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Source. Theorem 8, printed p. 1327 (published PDF).

Statement. Let $\mathcal A=(A_i)_{i\in I}$ be a finite indexed family and let $k,t\ge0$ be integers. There is a $k$-transversal of cardinality at least $t$ if and only if

$$k|A(J)|\ge |J| \tag{11}$$

and

$$|A(J)|\ge |J|+t-|I| \tag{12}$$

for every $J\subseteq I$.

Proof. In the free matroid on $S$, every set is independent and $r(X)=|X|$. Substituting this rank function into Theorem 5 turns its second inequality into (12), while its first inequality is (11). Its equivalence therefore proves the statement. $\square$

The $k=0$ and $I=\varnothing$ cases are exactly those dispatched in Theorem 5; no positivity assumption is added here.