I am studying connections on abstract manifolds. So far, I have read several equivalent definitions but I can't establish the equivalence between them on my own. The first definition is the Ehresmann connection that defines a connection on a manifold as a distribution of vector spaces completing the vertical space in the tangent space of the total space at each point. The second definition defines a connection on a manifold as a covariant derivative, i.e. a map ∇:Γ(E)→Γ(T∗M⊗E) where π:E→M is a vector bundle and there is a version of the Leibnitz rule as follows: ∇X(fs)=df⊗s+f⋅∇Xs for any section s and f∈C∞(M). I tried to write things in a chart to find out how the covariant derivation is induced from a given connection but I couldn't proceed forward. I think I haven't understood the definitions well. Would someone clarify how a distributional connection give us a connection one-form and how we can recover the horizontal space if we have a connection one-form?