reducible\], for every $ r = 1,2, \ldots, k $, we have that $$\label{lower bound}
\begin{split}
d_{R,r}(C) \geq \min \{ & d_{R,r_1}(C_1) + d_{R,r_2}(C_2) + \cdots + d_{R,r_l}(C_l) \\
 & \mid r = r_1 + r_2 + \cdots + r_l, 0 \leq r_i \leq k_i \},
\end{split}$$ and $$\label{upper bound}
\begin{split}
d_{R,r}(C) \leq \min \{ & d_{R,r_1}(C_1^\prime) + d_{R,r_2}(C_2^\prime) + \cdots + d_{R,r_l}(C_l^\prime) \\
 & \mid r = r_1 + r_2 + \cdots + r_l, 0 \leq r_i \leq k_i \}.
\end{split}$$

The proof can be found at the end of the section. We now elaborate on some particular cases of interest.

First, observe that the bound (\[lower bound\]) gives the bound (\[bound min distance\]) for the minimum rank distance (the case $ r=1 $), and the bound (\[upper bound\]) gives the following (immediate) upper bound: $$\label{upper bound min distance}
d_{R,1}(C) \leq \min \{ d_{R,1}(C_1^\prime), d_{R,1}(C_2^\prime), \ldots, d_{R,1}(C_l^\prime) \}.$$

The previous theorem also gives the following corollary for cartesian products:

\[exact grw product\] If $ C = C_1 \times C_2 \times \cdots \times C_l $ and $ 1 \leq r \leq k $, with notation as before, then $$\begin{split}
d_{R,r}(C) = \min \{ & d_{R,r_1}(C_1) + d_{R,r_2}(C_2) + \cdots + 