Let $n \in \mathbb{N}$, let $U \subset \mathbb{R}^n$ be an [open set](/page/Open%20Set), and let $m,m' \in \mathbb{R}$. Let $C_c^\infty(U)$ denote the space of smooth compactly supported complex-valued functions on $U$, and let $C^\infty(U)$ denote the space of smooth complex-valued functions on $U$. For each $r \in \mathbb{R}$, let $S^r(U \times \mathbb{R}^n)$ denote the Hörmander symbol class of order $r$ on $U \times \mathbb{R}^n$, let $\operatorname{Op}(q): C_c^\infty(U) \to C^\infty(U)$ denote a chosen properly supported local quantisation of a symbol $q \in S^r(U \times \mathbb{R}^n)$, and let $\Psi^r(U)$ denote the class of properly supported pseudodifferential operators $P: C_c^\infty(U) \to C^\infty(U)$ of order $r$ that admit a representation $P = \operatorname{Op}(q) + R$ with $q \in S^r(U \times \mathbb{R}^n)$ and $R: C_c^\infty(U) \to C^\infty(U)$ a properly supported smoothing operator. Assume the properly supported local calculus satisfies the symbolic composition theorem and that properly supported smoothing operators form a two-sided ideal for composition with properly supported pseudodifferential operators. If $A: C_c^\infty(U) \to C^\infty(U)$ and $B: C_c^\infty(U) \to C^\infty(U)$ satisfy $A \in \Psi^m(U)$ and $B \in \Psi^{m'}(U)$, then the composition $A \circ B: C_c^\infty(U) \to C^\infty(U)$ is well-defined and satisfies $A \circ B \in \Psi^{m+m'}(U)$. Consequently, $\Psi^0(U)$ is closed under composition, and hence is an algebra under operator composition.