The smoothness of Riemannian submersions with nonnegative sectional curvature

Let $M^n$ be a complete, non-compact and $C^{\mathrm{\infty }}$-smooth Riemannian manifold with nonnegative sectional curvature. Suppose $\text{Cal}S$ is a soul of $M^n$. Then any distance non-increasing retraction $\mathrm{\Psi }:M^n\to \text{Cal}S$ must give rise to a $C^{\mathrm{\infty }}$-smooth Riemannian submersion.