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Source. Theorem 10, printed p. 1327 (published PDF).

Statement. Let $U\subseteq S$ and let $k\ge0$ be an integer. The family $\mathcal A=(A_i)_{i\in I}$ has a $k$-transversal $X$ with $U\subseteq X$ if and only if, for every $J\subseteq I$,

$$k|A(J)|\ge |J| \tag{1}$$

and

$$|A(J)\cap U|\ge |J|+|U|-|I|. \tag{2}$$

Proof. Give $S$ the matroid whose elements in $U$ are free and whose elements outside $U$ are loops. Its rank function is

$$r_U(Y)=|Y\cap U|.$$

Since $r_U(Y)\le|U|$, the inequality $r_U(Y)\ge|U|$ is equivalent to $U\subseteq Y$. Apply Theorem 5 with $t=|U|$. Its first condition is (1), and its second condition is exactly (2). The equivalence in Theorem 5 proves both directions. $\square$

This argument includes $U=\varnothing$. It also includes $k=0$: if $I\ne\varnothing$, (1) fails, while for $I=\varnothing$ condition (2) forces $U=\varnothing$, the only possible support.