Matroid Base Substitution From Fundamental Circuit

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

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

Let $y \in S \setminus B$.

Let $\map C {y,B}$ denote the fundamental circuit of $y$ in $B$.

Let $x \in B$.

Then:
 * $\paren{B \setminus \set x} \cup \set y$ is a base of $M$ $x \in \map C{y,B}$