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?
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.