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		Proof: For each , let Lf denote the single-valued total language: . For each , let denote the collection of single-valued total languages .

		Let p1 be a recursive function that translates an index for a (partial) recursive function  y  into an index for the corresponding single-valued language L y . Formally, for all i, ,

		Similarly, let p2 be a recursive function that translates an index for a language L into an index for a partial recursive function represented by L. Formally, for all i, ,

		In the first condition of the above definition of  j p2(i)(x), we assume that the z used is the first appropriate one produced by some uniform enumeration of Wi.

		Let and let . It is easy to verify that if i is an index for an a-variant of f such that i does not make any convergent errors (with respect to f), then pl (i) is an index for an a-variant of the single-valued total language Lf. Similarly, if j is an index for an a-variant for the single-valued total language L, then p2(j) is an index for an a-variant of the recursive function represented by L.

		Choose an . (By Proposition 6.5, such an S exists.) We argue that .

		Claim: . Proof: Let M be a scientist that searches a text for an element of the form and keeps on emitting p1(z'), where z' is a program obtained by patching the convergent errors of z with respect to the function represented by the input language. Clearly, M TxtExm+1-identified . Hence, the claim follows.

		Claim: . Proof: Suppose by way of contradiction that a scientist M TxtExm-identifies . Let be such that t((x, y)) = <x, y> and t(#) = #. Extend t to by taking t( s ) to be the result of applying t to each element of  s . Now, let M' be a scientist such that, for each segment  s , M'( s ) = p2(M(t( s ))). Since M TxtExm-identifies , it follows that M' must Exm-identify S, a contradiction. Hence, the claim and part (a) follows.

		A similar proof works for part (b) using ASD* of Corollary 6.6.

		6.20 Corollary .