We study D0L sequences and their equality sets. If s = (s(n))n≥0 and t = (t(n))n≥0 are D0L sequences, their equality set is defined by E(s, t) = {n ≥ 0 | s(n) = t(n)}. It is an open problem whether such equality sets are always eventually periodic. Using methods developed by Ehrenfeucht and Rozenberg we show that a D0L equality set is eventually periodic if it contains at least one infinite arithmetic progression. As a main tool we use elementary morphisms introduced by Ehrenfeucht and Rozenberg.