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Source interface. The finite matroid definition on printed p. 147 and the arguments on pp. 150–152 (published PDF) use the following elementary consequences. They are expanded here, without importing a matroid representation theorem.

Statement. In a finite matroid:

1. Every independent subset of $A$ extends to a base of the restriction to $A$. If independent $I,J$ satisfy $|I|<|J|$, some $e\in J\setminus I$ makes $I\cup\{e\}$ independent.
2. Rank is monotone, $0\le r(A)\le |A|$, and $A$ is independent exactly when $r(A)=|A|$. Deleting one element lowers rank by at most one.
3. An element is a coloop exactly when it belongs to no circuit. Equivalently, $e$ is a coloop exactly when $r(E\setminus\{e\})=r(E)-1$.
4. A nonempty hereditary family on a finite set satisfying the augmentation assertion in part 1 satisfies the equal-size maximal-independent-set axiom.

Proof. To extend an independent set, add elements while independence is preserved. Finiteness ends this process at a maximal independent subset, whose size is $r(A)$ by the axiom.

If no element of $J\setminus I$ augments $I$, then $I$ is maximal independent in $I\cup J$. Extending $J$ inside the same set gives a maximal independent set of size at least $|J|>|I|$, a contradiction. This proves augmentation.

An independent subset of $A$ is also independent in any superset, proving rank monotonicity. The bounds and the independence criterion follow from the definition. Removing $e$ from a base leaves an independent set of size at least $r(E)-1$, so a one-element deletion lowers rank by at most one.

If a base $B$ omits $e$, then $B\cup\{e\}$ is dependent and, by finiteness, contains a circuit. That circuit contains $e$ because $B$ is independent. Thus an element in no circuit is a coloop. Conversely, if a circuit $C$ contains $e$, extend the independent set $C\setminus\{e\}$ to a maximal independent set $B$ of $E\setminus\{e\}$. The set $B\cup\{e\}$ contains $C$, so is dependent. Hence $B$ is also maximal independent in $E$, and is a base omitting $e$. This proves the circuit criterion. The rank criterion follows: a base omitting $e$ gives unchanged rank, whereas if all bases contain $e$, the deletion has rank strictly smaller, and hence exactly one less.

Finally, if two maximal independent subsets $I,J$ of the same set had $|I|<|J|$, augmentation would enlarge $I$ inside that set. This is impossible, and proves part 4. $\square$

These finite deductions are related to the rank-axiom deductions in Rado (1949). Here the starting axioms are the independent-set axioms stated by Edmonds–Fulkerson. Neither Rado's infinite selection principle nor the separate finite independent-representative theorem is an input.