= Solution
<Zorn lemma> states that a nonempty <partially ordered set> in which every <chain in a partial order> has an upper bound has a maximal element. Here is a proof from the <axiom of choice> and <Hartogs theorem>.
Suppose there is no maximal element. Every chain $C$ has a strict upper bound: choose an upper bound $b$ and then an element strictly greater than $b$. The family of sets of strict upper bounds is a set of nonempty subsets of the underlying set $P$. \b[The axiom of choice is used here] to choose, simultaneously for every chain $C$, one strict upper bound $b(C)$.
By <Hartogs theorem> there is an <ordinal> $\kappa$ admitting no injection into $P$. <Transfinite recursion> defines $x_\alpha$ for $\alpha<\kappa$ by
$$
x_\alpha=b(\{x_\beta:\beta<\alpha\}).
$$
At each stage the previous elements form a chain, and $x_\alpha$ is strictly above every one of them. Hence $\alpha\mapsto x_\alpha$ is an injection $\kappa\to P$, a contradiction. Therefore \b[a maximal element exists].
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