The regression of $X^{(1)}$ on $X^{(2)}$ is the conditional mean

If $\underline X\sim N_p(\underline\mu,\Sigma)$ is partitioned into $\underline X^{(1)}$ ($q$ components) and $\underline X^{(2)}$ ($p-q$ components), then the regression line of $\underline X^{(1)}$ on $\underline X^{(2)}$ will be equal to: