In Abraham, Marsden book (either "Foundation of Mechanics" or "Manifolds, tensor analysis and applications") it is stated, at the beginning of Frobenius' theorem proof, that a fiber of a distribution E in a manifold M has the form
$E_{(x,y)}={(u,f(x,y)·u), u \in \boldsymbol E, f(x,y)·u \in \boldsymbol F}$
where the model whole space is splited in E and F and $f:M \to L(\boldsymbol E, \boldsymbol F)$. My question is, how the existence of such $f$ is proved? Since a distribution is a subbundle, shouldn't the vectors on any fiber of E have just the form $(u,0) \in \boldsymbol E \times \boldsymbol F$?
EDIT: I have just found out that the statement above follows Serge Lang's book. Reading Lang my question remains equally unanswered. I see no reason to not simply assume that as a subbundle any vector field in the distribution belongs to $\boldsymbol E \times 0$