Independent Subset is Base if Cardinality Equals Rank of Matroid/Corollary

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

Let $X \subseteq S$ be any independent subset of $M$.

Let $\card X = \card B$.

Then:
 * $X$ is a base of $M$.

Proof
From All Bases of Matroid have same Cardinality:
 * $\card B = \map \rho S$

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

Hence:
 * $\card X = \map \rho S$

From Independent Subset is Base if Cardinality Equals Rank of Matroid:
 * $X$ is a base of $M$.