[proofplan] The proof is a direct verification from the defining seminorm estimates for $S_\delta(m)$. Differentiating a symbol simply shifts the multi-indices in the defining estimate, and the fixed shifted order contributes the factor $h^{-\delta(|\alpha|+|\beta|)}$. For products, the Leibniz formula expresses each derivative of $ab$ as a finite sum of products of derivatives of $a$ and $b$; the two symbol estimates multiply, and the powers of $h$ combine to the required delta loss. [/proofplan]

[step:Unpack the symbol estimates to be proved] We use the standard defining estimate for $S_\delta(m)$. A function $a:T^*\mathbb{R}^n \times (0,1] \to \mathbb{C}$ belongs to $S_\delta(m)$ precisely when, for every pair of multi-indices $\gamma,\eta \in \mathbb{N}_0^n$, there is a constant $C_{\gamma,\eta} > 0$ such that, for every $(x,\xi) \in T^*\mathbb{R}^n$ and every $h \in (0,1]$, \begin{align*} |\partial_x^\gamma \partial_\xi^\eta a(x,\xi;h)| \leq C_{\gamma,\eta} h^{-\delta(|\gamma|+|\eta|)} m(x,\xi). \end{align*} For a real number $\rho$, the notation $h^\rho S_\delta(m)$ means that $f \in h^\rho S_\delta(m)$ exactly when $h^{-\rho}f \in S_\delta(m)$. Equivalently, $f \in h^\rho S_\delta(m)$ if for every $\gamma,\eta \in \mathbb{N}_0^n$ there is a constant $C_{\gamma,\eta} > 0$ such that \begin{align*} |\partial_x^\gamma \partial_\xi^\eta f(x,\xi;h)| \leq C_{\gamma,\eta} h^{\rho-\delta(|\gamma|+|\eta|)} m(x,\xi). \end{align*} [/step]

[step:Shift the multi-indices to prove stability under differentiation] Fix multi-indices $\alpha,\beta \in \mathbb{N}_0^n$, and define $c_{\alpha,\beta}:T^*\mathbb{R}^n \times (0,1] \to \mathbb{C}$ by \begin{align*} c_{\alpha,\beta}(x,\xi;h) = \partial_x^\alpha \partial_\xi^\beta a(x,\xi;h). \end{align*} Let $\gamma,\eta \in \mathbb{N}_0^n$. Since $a \in S_\delta(m_1)$, the defining estimate applied with multi-indices $\alpha+\gamma$ and $\beta+\eta$ gives a constant $C_{\alpha+\gamma,\beta+\eta} > 0$ such that \begin{align*} |\partial_x^\gamma \partial_\xi^\eta c_{\alpha,\beta}(x,\xi;h)| = |\partial_x^{\alpha+\gamma}\partial_\xi^{\beta+\eta}a(x,\xi;h)|. \end{align*} Therefore \begin{align*} |\partial_x^\gamma \partial_\xi^\eta c_{\alpha,\beta}(x,\xi;h)| \leq C_{\alpha+\gamma,\beta+\eta} h^{-\delta(|\alpha+\gamma|+|\beta+\eta|)} m_1(x,\xi). \end{align*} Using $|\alpha+\gamma|=|\alpha|+|\gamma|$ and $|\beta+\eta|=|\beta|+|\eta|$, this becomes \begin{align*} |\partial_x^\gamma \partial_\xi^\eta c_{\alpha,\beta}(x,\xi;h)| \leq C_{\alpha+\gamma,\beta+\eta} h^{-\delta(|\alpha|+|\beta|)} h^{-\delta(|\gamma|+|\eta|)} m_1(x,\xi). \end{align*} This is exactly the defining estimate for $c_{\alpha,\beta} \in h^{-\delta(|\alpha|+|\beta|)}S_\delta(m_1)$. [/step]

