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I never understood the difference between

  • I prove 1+1=2
  • I prove that 1+1=2

Likewise umpty more similar sentence pairs. When is "that" mandatory, when is it only a matter of style, and which style is preferable?

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This is an example of a that-clause. You can find more information from the Cambridge Dictionary here. This is the verb + that-clause version using a reporting verb "prove". The same dictionary has an article about reporting verbs and "that" here.

Reading further in the second article, they provide examples of when "that" is left out:

We often leave out that after these verbs, especially in informal speaking. This is sometimes called zero-that. This is especially common after guess, think, hope and reckon.

Based on this and other information I found, I would suggest that you remove "that" from your sentences only when the intended meaning of the sentence can still be clearly discerned.

As for which style is preferable, I can't say definitively. I did find this paper which talks about the usage of both styles through the ages.

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