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The phrase "establish this" is correct and usable in written English
You can use it when referring to the act of setting up or confirming something specific. Example: "In order to move forward with the project, we need to establish this agreement between both parties."
Exact(60)
"We worked so hard to establish this place.
The writer quotes authorities to establish this fact.
To establish this way of treating experience required careful training.
Two major sets of observations helped to establish this view.
The officials said they had not yet decided where to establish this meeting place.
"Everyone wants to establish this is a Hatfield vs. McCoy," he said.
He had to do quite a lot of dtetctive work to establish this.
Trying to establish this information for the families is our first priority".
Figures available from the Office of Vital Statistics clearly establish this fact.
It should not have taken a deadly accident to establish this simple rule.
How important was it for you to establish this world before you introduced the surgery itself?
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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