Rank of Matroid Circuit is One Less Than Cardinality

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

Let $C \subseteq S$ be a circuit of $M$.

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

Then:
 * $\map \rho C = \card C -1$

Proof
By definition of a circuit:
 * $C$ is dependent

By matroid axiom $(\text I 1)$:
 * $C \ne \O$

Let $x \in C$.

From Leigh.Samphier/Sandbox/Set Difference of Matroid Circuit with Element is Maximal Independent:
 * $C \setminus \set x$ is a maximal independent subset of $C$

We have: