= Existence of transcendence bases
Apply <Zorn's lemma> to subsets of a <field extension> with <algebraic independence>, ordered by inclusion. The union of a chain is independent because any polynomial relation involves only finitely many distinct elements, all of which lie in one member of the chain. A maximal independent set $B$ is a <transcendence basis>: if any element were transcendental over $K(B)$, adjoining it would preserve independence, by clearing denominators, contradicting maximality. Tuple entries must be distinct in the set-based independence definition; otherwise the nonzero polynomial $X_1-X_2$ vanishes on a repeated pair and excludes every nonempty set.
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