be a properly supported scalar classical pseudodifferential operator of order $m\in\mathbb R$. Fix one local quantisation convention and use it in every coordinate chart.
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Let $(U,x)$ and $(V,y)$ be coordinate charts with $U\cap V\neq\varnothing$, and write the transition map as
\begin{align*}
a:x(U\cap V)\times\mathbb R^n\to\mathbb C
\end{align*}
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and
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\begin{align*}
b:y(U\cap V)\times\mathbb R^n\to\mathbb C
\end{align*}
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are complete local symbols for $A$ modulo $S^{-\infty}$ in the $x$- and $y$-coordinates, respectively, and let $a_m$ and $b_m$ denote their homogeneous degree-$m$ principal components. Then, for every $x_0\in x(U\cap V)$ and every $\eta\in\mathbb R^n\setminus\{0\}$,