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

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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 $M$.

$\blacksquare$