Assume that for each $t \geq 0$, we have a matrix $$K(t) = \int_0^t \mathrm{e}^{-Cs} D\, \mathrm{e}^{-C^\intercal s} \mathrm{d} s$$ where $D \in \mathbb{R}^{d \times d}$ is constant, symmetric, and positive semi-definite and that $C \in \mathbb{R}^{d \times d}$. It is claimed in some papers (without details unfortunately) that we have $$K_\infty = K(t) + \mathrm{e}^{-Ct}\,K_\infty\, \mathrm{e}^{-C^\intercal t} \label{1}\tag{1}$$ for all $t \geq 0$, where $$K_\infty = \int_0^\infty \mathrm{e}^{-Cs} D\, \mathrm{e}^{-C^\intercal s} \mathrm{d} s.$$ May I know how can we check the validity of equation \eqref{1} from basic matrix calculus?
Remark: For a given matrix $A \in \mathbb{R}^{d \times d}$, we denote its transpose by $A^\intercal$.
\begin{align} K_{\infty} - K(t) &= \int_{t}^{\infty} \mathrm{e}^{-Cs}D\mathrm{e}^{-C^\intercal s} \mathrm ds\\ &= \int_{0}^{\infty} \mathrm{e}^{-C(t+u)}D \mathrm{e}^{-C^\intercal (t+u)} \mathrm du & (u:=s-t)\\ &= \int_{0}^{\infty} \mathrm{e}^{-Ct}\mathrm{e}^{-Cu}D\mathrm{e}^{-C^\intercal u}\mathrm{e}^{-C^\intercal t} \mathrm du\\ &= \mathrm{e}^{-Ct}\left(\int_{0}^{\infty} \mathrm{e}^{-Cu}D\mathrm{e}^{-C^\intercal u} \mathrm du\right)\mathrm{e}^{-C^\intercal t}\\ &= \mathrm{e}^{-Ct}K_\infty \mathrm{e}^{-C^\intercal t} \end{align}