Paracompactness of spaces which have covering properties weaker than paracompactness

We prove that (i) a collectionwise normal, orthocompact, θm-refinable, [m,ℵ0]-submetacompact space is paracompact, (ii) a collectionwise normal, [∞,m]-paracompact [m,ℵ0]-submetacompact space is paracompact. This gives a sufficient condition for the paracompactness of para-Lindelöf, collectionwise normal spaces.