Lemma 3.3, p. 16 — along ≿ → ≿₀ ↠ ⊵ the menu-wise symmetric differences split disjointly

Let $X$ be a finite set and let $\succsim,\succsim_0,\trianglerighteq$ be acyclic orders on $X$ with $\succsim\to\succsim_0\twoheadrightarrow\trianglerighteq$ (that is, $\succsim_0$ is a one-step perturbation of $\succsim$ toward $\trianglerighteq$). Then for every $S\subseteq X$ the sets

are disjoint, and their union equals $M(S,\succsim)\triangle M(S,\trianglerighteq)$.

The lemma says that a one-step perturbation toward $\trianglerighteq$ never moves the set of maximal elements of a menu away from that of $\trianglerighteq$; summed over menus it gives the additivity (Axiom 1′) of every $D^\mu$.

Formalization Note $M(S,\cdot)$ is maxSet S, with S ranging over all finite subsets of $X$.