How to derive the Leibniz rule from this definition of a connection of a vector bundle?

Question

Suppose $M$ is a Riemannian manifold and $\pi\colon E\to M$ is a vector bundle of rank $n$. Let the vertical subbundle of $TE$ be \begin{equation} V=\ker(d\pi),\quad d\pi\colon TE\to \pi^*(TM) \end{equation} then we have an isomoephism between bundles over $E$ \begin{equation} V\cong\pi^*(E) \end{equation} and suppose we have a horizontal subbundle $H$ of $TE$ such that \begin{equation} TE=V\oplus H \end{equation} then we can define a connection $D$ from this decomposition: for any $X\in\Gamma(TM)$, $\sigma\in \Gamma(E)$, $D_X\sigma\in\Gamma(E)$ is defined \begin{equation} M\xrightarrow{X}TM\xrightarrow{d\sigma}\sigma^*(TE)\xrightarrow{\cong}\sigma^*(V)\oplus\sigma^*(H)\to\sigma^*(V)\xrightarrow{\cong}\sigma^*(\pi^*(E))\xrightarrow{\cong}E \end{equation}

Now I should check that $D$ satisfies:

1. $D_{fX+gY}\sigma=fD_X+gD_Y$
2. $D_X(\sigma+\tau)=D_X\sigma+D_X\tau$
3. $D_X(f\sigma)=df(X)\sigma+fD_X\sigma$

and the difficult part for me are point 2 and 3.

Thanks in advance.