=+CASE 4. rs , sass. By (7.30), A(r, s) = pq + qQ (2a - 1) - pqQ(2s - 1) - pqQ(2r).

From 0 ≤ 2a-1 =r +s -1 ≤ ;, follows Q(2a -1) =pQ(4a -2); and from :≤2a - = r+s- ≤ 1, follows Q(2a - 2) = p + qQ(4a - 2). Therefore, qQ(2a - 1) = pQ(2a-})-p2, and it follows that A(r, s) = p[q - p + Q(2a - 1) - qQ(2s - 1) - qQ(2r)].
If 2s - 1 ≤ 2r, the right side here is p[(q-p)(1-Q(2r)) +A(2s-1,2r)] ≥0.
If 2r ≤ 25 - 1, the right side is p[(q -p) (1 - Q(2s -1) + A(2r, 2s -1)] = 0.
This completes the proof of (7.28) and hence of Theorem 7.3.