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If a universal first-order theory in a first-order language with a constant entails an existential sentence with quantifier-free matrix, it entails a finite disjunction of closed-term instances of that matrix. Otherwise a first-order model of the first-order theory and all negated ground instances would have a ground-term substructure satisfying the first-order theory. An existential witness there is a tuple of closed-term values, contradicting its negated ground instance. Compactness theorem extracts finitely many instances.

Adjoin to $T$ the negations of all closed-term instances of $\theta$. If the resulting set had a first-order model $\mathcal{A}$, its ground-term substructure $\mathcal{B}$ from (ii) would still first-order model $T$. The assumed existential consequence therefore supplies a tuple $\vec{b}$ in $\mathcal{B}$ satisfying $\theta$. Each $b_j$ is the value of a closed term $t_j$.

Quantifier-free agreement between a substructure and its ambient first-order structure givescontradicting the adjoined negation of this very ground instance. Thusis unsatisfiable. Compactness theorem gives a finite unsatisfiable subset. Its finitely many ground negations imply that, for some finite list of first-order term tuples,If the finite subset already shows $T$ itself inconsistent and supplies no ground negations, choose any one closed-term tuple using the available constant; then any first-order sentence is a consequence of $T$, so $m=1$ suffices. Otherwise the selected list is nonempty. This also explains why the conclusion contains a finite disjunction rather than necessarily one universal choice of a witness first-order term.

The set of closed-term values is nonempty because the first-order language has a constant. It contains the value of every constant. If $f$ is an $r$-ary function symbol and $t_1,\dots ,t_r$ are closed terms, thenwhich is another closed-term value. Hence the set is closed under every operation. Restrict the operations and relations of $\mathcal{A}$ to it; this defines a substructure $\mathcal{B}$ whose carrier is exactly the specified set.

Part (i) now applies to the universal first-order theory, givingIt is the substructure generated by the first-order language's constants, often called the ground-term substructure. The source's constant-symbol assumption ensures its nonemptiness, even when no other elements can be generated.