We generalize a theorem of W. Benz by proving the following result: Let $H_{ heta}$ be a half space of a real Hilbert space with dimension $geq$ 3 and let Y be a real normed space which is strictly convex. If a distance $ ho$ > 0 is contractive and another distance N$ ho$ (N $geq$ 2) is extensive by a mapping f : $H_{ heta}$ longrightarrow Y, then the restriction f│$_{ heta}$ $H_{+}$$ ho$/2// is an isometry, where $H_{ heta}$＋$ ho$/2/ is also a half space which is a proper subset of $H_{ heta}$. Applying the above result, we also generalize a classical theorem of Beckman and Quarles.