Let $k$ be a field with $\operatorname{char}(k) \neq 2,3$, and let $a,b \in k$ be such that the short Weierstrass curve $E: y^2 = x^3 + ax + b$ is nonsingular over $k$. Let $E(k) := \{O\} \cup \{(x,y) \in k^2 : y^2 = x^3 + ax + b\}$, where $O$ denotes the point at infinity. If $P,Q \in E(k)$, then their sum under the chord-tangent group law satisfies $P+Q \in E(k)$.