Proof Explanation: Existence of "fg-chains" Axiom of Choice implies Zorn's Lemma

Question

I would be very glad if someone could explain the validity of a passage of this paper to me. In the proof of Lemma 3.3, "Fundamental Lemma", the notion of "fg-chains" is introduced and later on, we define the set S as the union of all "fg-chains". It is a crucial point in this proof that S is not empty, yet any kind of justification is omitted. So, my question is, why is S not empty (or, equivalently, how can one prove that a "fg-chain" exists?)

Answer 1

$\varnothing$ is a chain, so let $t=g(\varnothing)$, and let $C=\{f(t)\}$. Then $fg(C_{f(t)})=fg(\varnothing)=f(t)$, and $C$ is an $fg$-chain, since $C_{f(t)}$ is the only section of $C$.