Element of Horizontal Space as Horizontal Lift of Vector Field

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 $H_x$ be a horizontal tangent space of $\tilde M$ at $x$.

Let $\map {\mathfrak{X}} M$ be the space of smooth vector fields of $M$.

Then for every $x \in \tilde M$ and every $v \in H_x$ there is a vector field $X \in \map {\mathfrak{X}} M$ whose horizontal lift $\tilde X$ satisfies $\tilde X_x = v$.