Thus, if we take the first N 1 columns (documents) in X, it is possible to rearrange the rows of X with the result that there is some W 1 such that rows with index greater than W 1 have only zeroes in the columns up to N 1 .
we take a subset of N 1 documents and enumerate the words in such a way that the terms occurring in the first N 1 documents are enumerated 1, .