There is a result (for which I don't recall a reference, but you might prove it maybe with the smooth Serre-Swan theorem. See for example Nestruev, Smooth manifolds and observables), which says that for the sections of the pullback of a vector bundle $E\to M$ along a map $\phi:N\to M$ one has $$\Gamma(\phi^*(E))=C^\infty(N)\otimes_{C^\infty(M)}\Gamma (E)$$This is a canonical isomorphism obtained from the natural map $\Gamma(E)\to\Gamma(\phi^*(E))$. Combining this with the fact that the horizontal bundle in your case is canonically isomorphic with the pullback of $TB$ to $P$ would give you a proof.
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