Sion's mini-max theorem and Nash equilibrium in a multi-players game with two groups which is zero-sum and symmetric in each group

We consider the relation between Sion’s minimax theorem for a continuous function and a Nash equilibrium in a multi-players game with two groups which is zero-sum and symmetric in each group. We will show the following results.

1. The existence of Nash equilibrium which is symmetric in each group implies Sion’s minimax theorem with the coincidence of the maximin strategy and the minimax strategy for players in each group. %given the values of the strategic variables.
2. Sion’s minimax theorem with the coincidence of the maximin strategy and the minimax strategy for players in each group implies the existence of a Nash equilibrium which is symmetric in each group. Thus, they are equivalent. An example of such a game is a relative profit maximization game in each group under oligopoly with two groups such that firms in each group have the same cost functions and maximize their relative profits in each group, and the demand functions are symmetric for the firms in each group.