  A rigorous proof is presented of the boundedness of the entanglement entropy
of a block of spins for the ground state of the one-dimensional quantum Ising
model with sufficiently strong transverse field. This is proved by a refinement
of the arguments in the earlier work by the same authors (J. Statist. Phys. 131
(2008) 305-339). The proof is geometrical, and utilises a transformation to a
model of classical probability called the continuum random-cluster model. Our
method of proof is fairly robust, and applies also to certain disordered
systems.
