= Solution
The <conditional multivariate normal distribution> gives
$$
\boxed{f_*\mid\mathbf y,\theta\sim\mathcal N(m_n(x_*),v_n(x_*))},
$$
where
$$
\boxed{m_n(x_*)=k_*^TC^{-1}\mathbf y,\qquad v_n(x_*)=k_{**}-k_*^TC^{-1}k_*}.
$$
For a direct justification, form $R=f_*-k_*^TC^{-1}\mathbf y$. It is jointly <Gaussian> with $\mathbf y$, has zero <mean>, and satisfies
$$
\operatorname{Cov}(R,\mathbf y)=k_*^T-k_*^TC^{-1}C=0,\qquad\operatorname{Var}(R)=k_{**}-k_*^TC^{-1}k_*.
$$
The <independence of uncorrelated jointly normal variables> therefore makes $R$ independent of $\mathbf y$. Conditional on $\mathbf y$, only the displayed affine shift remains, proving the result. This is the latent-function <posterior variance>; a new observation $y_*=f_*+\sigma\epsilon_*$ with fresh independent noise has <predictive variance> $v_n(x_*)+\sigma^2$.
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