  With the XXZ spin chains as examples, we prove two theorems: (1) the
functional relations derived from the off-diagonal Bethe Ansatz scheme are the
sufficient and necessary conditions to characterize the complete spectrum of
the corresponding transfer matrix; (2) each eigenvalue of the transfer matrix
can be parameterized by a minimal inhomogeneous T-Q relation. These statements
hold for both with and without inhomogeneity. The proof can be generalized to
other finite-dimensional quantum integrable models.
