I can't verify an inequality in the paper: https://link.springer.com/content/pdf/10.1007/BF02110361.pdf
The inequality is at the end of the second page in the proof of the second theorem and is shown in this image: inequality
It is in the proof of showing that special flow over a transformation with an ND approximation is not mixing. The author makes a sequence that contradicts mixing. I tried various or probability inequalities, and considering the set $A$ as a long rectangle. This gives me the inequality: $\nu (T_{t(j)} A\cap A\cap Y_j')\geq \nu (A \cap Y_j')- \nu((T_{t(j)} A \cap Y_j')$ but I'm not sure how to proceed, or if this is the first step.
Thanks for your help.