[proofplan] The proof is a direct passage from the parametrix identities to the quotient algebra. Ellipticity guarantees that at least one properly supported two-sided parametrix exists, and an arbitrary properly supported parametrix $Q$ has left and right composition errors with $P$ lying in the properly supported smoothing ideal. Since the quotient algebra kills precisely that ideal, these two errors vanish after taking classes. Finally, if two parametrices are chosen, both represent the same two-sided inverse in the quotient algebra, so the inverse class is independent of the representative. [/proofplan]

[step:Fix an arbitrary properly supported parametrix with smoothing remainders]Let $P \in \Psi_{\mathrm{ps}}^m(U)$ be elliptic. The [elliptic parametrix theorem for pseudodifferential operators](/theorems/7696), applied to the elliptic properly supported operator $P$ on the [open set](/page/Open%20Set) $U \subset \mathbb{R}^n$, guarantees the existence of at least one properly supported parametrix of order $-m$. Now fix an arbitrary properly supported two-sided parametrix $Q \in \Psi_{\mathrm{ps}}^{-m}(U)$ for $P$. By the definition of such a parametrix, there exist operators $R_1, R_2 \in \Psi_{\mathrm{ps}}^{-\infty}(U)$ such that \begin{align*} QP = I + R_1 \end{align*} and \begin{align*} PQ = I + R_2. \end{align*} Here $I: C_c^\infty(U) \to C_c^\infty(U)$ denotes the identity operator. The compositions $QP$ and $PQ$ are defined in $\Psi_{\mathrm{ps}}^0(U)$ because the composition theorem for properly supported pseudodifferential operators applies to the properly supported factors $Q$ and $P$ and gives order $-m + m = 0$.[/step]

[guided]The theorem has two logically separate claims: first, ellipticity supplies parametrices, and second, every properly supported parametrix represents the same inverse in the quotient. Since $P \in \Psi_{\mathrm{ps}}^m(U)$ is elliptic and properly supported on the open set $U \subset \mathbb{R}^n$, the elliptic parametrix theorem for pseudodifferential operators guarantees that at least one properly supported parametrix of order $-m$ exists. For the quotient argument, however, we must not restrict attention to one constructed parametrix. We therefore fix an arbitrary properly supported two-sided parametrix $Q \in \Psi_{\mathrm{ps}}^{-m}(U)$ for $P$. By the definition of a two-sided parametrix, there are properly supported smoothing remainders $R_1, R_2 \in \Psi_{\mathrm{ps}}^{-\infty}(U)$ satisfying \begin{align*} QP = I + R_1 \end{align*} and \begin{align*} PQ = I + R_2. \end{align*} Here $I: C_c^\infty(U) \to C_c^\infty(U)$ is the identity operator. We also need to know that the products appearing in these identities belong to the algebra whose quotient we are taking. This is where proper support is used. The composition theorem for properly supported pseudodifferential operators says that the composition of properly supported pseudodifferential operators is again properly supported and has order equal to the sum of the orders. Since $Q$ has order $-m$ and $P$ has order $m$, both $QP$ and $PQ$ have order $0$ and lie in $\Psi_{\mathrm{ps}}^0(U) \subset \Psi_{\mathrm{ps}}^*(U)$. Thus the parametrix identities are identities inside the properly supported calculus, with remainders lying in the ideal $\Psi_{\mathrm{ps}}^{-\infty}(U)$.[/guided]

[step:Pass the parametrix identities to the quotient algebra] Let \begin{align*} \pi: \Psi_{\mathrm{ps}}^*(U) \to \mathcal{A}_{\mathrm{ps}}(U) \end{align*} be the quotient map, so $\pi(A) = [A]$ for $A \in \Psi_{\mathrm{ps}}^*(U)$. Since $R_1, R_2 \in \Psi_{\mathrm{ps}}^{-\infty}(U)$, their quotient classes vanish: \begin{align*} [R_1] = [0] \end{align*} and \begin{align*} [R_2] = [0]. \end{align*} Applying $\pi$ to $QP = I + R_1$ gives \begin{align*} [Q][P] = [QP] = [I + R_1] = [I]. \end{align*} Applying $\pi$ to $PQ = I + R_2$ gives \begin{align*} [P][Q] = [PQ] = [I + R_2] = [I]. \end{align*} Therefore $[Q]$ is both a left inverse and a right inverse of $[P]$ in $\mathcal{A}_{\mathrm{ps}}(U)$. [/step]

[step:Show that the inverse class is independent of the chosen parametrix] Let $Q_1, Q_2 \in \Psi_{\mathrm{ps}}^{-m}(U)$ be two properly supported parametrices for $P$. By the previous step, \begin{align*} [Q_1][P] = [I] \end{align*} and \begin{align*} [P][Q_2] = [I]. \end{align*} Using associativity of multiplication in the quotient algebra $\mathcal{A}_{\mathrm{ps}}(U)$, we compute \begin{align*} [Q_1] = [Q_1][I] = [Q_1]([P][Q_2]) = ([Q_1][P])[Q_2] = [I][Q_2] = [Q_2]. \end{align*} Thus any two properly supported parametrices represent the same element of the quotient. Hence the inverse $[P]^{-1}$ is well-defined and is represented by any properly supported parametrix $Q \in \Psi_{\mathrm{ps}}^{-m}(U)$. [/step]