Solution
= Solution
Put $b=\pi(p)$ and $u=(d\pi)_p v$. Choose a vector field $X$ near $b$ with $X(b)=u$, multiply it by a <smooth bump function>[bump function], and extend it by zero to $B$. At each $q\in E$, the restriction
$$
(d\pi)_q:H_qE\longrightarrow T_{\pi(q)}B
$$
is an isomorphism. Define $V(q)$ as the unique horizontal lift of $X(\pi(q))$. Smoothness of the horizontal distribution makes $V$ smooth, and uniqueness gives $V(p)=v$.