Set of Finite Character with Choice Function is of Type M

Theorem
Let $S$ be a set of sets with finite character.

Let there exist a choice function for $S$.

Then $S$ is of type $M$, that is:
 * every element of $S$ is a subset of a maximal element of $S$ under the subset relation.

Proof
Let $S$ be according to the hypothesis.

From Class of Finite Character is Closed under Chain Unions:
 * $S$ is closed under chain unions.

From Closed Set under Chain Unions with Choice Function is of Type $M$:
 * $S$ is of type $M$.