I am currently reading Kashiwara and Schapira's Sheaves on Manifolds, and in particular I'm in the section about the microlocalization. Right after this is defined, a theorem (4.3.2) containing a long list of properties of the microlocalization is stated without proof, and it is claimed to be a consequence of Theorem 4.2.3 (which in fact at first glance looks like an analogous list of properties of the specialization functor) and other results about the Fourier-Sato transform. I'm trying to understand how this proof should go, but it seems like I can't succeed. For example, for part (ii): it looks like they want to use Proposition 3.7.12 (ii) to pass from sections of $\mu_M(F)$ on a convex open cone $V$ to global sections supported on the polar set of $V$ of $\nu_M(F)$, and then use Theorem 4.2.3 (iii) to get the thesis, but the problem is that the polar set of a convex open cone has no reason to be closed (actually I might even have counterexamples for that)... So what am I missing? Can anybody help me? 

In general, it would also be appreciated if someone could give some intuition on what the microlocalization is doing (even without going too much into analysis, if possible), maybe with some examples of computations.