This paper continues the investigation into Krull-style dimensions in algebraic frames. Let $L$ be an algebraic frame. $dim\left(L\right)$ is the supremum of the lengths $k$ of sequences $p_0<p_1<\cdots <p_k$ of (proper) prime elements of $L$. Recently, Th. Coquand, H. Lombardi and M.-F. Roy have formulated a characterization which describes the dimension of $L$ in terms of the dimensions of certain boundary quotients of $L$. This paper gives a purely frame-theoretic proof of this result, at once generalizing it to frames which are not necessarily...