General Engineering Introduction/Error Analysis/Calculus of Error/addSubtract proof

Adding and subtracting proof
 * if $$y = x + z$$ or $$y = x - z$$ then $$ \delta_y= \sqrt{\delta_x^2 + \delta_z^2}$$

Algebra Proof
can not come up with one

Calculus Proof
start with the formula: dependent equals constant times independent $$y = x + z$$ then $$\delta_y = \sqrt{\left ( \delta_x \frac{\partial {\left ( x + z \right )}} \right ) ^2 + \left ( \delta_z \frac{\partial (x+z)}{\partial z} \right ) ^2} = \sqrt{\delta_x^2 + \delta_z^2}$$