in which i own someone then fold

[Noodles.]

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also i still don't really see how checkcalling the river is better than betfolding it.

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Because when you check-call you see the showdown. Betting will NEVER make him fold (see flop&turn line) so you will either fold or see the showdown for one bet. Check-calling always gets you to the showdown for one bet, so that is clearly better: it costs the same but there's a smaller chance of not seeing the showdown.