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    $\begingroup$ I was nearly finished writing up the same proof. +1 $\endgroup$ Commented Jul 19, 2023 at 9:18
  • $\begingroup$ @JaapScherphuis great minds... $\endgroup$ Commented Jul 19, 2023 at 10:15
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    $\begingroup$ ...thinks alike, but fools rarely differ. [don't forget that second part of the quote! XD although I believe in this case only the first case applies] $\endgroup$ Commented Jul 20, 2023 at 4:49