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Source. Theorem 4, statement on printed p. 1324 and proof on p. 1325 (published PDF).

Statement. Let $(S,M)$ be a finite matroid with rank function $r$, let $\mathcal A=(A_i)_{i\in I}$ be a finite indexed family, and let $p_i\ge0$ be integers. There is a $p$-transversal of $\mathcal A$ that is independent in $M$ if and only if

$$r(A(J))\ge p(J) \qquad(J\subseteq I). \tag{1}$$

Proof. Suppose $X=\bigsqcup_{i\in I}X_i$ is an independent $p$-transversal. For each $J\subseteq I$, the set

$$\bigsqcup_{i\in J}X_i\subseteq A(J)$$

is independent and has $p(J)$ elements. Thus (1) is necessary.

For sufficiency, form the replicated family $\mathcal A^p=(A^p_{i,h})_{(i,h)\in I^p}$ from the definitions. Let $K\subseteq I^p$, and let

$$J=\{i:\text{some }(i,h)\in K\}.$$

The union of the sets indexed by $K$ is $A(J)$, while $|K|\le p(J)$. Condition (1) therefore gives

$$r\left(\bigcup_{(i,h)\in K}A^p_{i,h}\right) =r(A(J))\ge p(J)\ge |K|.$$

Rado's finite independent-representative theorem supplies an independent full transversal of $\mathcal A^p$. Its range is a $p$-transversal of $\mathcal A$ by the replicated-family correspondence. This proves sufficiency. If $I^p$ is empty, the argument reduces to the empty independent set and all displayed conditions hold. $\square$

Proof boundary. The reduction for arbitrary subfamilies of copied indices is proved here. Rado's finite theorem is the exact unproved input recorded in external inputs.