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The phrase "the final proof" is grammatically correct and can be used in written English.
It refers to the ultimate evidence or confirmation of something. Example: The DNA analysis was the final proof that the suspect was guilty of the crime.
Exact(60)
Human flight is the final proof of a city's disintegration.
The final proof is The Great Stem Cell Compromise.
The UN samples will likely offer the final proof, but analysis could take several more weeks.
To Cheney, this was the final proof that Bush had lost his will.
"The final proof requires that you isolate the virus from the bats".
His photographs are the final proof that a stupendous human artefact has been lost forever.
The final proof is reversal of symptoms on a gluten-free diet.
(They don't read each other's work, apparently, until it is in the final proof stage).
They argue that we should not wait for the final proof.
That particular choice, he fumed, was the final proof of American perfidy.
Now, mathematicians said, the final proof of the elusive Taniyama conjecture is in sight.
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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