Gibbs sampling from conditional full posterior distribution I am reading the paper by Willemsen et al (2015), "A multivariate Bayesian model for embryonic growth", Statistics in Medicine, 34:8, 1351–1365
where they define the posterior distribution as,
\begin{multline}
p(\beta,\sigma^{2},\gamma,\Sigma_{\gamma}|y) \propto  \prod\limits_{ij}  N(y_{ij}|\gamma_{i2} + z^{T}_{ij}\beta_{j},\sigma^{2}) \prod\limits_{i}N(\gamma_{i} \mid 0,\Sigma_{\gamma})\\
\times \prod\limits_{j}N(\beta_{j}|0,\sigma^{2}_{\beta}\mathbf{I}_{5})
\times \prod\limits_{i}N(\sigma^{2}_{i}|\alpha_{\sigma},\beta_{\sigma})
IW (\Sigma_{\gamma}|\delta,\psi)
\end{multline}
where, 
$y_{ij} = \gamma_{i2}  + z^{T}_{ij}\beta + \epsilon_{ij}$
where, $z_{ij} = B(\exp(\gamma_{i3})(t_{ij} + \gamma_{i1} )) , i= 1, \ldots N; j= 1, \ldots n $,
$\gamma_{i} = (\gamma_{i1},\gamma_{i2}\gamma_{i3})$,
$\gamma_{i} \sim N_{3}(0, \Sigma_{(3*3)})$, $\epsilon_{ij} \sim N(0, \sigma^{2}) $
where, $z^{T}_{ij}$ is a spline function. 
I am trying to figure out the full conditional distribution(Gibbs sampling) for $\beta$.
They said the $\beta \sim N(\bar \mu,\bar\Sigma_{\beta})$,
where $\bar\mu = (\bar\Sigma_{\beta}\bar Z \tilde y)/\sigma^{2}$  and 
$\bar\Sigma_{\beta}=(\bar\Sigma^{-1}_{\beta}+ \bar Z^{T}\bar Z / \sigma^{2})^{-1}$ 
My question is how did they get that distribution of $\beta$? 
 A: \begin{align*}
p(\beta|\sigma^{2},\gamma,\Sigma_{\gamma},y) &\propto  \prod\limits_{ij}  N(y_{ij}|\gamma_{i2} + z^{T}_{ij}\beta_{j},\sigma^{2}) \prod\limits_{i}N(\gamma \mid 0,\Sigma_{\gamma})\\
&\qquad\times \prod\limits_{i}N(\beta_{i}|0,\sigma^{2}_{\beta}I_{p})
\times \prod\limits_{i}N(\sigma^{2}_{i}|\alpha_{\sigma},\beta_{\sigma})
\times IW (\Sigma_{\gamma}|\delta,\psi)\\
&\propto\prod\limits_{ij}  N(y_{ij}|\gamma_{i2} + z^{T}_{ij}\beta_{j},\sigma^{2}) \prod\limits_{i}N(\beta_{i}|0,\sigma^{2}_{\beta}I_{p})\\
&\propto\exp-\left\{||\mathbf{y}-\mathbf{\gamma}-\mathbf{Z}\mathbf{\beta}||^2 \right\}/2\sigma^2\times\exp-\left\{||\mathbf{\beta}||^2\right\}/2\sigma_\beta^2\\
&\propto\exp-\left\{\sigma^{-2}\mathbf{\beta}^\text{T}\mathbf{Z}^\text{T}\mathbf{Z}\mathbf{\beta}-2\sigma^{-2}(\mathbf{y}-\mathbf{\gamma})^\text{T}\mathbf{Z}\mathbf{\beta}+\sigma^{-2}_{\beta}\mathbf{\beta}^\text{T}\mathbf{\beta}
\right\}/2\\
&\propto\exp-\big\{\mathbf{\beta}^\text{T}\underbrace{[\sigma^{-2}\mathbf{Z}^\text{T}\mathbf{Z}+\sigma^{-2}_{\beta}I_p]}_{\bar\Sigma_{\beta}^{-1}}\mathbf{\beta}-2\sigma^{-2}(\mathbf{y}-\mathbf{\gamma})^\text{T}\mathbf{Z}\mathbf{\beta}
\big\}/2\\
&\propto\exp-\left\{\mathbf{\beta}^\text{T}\bar\Sigma_{\beta}^{-1}\mathbf{\beta}-2\sigma^{-2}\mathbf{\beta}^\text{T}\bar\Sigma_{\beta}^{-1}\bar\Sigma_{\beta}\mathbf{Z}^\text{T}(\mathbf{y}-\mathbf{\gamma})
\right\}/2\\
&\propto\exp-\left\{[\mathbf{\beta}-\sigma^{-2}\bar\Sigma_{\beta}\mathbf{Z}(\mathbf{y}-\mathbf{\gamma})]^\text{T}\bar\Sigma_{\beta}^{-1}\bar[\mathbf{\beta}-\sigma^{-2}\bar\Sigma_{\beta}\mathbf{Z}^\text{T}(\mathbf{y}-\mathbf{\gamma})]
\right\}/2\\
\end{align*}
