This paper presents an E -derivation analogue of a result on derivations due to van den Essen, Wright and Zhao. We prove that the image of a locally finite K - E -derivation of polynomial algebras in two variables over a field K of characteristic zero is a Mathieu subspace. This result together with that of van den Essen, Wright and Zhao confirms the LFED conjecture in the case of polynomial algebras in two variables. [ABSTRACT FROM AUTHOR]