Not Every Horizontal Vector Field is Horizontal Lift

Theorem
Let $\pi : \R^2 \to \R$ be the projection map such that:


 * $\map \pi {x, y} = x$

Let $W = y \partial_x$ be a smooth vector field on $\R^2$.

Then $W$ is horizontal, but there is no smooth vector field whose horizontal lift is equal to $W$.