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The phrase "been able to establish" is correct and usable in written English.
This phrase usually indicates that something that was previously not certain has been established or confirmed. For example, "The research team has been able to establish the cause of the disease."
Exact(60)
The trajectory we have been able to establish will not be realised," he said.
Psychiatrists have never been able to establish the line between mental health and mental illness.
They are partners, people with whom I have been able to establish a good relationship".
He's been able to establish an identity for himself outside of basketball".
She hasn't been able to establish herself over the years as much as others".
In these places, Democrats have been able to establish themselves as the safe and orderly party.
But all that his critics have been able to establish is that he has been consulted.
"They never have been able to establish that I have the money stashed away".
Nobody's been able to establish rule of law or end rampant corruption.
Not even the most ardent drug warriors have been able to establish an Osama-Sudafed link.
The police said they had not been able to establish any connection between the men.
More suggestions(15)
been able to install
been able to achieve
been possible to establish
been able to ascertain
been able to assess
been able to formulate
been able to reestablish
been used to establish
been allowed to establish
been able to acquire
been able to reach
been able to pursue
been able to introduce
been unable to establish
been able to conclude
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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