Subbundle of vector bundle over paracompact admits complement

This article describes a result applicable for real vector bundles over a paracompact Hausdorff space. In particular, the result is applicable for real vector bundles over a manifold, CW-space, or metrizable space

Statement

Any vector subbundle of a real vector bundle over a paracompact Hausdorff space, admits a direct complement. In other words, there is another subbundle such that at every point, the sum of the fibers for the two subbundles equals the fiber for the whole bundle.