Let $X$ be a smooth manifold, let $m \in \mathbb{R}$, and let $A \in \Psi_h^m(X)$ be a properly supported semiclassical pseudodifferential operator with full symbol
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\begin{align*}
a \in S_h^m(T^*X)
\end{align*}
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and elliptic principal symbol
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\begin{align*}
a_m: U \to \mathbb{C}
\end{align*}
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on an open set $U \subset T^*X$. Thus, for every compact set $K \subset U$, there are constants $c_K > 0$ and $h_K > 0$ such that
for every $\rho = (x,\xi) \in K$ and every $0 < h < h_K$.
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Then, for every open set $V \subset T^*X$ such that $\overline{V}$ is compact and $\overline{V} \subset U$, there exists a properly supported operator $B \in \Psi_h^{-m}(X)$ such that
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\begin{align*}
BA = I + R_L
\end{align*}
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microlocally on $V$, and
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\begin{align*}
AB = I + R_R
\end{align*}
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microlocally on $V$, where $R_L,R_R \in h^\infty \Psi_h^{-\infty}(X)$ microlocally on $V$. The principal symbol of $B$ on $V$ is $a_m^{-1}$.