Let $n \in \mathbb{N}$, let $0 \leq \delta < 1/2$, and let $m_1,m_2:T^*\mathbb{R}^n \to (0,\infty)$ be order functions. Define the pointwise product order function $m_1m_2:T^*\mathbb{R}^n \to (0,\infty)$ by $(m_1m_2)(x,\xi)=m_1(x,\xi)m_2(x,\xi)$. For the semiclassical delta symbol class $S_\delta(m)$ on $T^*\mathbb{R}^n \times (0,1]$, the following hold.

1. For every pair of multi-indices $\alpha,\beta \in \mathbb{N}_0^n$, if

$a:T^*\mathbb{R}^n \times (0,1] \to \mathbb{C}$ belongs to $S_\delta(m_1)$, then $\partial_x^\alpha \partial_\xi^\beta a$ belongs to $h^{-\delta(|\alpha|+|\beta|)}S_\delta(m_1)$.

2. If $a:T^*\mathbb{R}^n \times (0,1] \to \mathbb{C}$ belongs to $S_\delta(m_1)$ and

$b:T^*\mathbb{R}^n \times (0,1] \to \mathbb{C}$ belongs to $S_\delta(m_2)$, then the pointwise product $ab:T^*\mathbb{R}^n \times (0,1] \to \mathbb{C}$ belongs to $S_\delta(m_1m_2)$.