Let $U \subset \mathbb{R}^n$ be open. Let $(m_j)_{j=0}^{\infty}$ be a strictly decreasing sequence of [real numbers](/page/Real%20Numbers) with $m_j \to -\infty$, and for each $j \in \mathbb{N} \cup \{0\}$ let $a_j \in S^{m_j}_{1,0}(U \times \mathbb{R}^n)$.
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Then there exists a symbol $a \in S^{m_0}_{1,0}(U \times \mathbb{R}^n)$ such that, for every $N \in \mathbb{N} \cup \{0\}$,
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\begin{align*}
a - \sum_{j=0}^{N-1} a_j \in S^{m_N}_{1,0}(U \times \mathbb{R}^n),
\end{align*}
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with the convention that the empty sum is $0$ when $N=0$. Equivalently,
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\begin{align*}
a \sim \sum_{j=0}^{\infty} a_j
\end{align*}
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as a symbol asymptotic expansion.
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Moreover, if $a,b \in S^{m_0}_{1,0}(U \times \mathbb{R}^n)$ both satisfy
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\begin{align*}
a \sim \sum_{j=0}^{\infty} a_j
\end{align*}