\begin{definition}[Definition:Matroid/Definition 4]
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 A \subseteq S
        | mr= \text {all maximal subsets $Y \subseteq A$ with $Y \in \mathscr I$ have the same cardinality}
}}
{{end-axiom}}
\end{definition}