  We generalize a theorem by Khalfin, originally derived for the states $| F_1
> = | M^0>$, $| F_2 > = |\bar M^0>$, where $M^0$ is a neutral flavoured meson
(e.g., $K^0$ or $B_d^0$), by assuming CPT invariance. Dispensing with CPT
invariance and allowing for an arbitrary pair of orthogonal states $|
F_{1,2}>$, we show that any linear combinations $| P_a > = p_a | F_1 > + q_a |
F_2>$ and $| P_b > = p_b | F_1 > - q_b | F_2>$, if postulated to be
independently propagating in time, as in the Lee--Oehme--Yang Theory, must be
mutually orthogonal. This implies a reciprocity relation: equality of the
probabilities of the transitions $| F_1 > \leftrightarrow | F_2>$. Also implied
is another relation involving the coefficients $p_{a,b}$, $q_{a,b}$, which can
be interpreted as $\mathrm{Im} \theta = 0$, where $\theta$ is the
rephasing-invariant parameter describing CPT violation in $M^0 \bar M^0$ mixing
for Khalfin's choice of $| F_{1,2}>$. The states $| F_{1,2}>$ of \emph{our}
theorem need not form a particle-antiparticle pair, nor even be restricted to
particle physics.
