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The phrase "a simple fact of" is correct and usable in written English
It can be used to introduce a straightforward truth or reality about a situation or topic. Example: "It is a simple fact of life that we all face challenges at some point."
Exact(42)
It's a simple fact of life.
That's a simple fact of the art form.
That's just a simple fact of life in America".
That's a simple fact of cricket and results".
It's a simple fact of political reality that some states are less competitive battlegrounds than others.
But this is a simple fact of detective work; the means themselves are not for the squeamish.
Similar(17)
Implicit in the concept is the recognition of a simple fact: each of us is only human, capable of committing acts of good and bad in equal measure.
Yet Hester spoke so fearlessly, accepting as her due a simple fact, that a display of emotion on his part seemed out of place.
Start with one of the easiest clues, that gives you a simple fact matching two pieces of information together.
It is a simple fact that the majority of people in the world can speak.
These numbers are so big they cover up a simple fact: the vast majority of us still aren't rich.
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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