I don't understand this part of the proof on page 16 of the following paper.
http://www.maths.ed.ac.uk/~aar/papers/forman5.pdf
2026-03-26 07:44:13.1774511053
The main theorem of discrete Morse theory.
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By the assumption of the theorem, there are no critical values between a,b. But notice that according to the negation of (i) which you quoted, it follows that $\tau$ is a critical cell. Since its value is greater than that of $\sigma$ and it does not belong to [a,b], it has to be greater than b.
To see the deduction from Lemma 3.2, you can use induction on the dimension of $\tau$. Finally, the intersection is empty by the definition of $M(a)$ (Def. 3.1).