Matroid Induced by Affine Independence is Matroid

Theorem
Let $\R^n$ be the $n$-dimensional real Euclidean space.

Let $S = \set{x_1, \dots, x_r}$ be a finite subset of $\R^n$.

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

That is, $\mathscr I$ is the set of affinely 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 $(I3)$.