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		help show , we use the s-m-n theorem (Theorem 2.1) to define two limiting-recursive functions g and h as follows. For each i and x define:

		Since  l i.Ti is limiting recursive, it follows that one can construct a limiting-recursive g as above. For each , define

		and, for each i > k, define

		Since  l i.con(i, k), f, and g are limiting-recursive, it follows that one can construct a limiting-recursive h as above. (Note: One cannot, in general, limiting-recursively decide whether  Q k( j f(i)) is total. But this is not a difficulty since (13.1) affects only finitely many values of h.)

		13.9 Claim

		(a) For each i, if con(i, k), then  s g(i) is consistent with  Q k( j f(i)).

		(b) For each , . Thus, .

		(c) For each j > i, .

		Proof: By Claim 13.6(C) for , we have that con(i, k) implies

		Since , part (a) follows from the definition of g.

		Part (b) follows from part (a) and definition of h.

		For part (c), first note that, for , we have by (13.1) that  j h(i) is total, and hence, for such , part (c) clearly holds. So assume i > k and suppose  j h(i) ~  j h(j). By