Theorem 1

Let $\overline{\omega}_1, \ldots, \overline{\omega}_l$ be the fundamental weights corresponding to the chamber $C$, and, for $1 \leq i \leq l$, let $x_i$ be an element of $A[P]^W$ with $e^{\overline{\omega}_i}$ as its unique maximal term. Let

be the homomorphism from the polynomial algebra $\Lambda[X_1, \ldots, X_l]$ to $A[P]^W$ that takes $X_i$ to $x_i$. Then, the map $\varphi$ is an isomorphism.

Lemma 2 implies that the image under $\varphi$ of the monomial $X_1^{n_1} \ldots X_l^{n_l}$ is an element with unique maximal term $e^{n_1 \overline{\omega}_1 + \cdots + n_l \overline{\omega}_l}$. Since every element of $P \cap \overline{C}$ can be written uniquely in the form $n_1 \overline{\omega}_1 + \cdots + n_l \overline{\omega}_l$, Prop. 3 shows that the images under $\varphi$ of the monomials $X_1^{n_1} \ldots X_l^{n_l}$ are a basis of $A[P]^W$, hence the theorem.