(2) For [n.sup.2/3] [log.sup.3] n [is less than or equal to] m [is less than or equal to] n, t is at most 12[m.sup.3]/[n.sup.2] wvhp.

(3) For m [is less than or equal to] [n.sup.2/3] [log.sup.3] n, t is at most 12 [log.sup.9] n wvhp.

Since by Part (2) of Lemma 2.2.3, the number of nonsingletons in a random bag from [D.sub.m,n] is at most 4[m.sup.2]/n wvhp, when the experiment terminates all the m balls have been thrown wvhp.

Therefore, t' (and hence t) is at most 12[m.sup.3]/[n.sup.2] wvhp, establishing Part (2) of the lemma.

By Part (2), t [is less than or equal to] 12 [log.sup.9] n wvhp, establishing Part (3) of the lemma.

In Alg2(n, l, c), if i [is greater than or equal to] l, [s.sub.i-l][t.sub.i-1]/[t.sub.i-l] [is greater than or equal to] 2[n.sup.2/3] [log.sup.3] n and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], then wvhp,

Similarly, we can prove the desired lower bound on [s.sub.i] wvhp using the lower bound in Lemma 2.3.1.

In Alg2(n, l, c), if [s.sub.i-l][t.sub.i-1]/[t.sub.i-l] [is greater than or equal to] 2[n.sup.4/5] [log.sup.3] n and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], then wvhp,

In Alg2(n, l, c), if [t.sub.i-1] [is greater than or equal to] [log.sup.2] n, then [s.sub.i] [is less than or equal to] 3[s.sub.i-l][t.sub.i-1/[t.sub.i-l] wvhp. If [t.sub.i-1] [is less than or equal to] [log.sup.2] n, then [s.sub.i] [is less than or equal to] 3[s.sub.i-l]([log.sup.2] n)/[t.sub.i-l] wvhp.

If [t.sub.i-1] [is greater than or equal to] [log.sup.2] n, then we apply Lemma 2.1.4 with (s, t, s') = ([s.sub.i-l], [t.sub.i-l], 3[s.sub.i-l][t.sub.i-1]/[t.sub.i-l]) to establish that [s.sub.i] [is less than or equal to] 3[s.sub.i-l][t.sub.i-1]/[t.sub.i-l] wvhp. Similarly, Pr[[s.sub.i] [is less than or equal to] 3[s.sub.i-l]([log.sup.2] n)/[t.sub.i-l]] is equal to the probability that more than [t.sub.i-1] elements are selected from [T.sub.i-l] in a random selection of 3[s.sub.i-l]([log.sup.2] n)/[t.sub.i-l] elements from [s.sub.i-l].

Let condition C be such that there exist integers d and u satisfying d [is less than or equal to] m' [is less than or equal to] u wvhp.

(1) If d, u [is greater than or equal to] [n.sup.2/3] [log.sup.3] n, then [Delta]f(d) [is less than or equal to] t' [is less than or equal to] [Delta]f(u) wvhp.