Distinct Matroid Elements are Parallel iff Each is in Closure of Other/Lemma

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

Let $\set a$ and $\set b$ be independent.

Then $\set {a, b}$ is dependent :
 * $a \in \map \sigma {\set b}$

and
 * $b \in \map \sigma {\set a}$

where $\sigma: \powerset S \to \powerset S$ denotes the closure operator of $M$.