Rank of Matroid Circuit is One Less Than Cardinality/Lemma

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

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

Let $x \in C$.

Then:
 * $C \setminus \set x$ is an independent subset of $C$

Proof
From Set Difference is Subset:
 * $C \setminus \set x \subseteq C$

Because $x \in C$ and $x \notin C \setminus \set x$:
 * $C \setminus \set x \ne C$

Hence:
 * $C \setminus \set x \subsetneq C$

By definition of circuit:
 * $C$ is a minimal dependent subset

Hence:
 * $C \setminus \set x \in \mathscr I$