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The phrase "they have proven" is correct and usable in written English.
You can use it to talk about an idea or fact that has been confirmed or demonstrated. For example, "They have proven that climate change is real and is having an impact on our planet."
Exact(60)
But they have proven popular.
"They have proven they are a tribe.
So far they have proven unsuccessful.
"They have proven they are survivors".
In the event, they have proven to be all three.
They are Test animals and they have proven it.
"They have proven to be an unpredictable force".
They have proven themselves unworthy of their position.
They have proven that they were not simply war tourists.
"They have proven our innocence in this," Ms. Fernandez said.
We elect people because they have proven results.
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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