  This paper provides an E-theoretic proof of an exact form, due to E.
Troitsky, of the Mischenko-Fomenko Index Theorem for elliptic
pseudodifferential operators over a unital C*-algebra. The main ingredients in
the proof are the use of asymptotic morphisms of Connes and Higson, vector
bundle modification, a Baum-Douglas-type group, and a KK-argument of Kasparov.
