= Solution
For an action <sequence> $\pi$, let $T_\pi(B)$ be the <set> of all physical states obtainable by applying it to the initial <belief state> $B$. Under a deterministic model this is an image; for nondeterministic outcomes it is the <union> of all permitted images. For $B'\subseteq B$,
$$
T_\pi(B')\subseteq T_\pi(B).
$$
If each action is applicable to every current possibility from $B$, it is also applicable to every possibility from $B'$. If the final <belief state> for $B$ is contained in the physical goal <set> $G$, then $T_\pi(B')\subseteq G$. Thus \b[the same <plan> solves every <subset> of $B$].
Nothing comparable follows for an arbitrary <superset>. For example, let an action take $s_1$ to a goal and leave $s_2$ at a non-goal. It solves $\{s_1\}$ but not $\{s_1,s_2\}$. A larger <belief state> is solved by this particular <sequence> exactly when all added states also permit it and end in $G$. It may succeed on some <supersets> and fail on others; it cannot be guaranteed merely from success on $B$.
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