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Let $\vec{a}\in A^n$, with $A$ finite, and put $T=\mathrm{Th}(\mathcal{A})$. There are only finitely many competing tuples $\vec{b}\in A^n$. For each whose complete type differs from that of $\vec{a}$, choose a logical formula ${\varphi }_{\vec{b}}$ true at $\vec{a}$ and false at $\vec{b}$. Such a logical formula exists by the inequality of their complete types, negating a distinguishing logical formula if necessary.

Let $\theta$ be the conjunction of this finite list, or logical truth if the list is empty. It is true at $\vec{a}$, and every tuple satisfying it has the same partial type as $\vec{a}$. For each $\psi \in \mathrm{tp}(\vec{a})$, the first-order sentenceis therefore true in $\mathcal{A}$ and belongs to $T$. The logical formula $\theta$ is first-order consistent with $T$ because it is realized at $\vec{a}$. It is consequently a principal formula of a type of the partial type, proving every partial type realized in a finite first-order structure is principal. Finiteness of the first-order structure, rather than finiteness of the first-order language, is what makes the conjunction finite.

An $n$-type over a first-order theory $T$ is a set of logical formulas in the displayed $n$ free variables first-order consistent with $T$. A complete type chooses exactly one of $\varphi$ and $\neg \varphi$ for each such logical formula. Equivalently it is a maximal consistent such set. For parameter-free partial types the logical formulas use the original first-order language; partial types over parameters use additional names for those parameters.

A partial type $p(\vec{x})$ is principal if some logical formula $\theta (\vec{x})$ first-order consistent with $T$ implies every member of $p$ modulo $T$:Such a $\theta$ is a principal formula of a type of the partial type. For a complete type it isolates that partial type, so principal complete types are exactly isolated types.

For a tuple $\vec{a}$ in a first-order structure $\mathcal{A}$, its parameter-free partial type isIt is a complete type over $\mathrm{Th}(\mathcal{A})$ because the first-order structure decides the truth of every logical formula at that tuple.

A principal formula of a type is a logical formula first-order consistent with the base first-order theory that, modulo that first-order theory, implies every logical formula of the partial type. For a complete type it isolates a single point of the type space. A realized finite-tuple partial type in a finite first-order structure has such a logical formula by conjoining one distinguishing logical formula for each competing tuple with a different partial type.