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The phrase "establish cooperation" is correct and usable in written English
It typically is used to refer to the efforts of two or more parties to work together towards a common goal, such as "The two companies agreed to establish cooperation in order to develop new products for their respective markets."
Exact(18)
"Many countries are approaching us to establish cooperation on security matters.
The opposition is moving "minister by minister, institution by institution," trying to establish cooperation, a party leader said.
"We reiterated our disposition to establish cooperation mechanisms for international aid and assistance," Mr. Maduro wrote Tuesday on Twitter.
"We have to change our military structure radically, establish cooperation between our different types of forces and improve our reconnaissance," he said.
They discussed "a range of military issues, as well as ways to establish cooperation in the struggle against terror," according to the Turkish military.
Nyi Hla Nge, a retired rector at the university, said he had written a letter to the president of the Massachusetts Institute of Technology seeking to "establish cooperation".
Similar(42)
When children "feel felt" and cared for, they're quicker to recover and re-establish cooperation with their parents.
SunFarmer is also working to build the sustainability of Nepal's nascent solar industry by establishing cooperation partnerships with domestic solar providers.
It is oriented by a Negotiation Logic, aiming at establishing cooperation between units by using a suite of fundamental rules based on an objective function defined by price, quality and delivery parameters.
The introduction of mechatronic functions brings about implications for the engine management system (EMS) as a whole, which in turn influences established cooperation models between OEMs and their suppliers.
It's also a practical tool to evaluate already established cooperation agreements reflected in the cases presented in this paper.
More suggestions(15)
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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