Let $n \in \mathbb{N}$, let $U \subset \mathbb{R}^n$ be open, let $m,m' \in \mathbb{R}$, and consider the scalar properly supported local pseudodifferential calculus on $U$ defined by left quantization, with complex-valued local symbol classes $S^s(U \times \mathbb{R}^n)$ for $s \in \mathbb{R}$. If $P \in \Psi^m_{\mathrm{prop}}(U)$ and $Q \in \Psi^{m'}_{\mathrm{prop}}(U)$, then the composition $PQ$ is a well-defined properly supported operator and satisfies $PQ \in \Psi^{m+m'}_{\mathrm{prop}}(U)$. Moreover, if $\sigma_m(P) \in S^m(U \times \mathbb{R}^n)/S^{m-1}(U \times \mathbb{R}^n)$ and $\sigma_{m'}(Q) \in S^{m'}(U \times \mathbb{R}^n)/S^{m'-1}(U \times \mathbb{R}^n)$ denote the scalar principal symbol classes, then

\begin{align*} \sigma_{m+m'}(PQ) = \sigma_m(P)\sigma_{m'}(Q) \end{align*}

in the quotient space $S^{m+m'}(U \times \mathbb{R}^n)/S^{m+m'-1}(U \times \mathbb{R}^n)$, where the product on the right is induced by multiplication of symbol representatives.