 Shadow lemma for UV-compressions

Suppose that for every $x\in U$ there is a $y\in V$ such that $\mathcal{A}$ is fixed by the smaller compression $C_{U\setminus \{x\},V\setminus \{y\}}$. ThenIndeed, 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. Thusand compression preserves cardinality.