Independent Subset is Contained in Maximal Independent Subset

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

Let $A \subseteq S$.

Let $X \in \mathscr I$ such that $X \subseteq A$.

Then:
 * $\exists Y \in \mathscr I : X \subseteq Y \subseteq A : \size Y = \map \rho A$

where $\rho$ is the rank function on $M$.

Proof
By definition of the rank function on $M$:
 * $\size X \le \map \rho A$

Case 1 : $\size X = \map \rho A$
Let $\size X = \map \rho A$.

Let $Y = X$ and the result follows.

Case 2 : $\size X < \map \rho A$
Let $\size X < \map \rho A$.

By definition of the rank function on $M$:
 * $\map \rho A = \max \set{\size I : I \subseteq A \land I \in \mathscr I}$

From Max Operation Equals an Operand:
 * $\exists Z \in \mathscr I : Z \subseteq A$ and $\size Z = \map \rho A$

From Independent Set can be Augmented by Larger Independent Set:
 * $\exists Y' \subseteq Z \setminus X: \size {X \cup Y'} = \size Z$ and $X \cup Y' \in \mathscr I$

We have:

From Union of Subsets is Subset:
 * $X \cup Y' \subseteq A$

Let $Y = X \cup Y'$ and the result follows.

In either case, the result follows.