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In Donaldson and Kronheimer's book on the geometry of four manifolds, a brief review of connections on principal bundles is given. Three equivalent definition are stated:
1) Via horizontal subspaces,
2) Via connections $1$-forms,
3) Via covariant derivatives on an associated vector bundle.
The authors then proceed to sketch very briefly the implications $2) \rightarrow 1)$ and $3) \rightarrow 1) $. I am looking for more detailed proofs.

I have easily found references for the equivalences between $1)$ and $2)$. However I am having a harder time finding references for anything involving $3)$.
Here for instance, the implication $2) \rightarrow 3)$ is proved using local descriptions of connection $1$-forms and covariant derivatives.
On the other hand here, the same implication is proved using, basically the fact that connections are the same thing as parallel transport.
I can't find any proof of $3) \rightarrow 1)$ or $3) \rightarrow 2)$ except the sketch given in Donaldson and Kronheimer's book.

Can anyone help?

Ps: I realize that the same question was asked here. The question was answered. However I have some difficulties navigating in the concepts of the related link (connections on general fibre bundles).

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I don't have my copy handy, but I think this is pretty well explained in Volume 2 of Spivak's Comprehensive Introduction to Differential Geometry. –  Jack Lee Jul 27 at 19:41
Yes indeed it does, thank you! (Chapter 8 addendum 3 for anyone interested) –  Lepanais Jul 27 at 20:50

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