\begin{definition}[Definition:Matroid/Definition 2]
Let $M = \struct {S, \mathscr I}$ be an independence system.
$M$ is called a '''matroid on $S$''' {{iff}} $M$ also satisfies:
{{begin-axiom}}
{{axiom | n = \text I 3'
        | q = \forall U, V \in \mathscr I
        | mr= \size U = \size V + 1 \implies \exists x \in U \setminus V : V \cup \set x \in \mathscr I
}}
{{end-axiom}}
\end{definition}