Corollary

— Let E be an extension of a field K and $A \subset E$. The field $K(A)$ is the union of the fields $K(F)$ where F ranges over the set of all finite subsets of A.

For the set of fields $K(F)$ is directed by the relation $\subset$, because $F \subset F'$ implies $K(F) \subset K(F')$. The union L of these fields is thus a field containing $K \cup A$ and contained in $K(A)$, and hence identical with $K(A)$.