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**GIPC** I understand your point but the thing is, I have to somehow formally prove it.

So I see you did a little sandwich there and bounded R_1, but I can't just take this fact for granted, I have to somehow prove and show this for myself. So that is what I am trying to do, first find some map using some injective function f between 2^N to R_1 and then some map between R_1 to N^NcrossN^N. and actually, saying that R_1<N^NcrossN^N (but not <=)is not good enough because I wat to prove that R_1~P(N).

Unless there is some other way to prove that |R_1|=|P(N)| that i am missing?