In what follows, by retracing function we will always mean special retracing function: a partial recursive function f whose range is contained in its domain and which is associated with a unique point b such that for each x in the domain of f, f(x) x and for some n, fn (x) = b. Suppose R were a recursively enumerable set which was the union of n immune retraceable sets with n as small as possible. Then R must be infinite by definition of immunity. Let R=S1U .. USn where each Si is retraced by partial recursive function fi to basepoint bi. We observe that if T is any infinite recursive subset of R, for each i, Tn'Si is immune retraceable. For let B= {bi, b2, * * , bn} and T* = TUB. Then T* = S* U ... US* wrhere St = T*flSi. Now St is retraceable for it is retraced by the function f* where domain ft = T n Dom f, and xEDom f =>ft (x) is the largest element y of T* such that for some m f7(x) =y. Hence St could only be immune or finite, but the minimality of n insures that it must be immune. But B is finite so our conclusion holds without adjunction of B. Next, we use the previous observation to show that the intersection of the domains of the fi is infinite. For if this were not the case we could choose a largest subcollection of the fi whose domains had infinite intersection and enumerate the intersection of their domains. Tlhis infinite recursively enumerable subset of R contains only finitely imany members of the domain of some fj not in the subcollection hence only finitely many members of the corresponding Sj. But this set contains an infinite recursive subset with finite intersection with Sj contradicting our observation. Since the intersection of the domains of thefi is infinite recursively enumerable we may choose an infinite recursive subset of this intersection which (by our observation) is the union of n immune retraceable sets; or equivalently, assume that R is recursive and each fi has R as domain. If, for some n, fn(x) =y we will write i: x->y. If this happens for no n, we write i: x-f-y. We notice that if yE Sj then {z|j:zy} is finite. For if it were infinite, since if it is disjoint from Sj by the defi-