The equivariant covering homotopy property for differentiable G-fibre bundles

Let G be a compact Lie group, and X a diferentiable G-manifold. If p: E -> X is a differentiate fibre bundle, and G acts differentiably on E so that each g e G operates as a bundle map, then we call p a differentiate Gfibre bundle. We show that if p is a differentiate G-fibre bundle with Lie structure group or compact fibre, then it has the equivariant covering homotopy property. This generalizes the fact that a differentiable family of actions of a compact Lie group on a compact differentiable manifold is locally trivial.