  We present a short proof of the Jordan-H\"older theorem with uniqueness for
semimodular semilattice: Given two maximal chains in a semimodular semilattice
of finite height, they both have the same length. Moreover 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. Moreover, it shows
that the isomorphism is in some sense unique.
