  In this paper we construct such a set of `degenerate' Hamiltonians $\hat{H}$,
which differ by an `intrinsic' constant but represent different physical
systems yet possess the same ground state density. . Thus, although the proof
of Hohenberg-Kohn (HK) theorem is independent of whether the constant $C$ is
additive or intrinsic, its applicability is restricted to excluding the case of
the latter. This constitutes the corollary to the theorem.The corollary has
also been extended to the time-dependent version of the HK theorem.
