The AGM framework is the benchmark approach in belief change. Since the framework assumes an underlying logic containing classical Propositional Logic, it can not be applied to systems with a logic weaker than Propositional Logic. To remedy this limitation, several researchers have studied AGM-style contraction and revision under the Horn fragment of Propositional Logic (i.e., Horn logic). In this paper, we contribute to this line of research by investigating the Horn version of the AGM entrenchment-based contraction. The study is challenging as the construction of entrenchment-based contraction refers to arbitrary disjunctions which are not expressible under Horn logic. In order to adapt the construction to Horn logic, we make use of a Horn approximation technique called Horn strengthening. We provide a representation theorem for the newly constructed contraction which we refer to as entrenchment-based Horn contraction. Ideally, contractions defined under Horn logic (i.e., Horn contractions) should be as rational as AGM contraction. We propose the notion of Horn equivalence which intuitively captures the equivalence between Horn contraction and AGM contraction. We show that, under this notion, entrenchment-based Horn contraction is equivalent to a restricted form of entrenchment-based contraction.