This paper introduces a notion of partial type assignment on applicative term rewriting systems that is based on a combination of an essential intersection type assignment system, and the type assignment system as defined for ML \cite {Milner'78}, both extensions of Curry's type assignment system \cite {Curry-et-Feys'58}. Terms and rewrite rules will be written as trees, and type assignment will consists of assigning intersection types function symbols, and specifying the way in which types can be assigned to nodes and edges between nodes. The only constraints on this system are local: they are imposed by the relation between the type assigned to a node and those assigned to its incoming and out-going edges. In general, given an arbitrary typeable applicative term re\-writing system, the subject reduction property does not hold. We will formulate a sufficient but undecidable condition typeable rewrite rules should satisfy in order to obtain this property. Appeared as: @Inproceedings {Bakel-TLCA'93, Author = "S. van Bakel", Title = "Partial {I}ntersection {T}ype {A}ssignment in {A}pplicative {T}erm {R}ewriting {S}ystems", Booktitle = "Proceedings of TLCA '93. International Conference on Typed Lambda Calculi and Applications, {\rm Utrecht, the Netherlands}", Editor = "M. Bezem and J.F. Groote", Series = "Lecture Notes in Computer Science", Volume = "664", Publisher = "Springer-Verlag", Year = "1993", Pages = "29-44" }