Cardinality of Maximal Independent Subset Equals Rank of Set

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

Let $A \subseteq S$.

Let $X$ be a maximal independent subset of $A$.

Then:
 * $\card X = \map \rho A$

where $\rho$ is the rank function on $M$.

Proof
From Independent Subset is Contained in Maximal Independent Subset:
 * $\exists Y \in \mathscr I : X \subseteq Y \subseteq A : \card Y = \map \rho A$

By definition of a maximal independent Subset of $A$:
 * $X = Y$

The result follows.