Error propagation in combined linear models

I have a set of observed values ($y_{1Obs}$) and 3 predictive variables (n = 27). I use multiple linear regression to create a linear model:

$Z_1=alpha_0 + alpha_1 W_1 + alpha_2 X_1 + alpha_3 Y_1$

Where $Z_1$ is my response variable, $alpha_0$ is the intercept, $alpha_1 , alpha_2 , alpha_3$ are regression coefficients and $W_1, X_1, Y_1$ are predictive variables.

I also have a second set of observed values ($y_{2Obs}$) and 3 different predictive variables (n = 27). I again use multiple linear regression to create a second linear model of the same form:

$Z_2=beta_0 + beta_1 W_2 + beta_2 X_2 + beta_3 Y_2$

Where $Z_2$ is my response variable, $beta_0$ is the intercept, $beta_1 , beta_2 , beta_3$ are regression coefficients and $W_2, X_2, Y_2$ are predictive variables.

With 27 predicted values from each model I then calculate the predicted change between the two:

$Delta Z_{Pred}=Z_2 – Z_1$

While my observed change comes from the two sets of observed values used to create the two linear models:

$Delta Z_{Obs}=y_{2Obs} – y_{1Obs}$

My question is: what formula do I use to calculate the RMSE of my predictions ($Delta Z_{Pred}$) that will propagate the errors from predictions of $Z_1$ and $Z_2$?

I’ve tried using the following formula though I’m not convinced this is correct:

$RMSE_{Delta Z_{Pred}} = sqrt{{RMSE_1}^2 + {RMSE_2}^2}$

Where $RMSE_1$ and $RMSE_2$ are the RMSEs from the first two models shown above.