A binding-time analysis is correct if it always produces consistent binding-time information. Consistency prevents partial evaluators from "going wrong". A sufficient and decidable condition for consistency, called well-annotatedness, was first presented by Gomard and Jones. In this paper we prove that a weaker condition implies consistency. Our condition is decidable, subsumes the one of Gomard and Jones, and was first studied by Schwartzbach and the present author. Our result implies the correctness of the binding-time analysis of Mogensen, and it indicates the correctness of the core of the binding-time analyses of Bondorf and Consel. We also prove that all partial evaluators will on termination have eliminated all "eliminable"-marked parts of an input which satisfies our condition. This generalizes a result of Gomard. Our development is for the pure -calculus with explicit binding-time annotations. 1 Introduction A partial evaluator is an implementation of Kleene's S m n theorem....