Help defining a shift cipher mathematically

For an assignment question, I am trying to prove formally that double encryption with two keys by the shift-cipher encryption function results in a shift cipher as well. If the Shift-cipher encryption function is defined as:

$E_K(M)= (M + K) mod 26$

and suppose that we let $M = (x_1, ... , x_{m-1}, x_m)$ = an arbitrary but particular message of length m, where each element of M is a is an integer element of $Z_{26}$, and we pass M into the encryption function, resulting in: $(x_1 + K_1, ... , x_{m-1} + K_1, x_m + K_1) mod 26 = (y_{1}, ... y_{m})$ , where $1 \leq i \leq m, y_i \in Z_{26}$ .

Then my question is: what is it that makes $(y_{1}, ... y_{m})$ a shift cipher? And how is this generally expressed formally? When I know this I will be able to show that encrypting $(y_{1}, ... y_{m})$ also results in a shift cipher.

Right now all I can think of is that it is a shift cipher if it is a string where the index of each element is a number from 0-1, that was acquired by adding the index of each paintext element to the index of the key. Can anyone help clarify things for me?

My understanding is given that $E_K(x)=(x+K)~mod~26$ is the character by character encryption of a shift cipher, you want to extend it to words as the function $E'$ given by $$E'_K(x_1,\ldots,x_m)=(E_K(x_1),E_K(x_m)).$$ If you intended the key $K$ to change per character you can modify.