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An existential formula is an existential block followed by a quantifier-free formula, with remaining free variables serving as parameters. Truth is preserved by structure embeddings. Reflection of truth for every such formula defines an existential embedding.

A 1-embedding, or existential embedding, is a structure embedding $\alpha :A\to B$ satisfyingfor every quantifier-free $\theta$ and every parameter tuple $\mathbf{a}$ from $A$. The forward direction is automatic for an embedding; reflection of existential formulas is the extra property. For an inclusion this says that $A$ is simply closed in $B$. This is weaker than requiring an elementary embedding, which preserves all formulas.

Let $S$ consist of all universal-existential sentences entailed by $T$, and take $A_0\models S$. By (c), there is an existential embedding of $A_0$ into $B_0\models T$. By (b), $B_0$ embeds in $A_1$ with $A_0\preccurlyeq A_1$. After replacing structures by isomorphic copies, arrange these maps as inclusions. Since $A_1\equiv A_0$, it also satisfies $S$ and the construction can be repeated. This producesThe two interleaved chains have the same union $D$. The elementary chain theorem gives $A_0\preccurlyeq D$. The $B_i$ are an ordinary chain of models of $T$, so the assumed closure under unions of chains gives $D\models T$. Therefore $A_0\models T$. This provesConversely, universal-existential axioms are preserved in such unions: a finite universal tuple occurs in one stage, which already supplies its existential witnesses. The quantifier-free matrix has the same truth value in that stage and in the union. This also explains the inductive first-order theory characterization.