  Under the widely accepted but unproven assumption that the one-dimensional
$S=1$ antiferromagnetic Heisenberg model has a unique gapped ground state, we
prove that the model belongs to a nontrivial symmetry-protected topological
(SPT) phase. In other words, we rigorously rule out the possibility that the
model has a unique gapped ground state that is topologically trivial. To be
precise, we assume that the models on open finite chains with boundary magnetic
field have unique ground states with a uniform gap and prove that the ground
state of the infinite chain has a nontrivial topological index. This implies
the presence of a gapless edge excitation in the model on the half-infinite
chain and the existence of a topological phase transition in the model (9).
