 Suppose \(r\) stands for the action of trimming a piece of cake and putting it back with the main part of the cake, according to the Banach-Knaster algorithm, and suppose \(F(m,k)\) is the proposition that the main part of the cake is large enough for \(k\) people. Then surely \(F(m,n)\) holds at the beginning: the whole cake is large enough for the whole group to begin with. Moreover, Parikh (1983, 2002) uses his game logic to prove that \(F(m,k)\) is invariant under the action \(r\): If \(F(m,k)\) is true before \(r\), then it will still be true after \(r\) has occurred. Clearly, if one can show that \(F(m,k)\) continues to hold through the algorithm, for \(k\) running through \(n, \ldots, 1\), then this establishes that the division is  fair.[3]