I am confused about the covering lemmas in the captioned work and really hope to get some ideas here.

Firstly it is lemma 3.7. ([Image of Lemma 3.7][1]) (for convenience here is the lemma of this lemma ([Image of Lemma 3.6][2]))

I do not understand the ''$\supseteq$'' in the line 
''$J'\cap(J'-\ell)\supseteq \cdots$''. Also I am wondering if the following provides a counter-example of this line and this lemma:

Let $n=12$, $I=\{0,1,2,3,4,5,6,7,8\}$, $m=3$. Then $I'=\{0,4,8\}$ by the construction in Lemma 3.6. Let $J=\{0,2,4,6,8,10,12\}$, $\delta=0.5$. Then for all $i\in I$, $|[i,i+3]\cap J|=2\geq(1-\delta)m$. Take $\ell=1$. Then for all $J'\subseteq J$, we have $J'\cap(J'-\ell)\subseteq J\cap(J-\ell)=\emptyset$, $\bigcup_{i\in I'} J\cap[i,i+2]=\{0,2,4,6,8,10\}$, and $(1-\delta-\frac{\ell}{m})|I|=(\frac{1}{2}-\frac{1}{3})\cdot 9 >0$. 

Secondly, it is lemma 3.8.([Excerpt for Lemma 3.8][3]), which I am not sure how the marked inequality is obtained.

Since subsequent results depends on these lemmas and this paper is already acknowledged in the field, I think there should be an answer for resolving my confusion. 

Thanks for help!

  [1]: https://i.sstatic.net/Un5Xb.png
  [2]: https://i.sstatic.net/VU2jE.png
  [3]: https://i.sstatic.net/I9qyd.png