[proofplan] We invoke the Björner-Wachs recursive atom ordering theorem: a finite bounded graded poset equipped with a rooted recursive atom ordering admits a CL-labeling. The hypotheses in the theorem statement are exactly the rooted recursive atom-ordering hypotheses needed for that result. Once the CL-labeling is obtained, CL-shellability follows from the definition of CL-shellability for bounded graded posets. [/proofplan]

[step:Verify that the hypothesis is a rooted recursive atom ordering]Let $\hat{0}$ and $\hat{1}$ denote the minimum and maximum elements of $P$. A rooted interval means a pair $([x,y],\rho)$, where $x\le y$ in $P$ and $\rho$ is a saturated chain from $\hat{0}$ to $x$. For every rooted upper interval $([x,\hat{1}],\rho)$, the theorem hypothesis supplies a linear order of the atoms of $[x,\hat{1}]$; write this order as \begin{align*} b_1,b_2,\dots,b_m. \end{align*} For a rooted interval $([x,y],\rho)$, its atoms are precisely those $b_i$ satisfying $b_i\le y$, ordered by the same inherited order. The recursive hypotheses in the statement have two parts. First, after choosing an atom $b_j$ of $[x,\hat{1}]$, the order restricts recursively to the rooted upper interval $([b_j,\hat{1}],\rho\cup\{b_j\})$. Second, if $i<j$ and $v$ is a common upper bound of $b_i$ and $b_j$, then there is an atom $z$ of $[b_j,\hat{1}]$ with $z\le v$ and $b_k\le z$ for some $k<j$, and the rooted atom order above $b_j$ puts all atoms lying above earlier atoms before all atoms not lying above earlier atoms. These are exactly the Björner-Wachs rooted recursive atom-ordering axioms.[/step]

[guided]The point of checking rooted intervals is that Björner-Wachs recursive atom orderings are not just ordinary atom orders on intervals. The order of atoms above $x$ may depend on the saturated chain $\rho$ by which the interval is rooted at $x$. Fix a rooted interval $([x,y],\rho)$. The atom order on $[x,y]$ is inherited from the rooted upper interval $([x,\hat{1}],\rho)$ by keeping exactly those atoms below $y$. The recursive part of the hypothesis says that, after choosing an atom $b_j$, the same structure is present in the smaller rooted upper interval $([b_j,\hat{1}],\rho\cup\{b_j\})$. The priority condition is also part of the hypothesis: whenever an earlier atom $b_i$ and a later atom $b_j$ have a common upper bound $v$, there is an atom $z$ above $b_j$ with $z\le v$ that is already visible from an earlier atom $b_k$ with $k<j$, and the rooted order above $b_j$ places all such earlier-visible atoms first. Thus the data supplied in the theorem statement is precisely a rooted recursive atom ordering in the sense used by Björner and Wachs.[/guided]

[step:Apply the Björner-Wachs recursive atom ordering theorem] We use the following external result of Björner and Wachs: if a finite bounded graded poset is equipped with a rooted recursive atom ordering, then the Björner-Wachs construction gives a CL-labeling of that poset. In that CL-labeling, a maximal chain in a rooted interval is called rising when its label word is weakly increasing in the label order. The hypotheses of this external result match the present theorem. The poset $P$ is finite, bounded by $\hat{0}$ and $\hat{1}$, and graded by the stated rank function. The preceding step verifies that the orders assigned to the rooted upper intervals are exactly a rooted recursive atom ordering. Therefore the Björner-Wachs theorem gives a CL-labeling of $P$. [/step]

[step:Extract the CL condition supplied by the theorem]A CL-labeling is a chain-edge labeling for which every rooted interval has a unique rising maximal chain, and that rising chain is lexicographically first among all maximal chains in the rooted interval. Here rising means weakly increasing with respect to the label order. The Björner-Wachs theorem applied in the preceding step supplies exactly such a labeling for $P$.[/step]

[guided]The crucial point is that we are not comparing local integer positions from unrelated rooted intervals. Such a comparison would not be justified: the first atom in one rooted atom order and the first atom in a later rooted atom order need not have numerically comparable positions in any global sense. The Björner-Wachs theorem avoids this problem by using its own recursive chain-edge labeling construction and proves directly that the recursively selected chain is the unique weakly rising chain and is lexicographically first in every rooted interval. The theorem applies here because $P$ is finite, bounded, and graded, and because the preceding step checked the rooted recursive atom-ordering axioms. Therefore, for every rooted interval $([x,y],\rho)$, the CL-labeling supplied by Björner and Wachs has a unique weakly rising maximal chain in $[x,y]$, and that chain precedes every other maximal chain in lexicographic order.[/guided]

[step:Conclude that $P$ is CL-shellable] The previous step gives a CL-labeling of $P$. By the definition of CL-shellability, a finite bounded graded poset admitting a CL-labeling is CL-shellable. Therefore $P$ is CL-shellable. [/step]