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A model companion of $T$ is a model-complete theory $T^{\ast }$ in the same language with $T_{\forall }^{\ast }=T_{\forall }$. Equivalently each model of either theory embeds into a model of the other. Alternating these embeddings and using the elementary chain theorem proves uniqueness up to logical equivalence. A model companion need not be a syntactic extension of its original theory.

A model companion $T^{\ast }$ of a first-order theory $T$, in the same first-order language, is a model-complete theory with the same universal consequences of a theory as $T$:Equivalently, every model of $T$ embeds into a model of $T^{\ast }$, and every model of $T^{\ast }$ embeds into a model of $T$. The equivalence follows from the diagram embedding criterion for universal theories, which is an application of the compactness theorem. A model companion need not be a syntactic extension of $T$, and uniqueness is understood up to logical equivalence of theories.

A model-complete theory is one for which every embedding between models is elementary. If $(M_i)$ is an embedding chain of models, all transition embeddings are therefore elementary. The elementary chain theorem shows that the union is a model elementarily extending every $M_i$, so the theory is preserved under unions of embedding chains. Part (c) then gives an axiomatization by forall-exists sentences.