  We generalize Thomas-Yau's uniqueness theorem in two ways. We prove a
stronger statement for special Lagrangians and include minimal Lagrangians in
K\"ahler-Einstein manifold or more generally J-minimal Lagrangians introduced
by Lotay and Pacini. In every case the heart of the proof is to make certain
Hamiltonian perturbations. For this we use the method by Imagi, Joyce and
Oliveira dos Santos.
