Independent Set can be Augmented by Larger Independent Set

Theorem
Let $M = \struct{S, \mathscr I}$ be a matroid.

Let $X, Y \in \mathscr I$ such that:
 * $\size X < \size Y$

Then there exists $Z \subseteq Y \setminus X$ such that:
 * $X \cup Z \in \mathscr I$
 * $\size{X \cup Z} = \size Y$