  Nash equilibrium (NE) is a central concept in game theory. Here we prove
formally a published theorem on existence of an NE in two proof assistants, Coq
and Isabelle: starting from a game with finitely many outcomes, one may derive
a game by rewriting each of these outcomes with either of two basic outcomes,
namely that Player 1 wins or that Player 2 wins. If all ways of deriving such a
win/lose game lead to a game where one player has a winning strategy, the
original game also has a Nash equilibrium.
  This article makes three other contributions: first, while the original proof
invoked linear extension of strict partial orders, here we avoid it by
generalizing the relevant definition. Second, we notice that the theorem also
implies the existence of a secure equilibrium, a stronger version of NE that
was introduced for model checking. Third, we also notice that the constructive
proof of the theorem computes secure equilibria for non-zero-sum priority games
(generalizing parity games) in quasi-polynomial time.
