Jordan-Hölder Theorem with Uniqueness for Semimodular Lattices

In 2011 Czédli and Schmidt proved the strongest form of Jordan-Hölder theorem for lattices, which they called Jordan-Hölder theorem with uniqueness: Given two maximal chains in a semimodular lattice of finite height, they both have the same length and there is a unique bijection that takes the prime intervals of the first chain to the prime intervals of the second chain such that the interval and its image are up-and-down projective. The theorem generalizes the classical result that all composition series of a finite group have the same length and isomorphic factors and shows that the isomorphism is in some sense unique. The paper presents a simplified proof of the result.