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		good. The reader is directed to the exercises and Baliga, Jain, and Sharma [12] for more results.

		The following lemma gives information about the parameters a, b, c, and d, for which .

		8.50 Lemma Let a, b, c, be given. Suppose r, are such that the following hold: , , and . Then, .

		Proof: Let . Let C be the collection of functions that satisfy the following three conditions.

		1. .

		2. For each , the set {} is finite and .

		3. For each and x, y, , f(<i, < x, y>>) = f(<i, <x, z>>).

		For any a-noisy text G for , since 2r > a, there exist an x < r and an such that . Thus, using clauses 2 and 3 in the definition of C, it is easy to see that . The proof of is fairly complex and we direct the reader to Baliga, Jain, and Sharma [12] for the proof.

		Among other corollaries to this lemma we have the following.

		8.51 Corollary Let a, b, be such that and . Then, .

		Proof: Take r = c and  a  = b in the above lemma.

		8.52 Corollary Let a, b, be such that and . Then, .

		Proof: Take r =  a  = c in the above lemma.

		8.53 Corollary Let a, b, c be such that and . Then, .

		Proof: Take r =  a  = b in the above lemma.