= Szpilrajn extension theorem
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Every <partial order> has a <total order> extension. Apply <Zorn's lemma> to extensions ordered by inclusion: the union of a chain is a <partial order>. If a maximal extension leaves two elements incomparable, adjoining one comparison and taking its <transitive closure of a relation> creates no cycle, contradicting maximality. Quotienting a <preorder> by its <indifference relation> and lifting a <total order> gives a complete weak-order extension preserving its strict part.
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