We know that for any principal bundle $P$ with the Lie group$G$ and base manifold $M$，we can induced a connection on any of its associated bundle by the following formula：
$\pi ^*\nabla (s)=d (\hat s)+\omega •\hat s$ ,here $\pi$ the projection from $P$ to the base manifold $M$ ;$\omega$ the connection of $P$ ; $s$ the section of $P$, and$\hat s$ is the correspondent section of $s$ in the set of horizontal $G$-equivairant section of $P$, finally , $\nabla$ is the connection that we want.
To show the $\nabla$ exist ,it suffices to show the following there point:
1.RHS of the euqation is horizontal.
2.RHS of the equation is $G$-equivairant.
3.RHS of the equation satisfies the Leibniz rule.
I can show the first and the third ，but feel confused about the second .Can anyone give me a proof in details ？Thanks ！