Sion's minimax theorem and Nash equilibrium of symmetric multi-person zero-sum game

About a symmetric multi-person zero-sum game we will show the following results. 1. Sion's minimax theorem plus the coincidence of the maximin strategy and the minimax strategy are proved by the existence of a symmetric Nash equilibrium. 2. The existence of a symmetric Nash equilibrium is proved by Sion's minimax theorem plus the coincidence of the maximin strategy and the minimax strategy. Thus, they are equivalent. If a zero-sum game is asymmetric, maximin strategies and minimax strategies of players do not correspond to Nash equilibrium strategies. If it is symmetric, the maximin strategies and the minimax strategies constitute a Nash equilibrium. However, with only the minimax theorem there may exist an asymmetric equilibrium in a symmetric multi-person zero-sum game.