All Bases of Matroid have same Cardinality

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

Let $\rho : \powerset S \to \Z$ be the rank function of $M$.

Let $B$ be a base of $M$.

Then:
 * $\size B = \map \rho S$

That is, all bases of $M$ have the same cardinality, which is the rank of $S$.