  We investigate categoricity of abstract elementary classes without any
remnants of compactness (like non-definability of well ordering, existence of
E.M. models or existence of large cardinals). We prove (assuming a weak version
of GCH around lambda) that if K is categorical in lambda, lambda^+, LS(K) <=
lambda and 1 <= I(lambda^{++},K)< 2^{lambda^{++}} then K has a model in
lambda^{+++} .
