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Jun 3, 2021 at 17:04 history edited KP Hart CC BY-SA 4.0
Addendum
May 31, 2021 at 18:35 vote accept user254885
May 31, 2021 at 17:24 comment added user254885 Thank you! Yes, this is correct, and probably good for our purposes. Unfortunately, my second statement was misstated ($A'$ and $B'$ were intended to be arbitrary), again, but your answer still applies.
May 31, 2021 at 15:44 history edited KP Hart CC BY-SA 4.0
added 399 characters in body
May 31, 2021 at 15:32 comment added user254885 Indeed. Sorry. Now the question is stated correctly, thanks.
May 31, 2021 at 15:27 comment added KP Hart You did not require that.
May 31, 2021 at 15:12 comment added user254885 But why would such $U'\supset U$ and $V'\supset V$ be disjoint ?
May 31, 2021 at 14:47 history answered KP Hart CC BY-SA 4.0