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The phrase "admitting the fact" is correct and usable in written English.
It can be used when acknowledging or accepting a particular truth or reality in a discussion or argument.
Example: "After much consideration, I find myself admitting the fact that I was wrong about the project's timeline."
Alternatives: "acknowledging the reality" or "accepting the truth".
Exact(8)
I tried again, and realized that her hesitation concerned admitting the fact of drinking alcohol.
I lapse into mockney with the best of them, and avoid admitting the fact I was educated at Eton on the grounds that, nearly two decades since I last had anything to do with the place, it is for many people a more important marker of who I am than any of the offensive opinions I have developed since.
If Enron was too far gone for help at this point, Enron certainly wasn't admitting the fact.
"McDonald's is finally admitting the fact that it is no longer a growth company," said Allan F. Hickok, an analyst with US Bancorp Piper Jaffray.
"The would be admitting the fact that they bombed us first".
Cummings writes "We cannot continue to pour taxpayer dollars into buckets with holes," without ever admitting the fact that Congress itself created that bucket -- holes and all.
Similar(52)
Pietersen, in his self-absorption, could never admit the fact.
Not that everyone seemed willing to admit the fact.
They all admitted the fact but could not advance a sound reason why.
Samba-Panza admitted: "The fact that the Muslim community is leaving the country en masse is a real preoccupation for me and my government.
He told me that his words were misconstrued, but admitted, "The fact that I made those comments reflected that I was going through internal evaluations" — a rare acknowledgment of the existence of the unconscious.
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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