Abian's Fixed Point Theorem

(The range of f is not necessarily the subset of its domain). The proof of the sufficiency is by induction on the number of elements of D. A.Ma̧kowski and K.Wiśniewski [12] have shown that the assumption of finiteness is superfluous. 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 finiteness of D. We have simplified the proof from [12] which uses well-ordering principle and transfinite ordinals—our proof does not use these notions but otherwise is based on their idea (we employ choice functions).