Proposition 2

— If M and N are any two subsets of an extension of a field K, then $K(M \cup N) = K(M)(N) = K(N)(M)$.

For $K(M \cup N)$ contains $K(M)$ and N and hence $K(M)(N)$; since $K(M)(N)$ is a field containing $K \cup M \cup N$, it contains $K(M \cup N)$, whence the proposition.

We shall sometimes write $K(M, N)$ instead of $K(M \cup N)$.