Parallel Elements Depend on Same Subsets

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

Let $A \subseteq S$.

Let $x, y \in S$.

Let $x$ be parallel to $y$.

Then:
 * $x$ depends on $A$ $y$ depends on $A$

Proof
This follows directly from Closure of Subset Contains Parallel Elements.