In Saber Elaydi's book "An Introduction to Difference Equations", 3rd ed., Sec. 2.1 (page 59), the discrete analogue of the Fundamental Theorem of Calculus is stated: \begin{equation} \sum_{k=n_0}^{n-1} \Delta x(k) = x(n) - x(n_0) \end{equation} and \begin{equation} \Delta \left(\sum_{k=n_0}^{n-1}x(k) \right) = x(n) \end{equation} where $\Delta x(k) = x(k+1)-x(k)$ is the difference operator. I do not understand the second part. The $\Delta$ operator is linear so why does it not enter the sum? Also, even if the sum is evaluated first, and then the difference is taken, the result is the same as in the first equation. Is there any missing notation on which $x's$ the $\Delta$ should act on?
Thank you in advance for the support.