Equivalence of Definitions of Matroid/Definition 1 implies Definition 4

Theorem
Let $M = \struct {S, \mathscr I}$ be an independence system.

Let $M$ also satisfy:

Then $M$ satisfies:

Proof
Let $M$ satisfy condition $(\text I 3)$.

Let $A \subseteq S$.

Let $Y_1, Y_2$ be maximal independent subsets of $A$.

, let:
 * $\size {Y_2} \le \size {Y_1}$


 * $\size {Y_2} < \size {Y_1}$
 * $\size {Y_2} < \size {Y_1}$

By condition $(\text I 3)$:
 * $\exists y \in Y_1 \setminus Y_2 : Y_2 \cup \set y \in \mathscr I$

From Union of Subsets is Subset:
 * $Y_2 \cup \set y \subseteq A$

This contradicts the maximality of $Y_2$.

Then:
 * $\size {Y_2} = \size {Y_1}$

It follows that $M$ satisifies $(\text I 3''')$.