Lemma 15. Assuming that:

• $U,V\subset X$

• $|U|=|V|$

• $U\cap V=\varnothing$

• $\max U<\max V$

• $A\subset X^{(r)}$

• $$\forall u\in U \exists v\in V\text{ such that }A\text{ is }(U-u,V-v)\text{-compressed}\text{(∗)}$$