Matroid Unique Circuit Property/Corollary

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

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

Let $x \in S \setminus B$.

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

That is, $C$ is the fundamental circuit of $x$ in $B$.

Proof
From Union of Matroid Base with Element of Complement is Dependent:
 * $B \cup \set x$ is dependent.

From Matroid Unique Circuit Property there exists a unique circuit $C$ such that:
 * $x \in C \subseteq B \cup \set x$