A proof of the Kramers degeneracy of transmission eigenvalues from antisymmetry of the scattering matrix

In time-reversal-symmetric systems with half integral spins (or more concretely, systems with an antiunitary symmetry that squares to - 1 and commutes with the Hamiltonian) the transmission eigenvalues of the scattering matrix come in pairs. We present a proof of this fact that is valid for both even and odd number of modes and relies solely on the antisymmetry of the scattering matrix imposed by time reversal symmetry.