How could I get this lemma about stoquastic hamiltonian in the paper "Complexity of stoquastic frustration-free Hamiltonians"

Question

In the paper Complexity of stoquastic frustration-free Hamiltonians, I was confused about the derivation of Lemma 4.5: How could we get $\delta Tr(O(I-\Pi_a)) \leq Tr(OH_a)$ given that $\delta (I-\Pi_a) \leq H_a$? (I think the $\leq$ sign here should be the partial order for positive semidefinite matrix)