Prove that A_{i j} occurs if and only if the first pivot chosen from \ell_{i}, \ell_{i+1}, \ldots, \ell_{j} is either \ell_{i} or \ell_{j}.
Prove that A_{i j} occurs if and only if the first pivot chosen from \ell_{i}, \ell_{i+1}, \ldots, \ell_{j} is either \ell_{i} or \ell_{j}.