I'm working on a problem,
Let $\mathcal{E}:y^2=x^3-2x+1$ be an elliptic curve over the field $\mathbb{F}_5$, and let $P=(0,1)$ be among the points on $\mathcal{E}$. Find the equation of the line on which $P$, $2P$ and $4P$ all lie. Deduce that $7P=O$, the neutral element of the group law (point at infinity).
I get $P=(0,1),2P=(1,3),4P=(3,2)$ all lie on $y=2x+1$, in agreement with the given solution. In showing $7P=O$, the solution uses the statement
The three intersection points with a line add up to $O$
without further explanation. I can't find this mentioned in the course notes and I can't figure out why this should be. Could someone explain?