How to cancel a linearly ordered exponent

In this paper, XY denotes the set of monotone functions from the partially ordered set Y into the partially ordered set X, ordered by everywhere inclusion. Several cancellation and refinement theorems for this operation on po-sets were announced in [2], and will be proved in [3]. The purpose of this short paper is to establish the following theorem. The proof is a good example of the circle of ideas elaborated more fully in [3].