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My attempt is:

Suppose $s^D$ is a strictly dominant strategy equilibrium. Then by definition $s_i^D$ $\in$ $S_i$ is a strict dominant strategy for all $i \in N$. This means that all $s_{-i}$ are strictly dominated by $s_i^D$, and hence there cannot exist any other strictly dominant strategy equilibrium, because that would mean that $s^D$ is strictly dominated by some $s_{-i} \in S_{-i}$, which is a contradiction.

Could someone tell me whether this is correct, and if not then hint me?

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