Blow up of mean curvature in mean curvature flow

Question

When I read the proof of Theorem 1.1 of Blow-up of the mean curvature at the first singular time of the mean curvature flow I can't understand why there are only two things are becoming strictly mean convex or into iterative process. In fact, from the 3.2 3.3 3.4 3.5, I think we always can go into the iterative process when the initial condition is meet. The mean convex is redundant, since the second situation makes it does not blow up. How to understand it ?