The monotonicity property has a natural version for two measures, saying that μ 1 conditionally pointwise dominates μ 2. It is again easy to see that if μ 1 and μ 2 satisfy the lattice-type condition of the Holley inequality, then μ 1 conditionally pointwise dominates μ 2. On the other hand, a Markov chain coupling argument similar to the above, but now without invoking the Harris inequality, shows that conditional pointwise domination, in fact, implies stochastically domination. Stochastic domination is equivalent to saying that ⟨ ⟨ f ⟩ ⟩ 1 ≥ ≥ ⟨ ⟨ f ⟩ ⟩ 2 for all increasing ƒ, thus we get a proof of the Holley inequality. (And thus also a proof of the FKG inequality, without using the Harris inequality.)