What the heck does this definition of a regular partition $P$ of $R$ mean? I follow what it is saying until we get to the last part, "the $k_1\cdot k_2\cdot \cdots \cdot k_n$ subrectangles of the form $(a_{1,j_1},a_{1,j_i+1})\times(a_{2,j_2},a_{2,j_2+1})\times\cdots\times(a_{n,j_n},a_{n,j_n+1})$ with $0\le j_i\le k_i$ for $1\le i\le n$", at which point I am completely befuddled.