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Source. Theorem 1b, printed p. 150 (published PDF).

Statement. Let $M=(E,\mathcal F)$ be finite, $k\ge1$, and $0\le n_t\le r(E)$ be integers. Independent sets $I_t$ of exact sizes $n_t$ cover $E$ if and only if

$$|A|\le\sum_{t=1}^k\min(n_t,r(A)) =\sum_{j=1}^{r(A)}n_j^* \qquad(A\subseteq E), \tag{1}$$

where $n_j^*=|\{t:n_t\ge j\}|$.

Proof. If the cover exists, each $A\cap I_t$ is independent, with size at most both $n_t$ and $r(A)$. Counting their union proves (1). The displayed equality is the conjugate-counting identity in the maximum-union formula.

Conversely, truncate $M$ at $n_t$ to obtain matroid $M_t$ of rank $\min(n_t,r(A))$, by Lemma 1. Condition (1) is exactly Theorem 1c for these matroids, so it yields a partition into independent sets $J_t$ with $|J_t|\le n_t$. Extend each $J_t$ within $M$ to size $n_t$, which is possible because $n_t\le r(E)$. The extensions cover $E$; disjointness is not required. $\square$