Kalman filter: the bayesian approach derivation some clarifications

Question

I'm reading the book Methods and algorithms for signal processing from Moon Stirling at page 592 there is a derivation of Kalman filter using the Bayesian approach. I have some issues in understanding the propagate step which is the following: $ p(\textbf{x}_{k}|\textbf{z}_{1:k-1}) = \int{p(\textbf{x}_k|\textbf{x}_{k-1})p(\textbf{x}_{k-1}|\textbf{z}_{1:k-1})d\textbf{x}_{k-1}}$. The book says under some assumption: $ x_{k-1} | z_{1:k-1} \sim \mathcal{N}(m_{k-1|k-1}), P_{k-1|k-1})$ and
$x_{k}|x_{k-1} \sim \mathcal{N}(F_k x_{k-1}, Q_k)$ And inserting these in the prediction step and performing the integration (which involves expanding and completing the square) we find that $x_{k|k-1}$ is a Gaussian with mean $F_k m_{k-1|k-1}$ and variance $F_k P_{k|k} F_k ^{T} + Q_k$. I don't understand when I insert that two gaussian in prediction formula how is possible to get that mean and that variance, there we have the product of two gaussian distribution but with different random variables maybe that's a gaussian? but then we have an integral of the resulting distribution. Can someone clarify me this? Thank you