Idempotent-consistent semigroups are defined by the property that each idempotent in a homomorphic image of a semigroup has an idempotent pre-image. In a manner this property is an another formulation for a well known Lallement's lemma. In this paper on an arbitrary semigroup we introduce a system of congruence relations and using them we give a new version of the proof of Lallement's lemma.