To be pedantic, interlining is more than just through checking of luggage (although around many forums, colloquially that is what it is taken to mean). Interlining also involves ticket issuing, service recovery, airline invoicing etc. But usually around here people take it to mean through checking of luggage. But again to be pedantic, through checking of luggage on separate tickets in a separate reservations (PNRs) is simply through checking of luggage, it is not interlining, and therefore depends on the policies of the first carrier and whether or not they have an agreement to through check to the second carrier (if they have an interline agreement they would).