I must apologize in advance for the mathematical gaps in setting up the notation and for being unable to provide a copy of the literature pertaining to this question.
I have the following questions :
Q1. I am having trouble understanding the proof of Theorem 9.7.10 of the book by Kashiwara-Schapira titled Sheaves on manifolds. Let us recall some notations. Let $X$ be a real analytic manifold and let $\mathrm{D}^b_{\mathbb{R}-c}(X)$ denote the derived category of cohomologcally $\mathbb{R}$-constructible objects. Let $\mathrm{K}_{\mathbb{R}-c}(X)$ be the Grothendieck group associated to the derived category $\mathrm{D}^b_{\mathbb{R}-c}(X)$. Let $\mathcal{L}_X$ be the sheaf of Lagrangian cycles. Let $\pi : T^*X \rightarrow X$ be the cotangent bundle of $X$. Then there is the following map
$$\mathrm{CC} : \mathrm{K}_{\mathbb{R}-c}(X) \rightarrow \mathrm{H}^0(T^*X, \mathcal{L}_X).$$
Then Theorem 9.7.10 from the book Sheaves on manifolds states that : The map $\mathrm{CC}$ is an isomorphism.
I have trouble understanding the part (a) of the proof. To be more precise given a Lagrangian cycle $\lambda$ with support $\Lambda$ a closed conic isotropic subset, why is it possible to find $F \in \mathrm{D}^b_{\mathbb{R}-c}(X)$ such that for a suitably chosen subanalytic submanifold $Y$ of $X$ (Here $Y$ is such that $Y$ is a dense open subset of $\pi(\Lambda)$ such that $\Lambda \cap \pi^{-1}(Y) \subset T^*_YX$) we have $\mathrm{CC}(F) = \lambda$ on $\pi^{-1}(Y)$.
Q2. Example of a Lagrangian cycle such that it is not the characteristic cycle of any $\mathbb{R}$-constructible sheaf?