Affine Sets are Translates of Subspaces

Let $X$ be a vector space and let $\Omega\subset X$ be nonempty.

Lemma: The set $\Omega$ is affine if and only if for every $\omega\in\Omega$, the translate

is a linear subspace of $X$.

Equivalently, $\Omega$ is affine iff there exist $\omega\in X$ and a subspace $L\subset X$ such that $\Omega=\omega+L$.

Context: This lemma explains why affine sets are often called “affine subspaces”: they are precisely translates of linear subspaces.