Strictly dominated strategy

In a game with $n$ players where player $i$ has choice of strategies in $S_i$ and has utility function $U_i$. For a player $i$ strategy $s^{+} \in S_i$ strictly dominates $s^- \in S_i$ if and only if for all $(s_1, \ldots, s_{i-1}, s_{i+1}, \ldots, s_n) \in S_1 \times \ldots \times S_{i-1} \times S_{i+1} \times \ldots \times S_n$ we have

$$U_i(s_1, \ldots, s_{i-1}, s^+, s_{i+1}, \ldots, s_n) \geq U_i(s_1, \ldots, s_{i-1}, s^-, s_{i+1}, \ldots, s_n) $$