= Directed limit of elementary embeddings
A directed system of <first-order structures> and coherent <elementary embeddings> has a direct limit into which every member embeds elementarily. Represent an element by a pair consisting of a stage and an element at that stage; identify two representatives when their images agree at a common later stage. Interpret <functions> and relations at a common stage. <Mathematical induction> on <first-order formulas> proves elementarity: for an existential <first-order formula> true at the limit, its witness and parameters occur at a common stage, and elementarity pulls truth back to the stage containing the original parameters. This generalizes the <elementary chain theorem> beyond linearly ordered diagrams.
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