Dieudonné's theorem

Statement

Let ${\displaystyle X}$ be a locally convex space and ${\displaystyle A,B\subset X}$ nonempty closed convex sets. If either ${\displaystyle A}$ or ${\displaystyle B}$ is locally compact and ${\displaystyle \mathrm{recc}(A)\cap \mathrm{recc}(B)}$ (where ${\displaystyle \mathrm{recc}}$ gives the recession cone) is a linear subspace, then ${\displaystyle A-B}$ is closed.