I don't know what more you need. We gave you the statement of a theorem, the proof of it, and a quite precise proof of how it applies... In saying you already knew both the theorem and the proof for the

function, you make me feel like my dedicated posts were somewhat unpredictably useless to you... I would have appreciated your being more specific.

Anyway, there's not much of another proof than the one I gave, and you can see that properties a) and b) provide the

*core *of the proof of the domination. They respectively give both parts of the dominating function. I needed a simple additional ingredient that comes from the fact that the domination must hold for

is a slab

, so that I had to replace (simply by monotonicity)

by

or

depending if

or

. Since

(and

) satisfy the same condition like

, namely

, I could use property a) for

instead of

. So a) and b) are almost the only crucial properties needed; they can help you write the proof but don't make one by themselves of course.