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Source. Theorem 1d, printed p. 152 (published PDF).

Statement. Let $J_1,\ldots,J_k$ be pairwise disjoint independent sets of a finite matroid $M=(E,\mathcal F)$, with $k\ge1$, and put $E'=E\setminus\bigcup_iJ_i$. There is an indexed partition into independent sets $I_i$ such that $J_i\subseteq I_i$ if and only if

$$|A|\le\sum_{i=1}^k\bigl(r(A\cup J_i)-r(J_i)\bigr) \qquad(A\subseteq E'). \tag{1}$$

Proof. For each $i$, contract the independent seed $J_i$ and restrict the resulting matroid to $E'$. By the contraction formula, the resulting matroid $M_i$ has rank

$$r_i(A)=r(A\cup J_i)-r(J_i)\qquad(A\subseteq E').$$

Its independent subsets $K\subseteq E'$ are precisely those for which $K\cup J_i$ is independent in $M$.

An independent partition $(I_i)$ extending the seeds gives a partition $K_i=I_i\setminus J_i$ of $E'$ into independent sets of the respective $M_i$: disjointness of the $I_i$ prevents a part from containing any other seed. Conversely such a partition $(K_i)$ gives $I_i=K_i\cup J_i$, a disjoint independent cover of $E$, because the seeds are pairwise disjoint.

Theorem 1c applied on the common ground set $E'$ now gives exactly (1). This argument includes empty seeds and $E'=\varnothing$. $\square$