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		10.34 Claim .

		Proof: Suppose by way of contradiction that some scientist M ApaEx*-identifies . We show that the existence of such an M implies that is Ex*-identifiable, contradicting Corollary 6.11. First, let p* be a program such that

		Clearly, for each , we have that and . Hence,  j p* is 1-conforming with each member of . Next, for each f, define

		So, if , then . Also let r be a recursive function such that, for all i and j,  j r(i)(j) =  j i(min(Sj)). So if  j i = f', for some , then  j r(i) = f. Finally, let M' be a scientist such that, for each f,

		As the reader may check, it follows from our assumption on M and the above definitions that M' must . Thus we have the desired contradiction.

		10.35 Claim Suppose . Then .

		Proof: Let p be a recursive function such that, for each s and x,

		Also, let M be a scientist such that, for all s and  s , M(s,  s ) = p(s).

		Fix some and suppose that s is such that  j s is uniformly d-conforming to f. Then it easily follows that, for each j, , and hence,  j p(s) = f. Therefore, M UapdEx-identifies f.

		Let us contrast the preceding result with the following proposition, showing that Uap-type additional information cannot, in general, compensate for a higher density Ap-type additional information. The proof of this proposition is left to Exercise 10-18.