 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.