Let $\pi:E\to M$ be a smooth vector bundle with a connection. Let $v\in T_pM$ and $v^e$ be its horizontal lift to $e\in\pi^{-1}(\{p\})$. If $e_1,e_2\in\pi^{-1}(\{p\})$ then I want to show that $v^{e_1+e_2}=v^{e_1}+_*v^{e_2}$ where $+_*$ is the pushforward of the addition map $+:E\times E\to E$ on the fibers.
By uniqueness of horizontal lifts, I need to show that $\pi_*(v^{e_1}+_*v^{e_2})(f)=v(f)$ for any smooth function $f$ on $M$. What I get is
$ \begin{align*} \pi_*(v^{e_1}+_*v^{e_2})(f)&=(v^{e_1}+_*v^{e_2})(f\circ\pi) \\ &=(v^{e_1},v^{e_2})(f\circ\pi\circ+) \\ &=v^{e_1}(f\circ\pi\circ+\vert_{\pi^{-1}(\{p\})\times\{e_2\}})+v^{e_2}(f\circ\pi\circ+\vert_{\{e_1\}\times\pi^{-1}(\{p\})}) \\ &=v^{e_1}(f\circ\pi)+v^{e_2}(f\circ\pi) \\ &=v(f)+v(f) \\ &=2v(f). \end{align*} $
The 4th line is from the fact that restriction of the addition map sends points to the same fiber. My answer results in a factor of $2$ that shouldn't be there but I cannot figure out what went wrong, although I suspect that the 3rd line is the culprit. I appreciate any help.