I am trying to understand Kalman-Filtering in the recursive bayesian sense and for that I need to get an understanding why the product of two multivariate gaussian normal distributions is again of gaussian shape (it may actually not be a pdf as the scaling may be wrong).
The univariate case is derived there:

However, the multivariate case is too much for me to derive. Can somebody give me a hint on the calculation or point me to some resources?