= Canonical models of complete theories are prime
Let two <first-order models> satisfy a <complete theory> and suppose every element of the first is the value of a <closed term>. Send each such value to the value of the same <first-order term> in the second <first-order model>. Completeness decides all <logical equality> <first-order sentences>, making this map well-defined and <injective>. It also decides every <logical formula> with <closed terms> substituted, making the map elementary. Thus the first <first-order model> is prime.
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