= Solution
Use the usual state-space assumption that the prior $X_0$ is independent of future process and observation noises. Suppose inductively $X_t\mid\mathcal Y_t\sim N(\hat X_t,V_t)$. Prediction through the linear dynamics gives $X_{t+1}\mid\mathcal Y_t\sim N(A\hat X_t,M_t)$, where $M_t=AV_tA^T+\Sigma_\varepsilon$. The pair $(X_{t+1},Y_{t+1})$ is conditionally <multivariate normal>, with means $A\hat X_t,CA\hat X_t$, cross covariance $M_tC^T$, and observation covariance $R_t=\Sigma_\eta+CM_tC^T$. Applying part (a) gives the <Kalman filter>
$$
\boxed{\begin{aligned}
\hat X_{t+1}&=A\hat X_t+M_tC^TR_t^{-1}(Y_{t+1}-CA\hat X_t),\\
V_{t+1}&=M_t-M_tC^TR_t^{-1}CM_t.
\end{aligned}}
$$
This proves the inductive Gaussian assertion. The inverse requires nonsingular innovation covariance, as in the printed formula. Importantly, prediction uses $A\hat X_t$: the TeX conversion drops this $A$ in one place, whereas the original PDF includes it.
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