Every chain in a proset is contained in a maximal chain.
We will use Zorn's lemma. Let be a proset and let be a chain. Consider the collection of chains in that contain , ordered by inclusion. If is a family totally ordered by inclusion, then the union , with the order coming from , is also totally ordered: any two elements are comparable in . The hypotheses for Zorn’s lemma therefore obtain on , and we conclude that has a maximal element, which is clearly maximal in the collection of all chains.
Conversely, suppose that the Hausdorff maximal principle holds; we will prove Zorn’s lemma. Suppose given a poset (or preorder) such that every chain in has an upper bound. Since is a chain, the Hausdorff maximal principle implies that contains a maximal chain ; let be an upper bound of . Then is maximal: if , then by maximality of ; therefore and hence since is an upper bound of .