Equivalence of Definitions of Matroid/Definition 2 implies Definition 3

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

Let $M$ also satisfy:

Then $M$ satisfies:

Proof
From Independent Set can be Augmented by Larger Independent Set it follows that if $M$ satisfies condition $(\text I 3')$ then $M$ satisfies condition $(\text I 3'')$.