I have a simple question. Let $A=\{a,b,c,...\}$ be a set and $>$ a total strict order on $2^A$. Total strict order means that for any two subsets of $A$, say $S$ and $S'$, either $S>S'$ or $S'>S$ but not both.
The order has the following property (sometimes called substitutability). For any subsets $S,S'\subset 2^A$, this is any two sets of elements of $A$ for which $S>S'$,
$x \in C(S \cup \{x\}$) implies $x \in C(S' \cup \{x\}$)
where $C(B) \subseteq B$ denotes the unique best subset chosen by $>$ when only $B$ is available (it is unique because $>$ is strict, i.e. no indifferences are allowed).
A paper I am reading claims that it is obvious that, if I have $a>b>c$, it follows that $\{a,b\} > \{a,c\}$, given that $>$ is substitutable.
Can somebody explain me why is this obvious? I just don't see it. Thanks. The paper is here: people.hss.caltech.edu/~fede/published/echen-oviedo-TE.pdf. The question is in Example 6.8.
Edit: Or why the following order does not satisfy the substitutability condition: $\{a\} > \{a,c\} > \{a,b\} > \{a,b,c\} > \{b\} > \{b,c\} > \{c\}$