Unable to understand how the paper simplifies the covariance matrix - Kalman filter
In order to derive equation (20) you can use the following steps:


From substitution of  equation (16) into equation (18)


$$  P_n = P_{n|n-1} - K_nP^t_{x_ny_n}$$


Now plugging right side equation of (19) into the last equation


$$  P_n = P_{n|n-1} - K_n(P_{y_ny_n} - R_n)=$$
$$ =P_{n|n-1} - K_nP_{y_ny_n} + K_nR_n $$


Now Pluging (16) again we know that
$$ K_nP_{y_ny_n} = P_{x_ny_n}$$
putting this back we get 


$$ P_n=P_{n|n-1} - P_{x_ny_n} + K_nR_n $$


Using left side equation of (19) we get the desired result
$$P_n = K_nR_n$$