Matroid Induced by Algebraic Independence is Matroid

Theorem
Let $L / K$ be a field extension.

Let $S \subseteq L$ be a finite subset of $L$.

Let $\struct{S, \mathscr I}$ be the matroid induced by algebraic independence over $K$ on $S$.

That is, $\mathscr I$ is the set of algebraically independent subsets of $S$.

Then $\struct{S, \mathscr I}$ is a matroid.

Proof
It needs to be shown that $\mathscr I$ satisfies the matroid axioms $(I1)$, $(I2)$ and $(I3)$.