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Like the well-ordering principle, Zorn's Lemma is equivalent to the axiom of choice, in the sense that either one together with the Zermelo-Fraenkel axioms (see set theory) is sufficient to prove the other. It is probably the most useful of all equivalents of the axiom of choice and occurs in the proofs of several theorems of crucial importance, for instance the [Hahn-Banach theorem]? in [functional analysis]?, the theorem that every vector space has a basis, and the theorem that every field has an [algebraic closure]?.
Zorn's Lemma is named after Max Zorn.