We present a proposal for the design of the higher-order extension of CASL. For each design step, we have tried to find the best of several possible alternatives, give a motivation for the preferred alternative and argue why the other alternatives are not taken. This note discusses function spaces, product types, partial functions, predicates, subsorting, signature morphisms and \-abstraction. A forthcoming note will be devoted to polymorphism, type constructors and dependent types. A companion note, replacing L-2, will explain the formal details of the proposal made in this note.