Equivalence of Definitions of Matroid Rank Axioms/Condition 3 Implies Condition 2

Theorem
Let $S$ be a finite set.

Let $\rho$ is the rank function of a matroid $M = \struct{S, \mathscr I}$.

Then $\rho$ satisfies definition 2 of the rank axioms:

$\rho$ satisfies $(\text R 1')$
This follows immediately from Bounds for Rank of Subset.

$\rho$ satisfies $(\text R 2')$
This follows immediately from Rank Function is Increasing.

$\rho$ satisfies $(\text R 3')$
Let $X, Y \subseteq S$.

Let $A$ be a maximal independent subset of $X \cap Y$.

From Cardinality of Maximal Independent Subset Equals Rank of Set:
 * $\card A = \map \rho {X \cap Y}$

From Independent Subset is Contained in Maximal Independent Subset:
 * $\exists B \subseteq X \setminus Y : A \cup B$ is a maximal independent subset of $X$

Simiarly from Independent Subset is Contained in Maximal Independent Subset:
 * $\exists C \subseteq Y \setminus X : \paren{A \cup B} \cup C$ is a maximal independent subset of $X \cup Y$

and
 * $\card{A \cup B \cup C} = \map \rho {X \cup Y}$

By matroid axiom $(\text I 2)$:
 * $A \cup C$ is an independent subset of $Y$.

We have: