In ECC, how do I prove that point addition is commutative?
If I am not mistaken, $y_3 = y'_3$:

$y_3 = \frac{y_2 - y_1}{x_2 - x_1}(x_1 - x_3) - y_1 = \frac{y_2 x_1 - y_2 x_3 - y_1 x_1 + y_1 x_3 - y_1 x_2 + y_1 x_1}{x_2 - x_1} = \frac{y_2 x_1 - y_2 x_3 + y_1 x_3 - y_1 x_2}{x_2 - x_1}$

$y'_3= \frac{y_1-y_2}{x_1-x_2}(x_2-x_3) - y_2 = \frac{y_1 x_2 - y_1 x_3 - y_2 x_2 + y_2 x_3 - y_2 x_1 + y_2 x_2}{x_1-x_2} = \frac{y_2 x_1 - y_2 x_3 + y_1 x_3 - y_1 x_2}{x_2 - x_1}$