prove that every answer set of program(pi) is closed under(pi)
plz i need the answer
Further, proof theory is one of the four main branches of mathematical logic (see "Handbook of Mathematical Logic" edited by Barwise). Logic programming (LP) is a quite small topic in proof theory. Concepts like "product of a program" and "closed under a program," which are particular to LP and possibly to a particular source about LP, are in no way basics of logic.
I've studied logic programming, but it was quite a few years ago, and I can't help with your question without being reminded of the definitions.