I see from this question that the product of two multivariate Gaussians is $N(\mu,\Sigma)$ where:
$\mu = \Sigma_2(\Sigma_1+\Sigma_2)^{-1}\mu_1 +\Sigma_1(\Sigma_1+\Sigma_2)^{-1}\mu_2$
$\Sigma =(\Sigma_1^{-1}+\Sigma_2^{-1})^{-1}$
I've also seen the following result,
$(\Sigma_1^{-1}+\Sigma_2^{-1})^{-1}= \Sigma_1-\Sigma_1(\Sigma_1+\Sigma_2)^{-1}\Sigma_1.$
My question is how is this result derived?
