Abian's Fixed Point Theorem 1

A. Abian 1] proved the following theorem: Let f be a mapping from a nite set D. Then f has a xed point if and only if D is not a union of three mutually disjoint sets A, B and C such that A \ fA] = B \ fB] = C \ fC] = ;: (The range of f is not necessarily the subset of its domain). The proof of the suuciency is by induction on the number of elements of D. A. M akowski and K. Wi sniewski 13] shown that the assumption of niteness is superruous. They proved their version of the theorem for f being a function from D into D. In the proof the required partition was constructed and the construction used the axiom of choice. Their main point was to demonstrate that the use of this axiom in the proof is essential. We have proved in Mizar the generalized version of Abian's theorem, i.e. without assuming niteness of D. We have simpliied the proof from 13] which uses well-ordering principle and transsnite ordinals|our proof does not use these notions but otherwise is based on their idea (we employ choice functions). and 16] provide the notation and terminology for this paper. will be sets, s 1 will be a family of subsets of E, f will be a function from E into E, and k, l, n will be natural numbers. Let i be an integer. We say that i is even if and only if: (Def. 1) There exists an integer j such that i = 2 j: We introduce i is odd as an antonym of i is even. Let n be a natural number. Let us observe that n is even if and only if: (Def. 2) There exists k such that n = 2 k: We introduce n is odd as an antonym of n is even. One can verify the following observations: there exists a natural number which is even, there exists a natural number which is odd, there exists an integer which is even, and there exists an integer which is odd. One can prove the following proposition (1) For every integer i holds i is odd ii there exists an integer j such that i = 2 j + 1: