In bayesian estimation, when the model and plant noise is hold , the optimal estimator is Kalman filter. but I am wondering is there any literature that could prove the following gaussian identities?
$$ N(z; Hx, R)N(x; y, P) = N(z; Hy, C)N(x; e, E) $$ which $$ C=R+HPH^T$$ $$ E^{-1}=P^{-1}+H^TR^{-1}H, E^{-1}e=P^{-1}y+H^TR^{-1}z $$