism.

From Definition 4 of a homeomorphism, $T_{-x}$ is therefore a closed mapping.

Hence $T_{-x} \sqbrk F = F + x$ is closed.