Let $X(t)$ and $Y(t)$ be two orthogonal processes with power spectral densities $$S_{xx}(f) = S_{yy}(f)=\begin{cases} 1-\lvert f\rvert, & \lvert f\rvert<1 \\[1ex] 0,& \text{otherwise} \end{cases}$$
Define a new process $Z(t) = Y(t)-X(t-1)$. Determine and sketch the power spectral density $S_{zz}(f)$.
The solution is the following: The autocorrelation function of the process is $$R_{zz}= E \bigg\{\big[Y(t-\tau)-X(t+\tau-1)\big]\big[Y(t)-X(t-1)\big] \bigg\} $$ And now comes the part which I do not understand at all. \begin{align} R_{zz}&=R_{yy}(\tau)-R_{yx}(\tau+1)-R_{xy}(\tau-1)+R_{xx}(\tau)\\ &=R_{yy}(\tau) + R_{xx}(\tau)\ \quad\text{since}\quad R_{yx} = R_{xy} = 0 \quad\text{from orthogonality.} \end{align}
Therefore, $S_{zz}(f)=2S_{yy}(f)=2S_{xx}(f)$ as shown below.
Can anyone explain it like I am five?