[step:Apply the multi-index Leibniz formula to prove stability under multiplication]Since $m_1$ and $m_2$ are order functions, their pointwise product $m_1m_2:T^*\mathbb{R}^n \to (0,\infty)$ is the order function defined by $(m_1m_2)(x,\xi)=m_1(x,\xi)m_2(x,\xi)$. Let $p:T^*\mathbb{R}^n \times (0,1] \to \mathbb{C}$ be the pointwise product defined by \begin{align*} p(x,\xi;h)=a(x,\xi;h)b(x,\xi;h). \end{align*} Fix $\alpha,\beta \in \mathbb{N}_0^n$. The multi-index Leibniz formula gives \begin{align*} \partial_x^\alpha \partial_\xi^\beta p(x,\xi;h)=\sum_{\mu\leq\alpha}\sum_{\nu\leq\beta}\binom{\alpha}{\mu}\binom{\beta}{\nu}(\partial_x^\mu\partial_\xi^\nu a)(x,\xi;h)(\partial_x^{\alpha-\mu}\partial_\xi^{\beta-\nu} b)(x,\xi;h). \end{align*} Since $a\in S_\delta(m_1)$ and $b\in S_\delta(m_2)$, for each pair $(\mu,\nu)$ appearing in the finite sum there are constants $A_{\mu,\nu}>0$ and $B_{\alpha-\mu,\beta-\nu}>0$ such that \begin{align*} |\partial_x^\mu\partial_\xi^\nu a(x,\xi;h)| \leq A_{\mu,\nu} h^{-\delta(|\mu|+|\nu|)}m_1(x,\xi) \end{align*} and \begin{align*} |\partial_x^{\alpha-\mu}\partial_\xi^{\beta-\nu} b(x,\xi;h)| \leq B_{\alpha-\mu,\beta-\nu} h^{-\delta(|\alpha-\mu|+|\beta-\nu|)}m_2(x,\xi). \end{align*} Multiplying these estimates and using \begin{align*} |\mu|+|\alpha-\mu|+|\nu|+|\beta-\nu|=|\alpha|+|\beta| \end{align*} gives \begin{align*} |(\partial_x^\mu\partial_\xi^\nu a)(x,\xi;h)(\partial_x^{\alpha-\mu}\partial_\xi^{\beta-\nu} b)(x,\xi;h)| \leq A_{\mu,\nu}B_{\alpha-\mu,\beta-\nu} h^{-\delta(|\alpha|+|\beta|)}m_1(x,\xi)m_2(x,\xi). \end{align*} Define the finite constant \begin{align*} C_{\alpha,\beta}=\sum_{\mu\leq\alpha}\sum_{\nu\leq\beta}\binom{\alpha}{\mu}\binom{\beta}{\nu}A_{\mu,\nu}B_{\alpha-\mu,\beta-\nu}. \end{align*} Taking absolute values in the Leibniz formula and applying the triangle inequality yields \begin{align*} |\partial_x^\alpha \partial_\xi^\beta p(x,\xi;h)| \leq C_{\alpha,\beta}h^{-\delta(|\alpha|+|\beta|)}m_1(x,\xi)m_2(x,\xi). \end{align*} This is the defining estimate for $p\in S_\delta(m_1m_2)$.[/step]

