\begin{definition}[Definition:Matroid/Definition 3]
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 V < \size U \implies \exists Z \subseteq U \setminus V : \paren {V \cup Z \in \mathscr I} \land \paren {\size {V \cup Z} = \size U}
}}
{{end-axiom}}
\end{definition}