  This paper proposes an algorithm to rigorously estimate the difference
quotient of multiple eigenvalues with respect to domain deformation. This is
achieved by deriving a novel difference quotient formula and applying a newly
developed guaranteed computation method for eigenvalue problems. Using the
proposed algorithm, a computer-assisted proof is provided to validate the
simplicity of the second Dirichlet eigenvalue on nearly equilateral triangles.
This proof partially solves Conjecture 6.47 in A. Henrot's book ``Shape
Optimization and Spectral Theory".