[guided]We need to prove that the product has the same symbol estimates, with the product order function $m_1m_2:T^*\mathbb{R}^n \to (0,\infty)$ defined by \begin{align*} (m_1m_2)(x,\xi)=m_1(x,\xi)m_2(x,\xi). \end{align*} Define the product symbol $p:T^*\mathbb{R}^n \times (0,1] \to \mathbb{C}$ by \begin{align*} p(x,\xi;h)=a(x,\xi;h)b(x,\xi;h). \end{align*} To show $p\in S_\delta(m_1m_2)$, we must estimate every mixed derivative $\partial_x^\alpha\partial_\xi^\beta p$. Fix multi-indices $\alpha,\beta \in \mathbb{N}_0^n$. The correct tool is the multi-index Leibniz formula, because a derivative of a product is a finite sum over all ways to distribute the $x$-derivatives and $\xi$-derivatives between the two factors: \begin{align*} \partial_x^\alpha \partial_\xi^\beta p(x,\xi;h)=\sum_{\mu\leq\alpha}\sum_{\nu\leq\beta}\binom{\alpha}{\mu}\binom{\beta}{\nu}(\partial_x^\mu\partial_\xi^\nu a)(x,\xi;h)(\partial_x^{\alpha-\mu}\partial_\xi^{\beta-\nu} b)(x,\xi;h). \end{align*} Here $\mu\leq\alpha$ means $\mu_i\leq\alpha_i$ for every component $i\in\{1,\dots,n\}$, and similarly $\nu\leq\beta$. The sums are finite because there are only finitely many such multi-indices. Now we estimate one summand. Since $a\in S_\delta(m_1)$, the defining estimate gives a constant $A_{\mu,\nu}>0$ such that \begin{align*} |\partial_x^\mu\partial_\xi^\nu a(x,\xi;h)| \leq A_{\mu,\nu} h^{-\delta(|\mu|+|\nu|)}m_1(x,\xi). \end{align*} Since $b\in S_\delta(m_2)$, the defining estimate gives a constant $B_{\alpha-\mu,\beta-\nu}>0$ such that \begin{align*} |\partial_x^{\alpha-\mu}\partial_\xi^{\beta-\nu} b(x,\xi;h)| \leq B_{\alpha-\mu,\beta-\nu} h^{-\delta(|\alpha-\mu|+|\beta-\nu|)}m_2(x,\xi). \end{align*} Multiplying the two inequalities gives \begin{align*} |(\partial_x^\mu\partial_\xi^\nu a)(x,\xi;h)(\partial_x^{\alpha-\mu}\partial_\xi^{\beta-\nu} b)(x,\xi;h)| \leq A_{\mu,\nu}B_{\alpha-\mu,\beta-\nu} h^{-\delta(|\mu|+|\nu|+|\alpha-\mu|+|\beta-\nu|)}m_1(x,\xi)m_2(x,\xi). \end{align*} The exponent is exactly the exponent required for an $(\alpha,\beta)$ derivative, because multi-index length is additive under this splitting: \begin{align*} |\mu|+|\alpha-\mu|+|\nu|+|\beta-\nu|=|\alpha|+|\beta|. \end{align*} Thus each summand is bounded by a constant times \begin{align*} h^{-\delta(|\alpha|+|\beta|)}m_1(x,\xi)m_2(x,\xi). \end{align*} Because the Leibniz formula contains only finitely many summands, we may collect their constants into one finite constant: \begin{align*} C_{\alpha,\beta}=\sum_{\mu\leq\alpha}\sum_{\nu\leq\beta}\binom{\alpha}{\mu}\binom{\beta}{\nu}A_{\mu,\nu}B_{\alpha-\mu,\beta-\nu}. \end{align*} Taking absolute values in the Leibniz formula and applying the triangle inequality gives \begin{align*} |\partial_x^\alpha \partial_\xi^\beta p(x,\xi;h)| \leq C_{\alpha,\beta}h^{-\delta(|\alpha|+|\beta|)}m_1(x,\xi)m_2(x,\xi). \end{align*} This is precisely the defining estimate for $p\in S_\delta(m_1m_2)$. Since $\alpha$ and $\beta$ were arbitrary, the product $ab$ belongs to $S_\delta(m_1m_2)$.[/guided]

[step:Conclude both stability assertions] The differentiation estimate was proved for arbitrary $\alpha,\beta \in \mathbb{N}_0^n$, so every fixed mixed derivative $\partial_x^\alpha\partial_\xi^\beta a$ lies in $h^{-\delta(|\alpha|+|\beta|)}S_\delta(m_1)$. The product estimate was proved for arbitrary derivatives of $ab$, so $ab\in S_\delta(m_1m_2)$. These are exactly the two asserted stability properties. [/step]