Let $M_1 = (E_1,\mathcal{I}_1)$ and $M_2 = (E_2,\mathcal{I}_2)$ be matroids with disjoint ground sets $E_1 \cap E_2 = \varnothing$. Let $M = M_1 \oplus M_2$ be their direct sum on the ground set $E = E_1 \cup E_2$, so that a subset $I \subset E$ is independent in $M$ if and only if $I \cap E_1$ is independent in $M_1$ and $I \cap E_2$ is independent in $M_2$. Then for every subset $A \subset E$,

\begin{align*} r_M(A)=r_{M_1}(A\cap E_1)+r_{M_2}(A\cap E_2). \end{align*}