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The phrase "the proof that" is correct and usable in written English.
You would use it to introduce the evidence that validates or verifies something. For example: "The proof that diamonds are the hardest substance on Earth is in its incredible durability."
Exact(59)
"There is no value in us going to the courts without the proof that is beyond doubt," Mr. Chamisa said.
This complete the proof that.
This completes the proof that (4.19) solves Eq. (4.17).
This is the proof that it works".
The proof that this is unique.
Jokowi is the proof that democracy works.
And the proof that crime pays?
We're the proof that something has changed".
We have the proof that this agreement existed.
"Where's the proof that the fishery is collapsing?
Where's the proof that our reason is any good?
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Since I tried Ludwig back in 2017, I have been constantly using it in both editing and translation. Ever since, I suggest it to my translators at ProSciEditing.

Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com