Union of Non-Disjoint Convex Sets is Convex Set

Theorem
Let $\left({S, \preccurlyeq}\right)$ be an ordered set.

Let $\mathcal C$ be a set of convex sets of $S$ such that their intersection is non-empty:
 * $\displaystyle \bigcap \mathcal C \ne \varnothing$

Then the union $\displaystyle \bigcup \mathcal C$ is also convex.