Corollary

Let L be a commutative field, K a subfield of L, $x = (x_i)_{i \in I}$ a family of elements of L, M the set consisting of the $x_i$, U the set of all $f \in K((X_i)_{i \in I})$ such that x is substitutable in $f$ and $\varphi$ the homomorphism $f \mapsto f(x)$ of U into L. Then $\varphi(U)$ is the subfield of L generated by $KUM$.

Let L' be the subfield of L generated by $KUM$. We have

and $\varphi(U)$ is a subring of L. Now Prop. 2 implies that $\varphi(U)$ is a subfield of L, whence $\varphi(U) = L'$.

Let $f \in K((X_i)_{i \in I})$ and let $(g_i)_{i \in I}$ be a family of elements of $K((Y_l)_{l \in L})$. If (g_i) is substitutable in f, then f((g_i)) is an element of K((Y_l)_{l \in L}). In particular, (X_i)_{i \in I} is substitutable in f and f = f((X_i)_{i \in I}).