Choice Function Exists for Set of Well-Ordered Sets

Theorem
Let $\mathbb S$ be a set of sets such that:
 * $\forall S \in \mathbb S: S \ne \varnothing$

that is, none of the sets in $\mathbb S$ may be empty.

Let every element of $\mathbb S$ be well-ordered.

Then there exists a choice function $f: \mathbb S \to \bigcup \mathbb S$ defined as:
 * $\forall S \in \mathbb S: \exists x \in S: f \left({S}\right) = x$

Thus, if every member of $\mathbb S$ is a well-ordered, then we can create a choice function $f$ defined as:
 * $\forall S \in \mathbb S: f \left({S}\right) = \inf \left({S}\right)$

True, we may be making infinitely many choices, but we have a rule for doing so.