Smooth Vector Field has Unique Smooth Horizontal Lift

Theorem
Let $\tilde M, M$ be smooth manifolds.

Let $\pi : \tilde M \to M$ be a smooth submersion.

Let $\tilde g$ be a Riemannian metric on $\tilde M$.

Let $W$ be a smooth vector field on $M$.

Then $W$ has the unique smooth horizontal lift to $\tilde M$.