Continuous mappings of open sets in a Banach space

If $\Gamma$ is a bounded open set of a Banach space ($B$), $\varphi$ is a completely continuous mapping of $\Gamma$ into the same space ($B$), and $E-\varphi\equiv\Phi$, where E is the identity transformation, is a uniformly fading mapping of $\Gamma$ into the Banach space, then the order of $\Phi$ on $\Gamma$ equals $\pm1$ at every point $y$ of $\Phi\Gamma$.