Parallel Subspace to an Affine Set is Ω−Ω

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

Proposition: The set $\Omega$ is parallel to a unique linear subspace $L\subset X$, and this subspace is

Context: The subspace $\Omega-\Omega$ captures the “direction” of the affine set. It is the appropriate linear object used to define hyperplanes as affine sets of codimension one.