The question is about this problem: (Zero XOR Subset)-less.

The problem's editorial describes the solution as follows:

First let . If we divided the array into k segments, where the right boundaries of these segments are (in increasing order) i₁, i₂, …, i_k (i_k must be n), Every segments subset-xor can be represented as pr[i_j] subset-xor. Therefore besides maximizing k, we want to guarantee that the any pr[i_j] non-empty subset has a non-zero XOR-sum. So we calculate the basis size of pr[1], pr[2], …, pr[n] numbers (binary vectors) under GF(2). Because the basis size will represent the maximum k such that all pr[i_j] are linearly independent under GF(2) (no pr[i_j] non-empty subset has a zero XOR-sum).

My question is: from the calculated basis size k, we know that we can choose k linearly independent (under GF(2)) pr[i_j] values, but how do we guarantee that we can choose these values such that i_k = n (as the last right boundary of our chosen segments has to be n)?

Thanks in advance.