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The phrase "established institutions" is correct and can be used in written English.
It refers to longstanding organizations or systems that have been in place for a significant period of time. Example: The country's economy relies heavily on the stability and functioning of its established institutions, such as the central bank and regulatory agencies.
Exact(53)
Even established institutions have trouble.
In the long term that advantages established institutions and students from advantaged backgrounds".
"Frustrations are leading people to question established institutions and international norms," she said.
Popular attitudes and established institutions and procedures are all in a state of flux.
Foucault advocated resistance to the political status quo and the power of established institutions.
Most residencies are in urban areas, which have established institutions large enough to support the programs.
It is harder for the top-down, established institutions to embrace the new innovative ethos.
Similar(4)
Hispanic Society of America Closer to the top of Manhattan, two long-established institutions give consistent pleasure.
On that score, San Francisco seems better placed than ever to challenge the longer-established institutions.
Meanwhile, collaboration between top-tier Asian universities and less-established institutions in the region was lagging.
Long-established institutions and practices have evolved over many generations and are thus, say conservatives, "tried and tested".
Related(17)
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stated institutions
organized institutions
established foundations
tested institutions
established establishments
pursued institutions
elected institutions
erected institutions
partnership institutions
formalized institutions
determine institutions
formation institutions
established models
established starters
established players
held institutions
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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