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I find that when proving a theorem by the method of contradiction, some authors start by saying "assume to the contrary" while some use "assume on the contrary". According to this post, it seems that "to the contrary" and "on the contrary" have subtle differences. So my question is which one should I use, "assume to the contrary" or "assume on the contrary"?

Below is an example.

Theorem. A is equal to B.

Proof: Assume on/to the contrary that A does not equal B. ...

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  • Can you give the full sentence you want to use it in?
    – Laurel
    Commented Feb 27, 2018 at 4:17
  • I'd probably go with just "Assume that A does not equal..." But forced to include the contrary I'd use "Assume the contrary- "
    – Jim
    Commented Feb 27, 2018 at 4:24

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