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Source. Theorem 2b, printed p. 150 (published PDF).

Statement. Let $M=(E,\mathcal F)$ be finite and let $0\le n_t\le r(E)$, $1\le t\le k$, be integers, with $k\ge1$. There are pairwise disjoint independent sets $I_t$ with $|I_t|=n_t$ if and only if

$$\begin{aligned} |A| &\ge\sum_t\bigl(n_t-\min(n_t,r(E\setminus A))\bigr)\\ &=\sum_{j=r(E\setminus A)+1}^{r(E)}n_j^* \qquad(A\subseteq E). \tag{1} \end{aligned}$$

Proof. The truncation $M_t$ at $n_t$ has rank $n_t$ on $E$, and rank $\min(n_t,r(B))$ on a subset $B$. Its bases are exactly the independent sets of $M$ of size $n_t$. Theorem 2c for the family $(M_t)$ therefore gives precisely the first inequality in (1). For the equality, count for each $t$ the integers $r(E\setminus A)<j\le n_t$; the upper bound $n_t\le r(E)$ permits the common upper limit $r(E)$. This proves both directions, including $n_t=0$. $\square$