Matroid Unique Circuit Property

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

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

Let $x \in S$ such that:
 * $X \cup \set x$ is a dependent subset of $M$.

Then there exists a unique circuit $C$ such that:
 * $x \in C \subseteq X \cup \set x$