Orthocompactness implies Δ-paracompactness for the product of a Δ-paracompact normal space and a compact space

Δ-paracompactness is a topological property introduced by Buzyakova. It is known that for a monotonically normal space X and for a compact space K, if X×K is orthocompact, then X×K is normal and Δ-paracompact. We will show that the assumption for X can be weakened as X is a normal and Δ-paracompact space. To prove this fact, we use a concept of Δn-paracompactness. It is a generalization of Δ-paracompactness, actually Δ2-paracompactness is equivalent with Δ-paracompactness. For each n≤m<ω, Δm-paracompactness implies Δn-paracompactness. We prove that in the class of orthocompact and normal spaces, Δ-paracompactness implies Δn-paracompactness for every n∈ω.