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"We have established" is a correct and usable phrase in written English.
You can use it when you want to describe something that has been done or confirmed, or when you need to indicate a change in status. For example, "We have established a new policy for protecting customer data."
Exact(60)
We have established that this is a Jersey company.
We have established a college scholarship in David's name.
We have established ourselves as an educational facility".
"We have established the flow of money," he said.
We have established the officer traing academy in Kabul.
We have established certain parameters for these debauches.
"We have established that everyone will get paid," Mr. Smellie said.
We have established joint ventures in Canada, India and the Philippines, one in each country.
"We have established a certain credibility on the fashion side," he said.
"We have established that our students are interested in these events," he said.
"We have established that this was a false alarm and the man is the only suspect.
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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