  We mathematically show an equality between the index of a Dirac operator on a
flat continuum torus and the $\eta$ invariant of the Wilson Dirac operator with
a negative mass when the lattice spacing is sufficiently small. Unlike the
standard approach, our formulation using $K$-theory does not require the
Ginsparg-Wilson relation or the modified chiral symmetry on the lattice. We
prove that a one-parameter family of continuum massive Dirac operators and the
corresponding Wilson Dirac operators belong to the same equivalence class of
the $K^1$ group at a finite lattice spacing. Their indices, which are evaluated
by the spectral flow or equivalently by the $\eta$ invariant at finite masses,
are proved to be equal.
