Maximal algebraically independent subset and transcendence basis

Question

I'm studying transcendence basis and I got stuck with the following problem:

Let $K$ be a field and $E$ its extension. Let $S$ be a subset of $E$ such that $E$ is algebraic with respect to $K(S)$. Why do the every maximal subset of $S$ with respect to $K$ is a transcendence basis of $E$ with respect to $K$?

Answer 1

Let $M$ be a maximal algebraically independent subset of $S$. Then $K(S)$ is algebraic over $K(M)$ by the maximality of $M$.

As the composition of algebraic extension is algebraic, we get that $E$ is algebraic over $K(M)$, completing the argument.