= Shadow lemma for UV-compressions
{c}
Suppose that for every $x\in U$ there is a $y\in V$ such that $\mathcal A$ is fixed by the smaller compression $C_{U\setminus\{x\},V\setminus\{y\}}$. Then
$$
\left|\partial C_{U,V}(\mathcal A)\right|\leq|\partial\mathcal A|.
$$
Indeed, deleting an element outside $U$ from a newly compressed member gives the $UV$-compression of an old shadow member. If the deleted element is $x\in U$, stability under the chosen smaller compression shows that the resulting set already belongs to the old <lower shadow>. Thus
$$
\partial C_{U,V}(\mathcal A)\subseteq C_{U,V}(\partial\mathcal A),
$$
and compression preserves cardinality.
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