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The phrase "entrenched within" is correct and usable in written English.
You can use it to describe a situation that is deeply rooted or firmly established. For example, you could say: "Political divisions are deeply entrenched within our society."
Exact(39)
Flash forward a few years and R&B has clearly become entrenched within the musical landscape.
A fourth doubter is Green's sister, Susan Hilton, who said: 'I felt helpless to do anything to rescue her as she was entrenched within the sect.
Mayville said he had seen no reports of civilian casualties, even as Isis is entrenched within civilian populations in cities like Raqqa.
Big business customers, though, picked up the slack, showing how Microsoft's products are entrenched within one of the most lucrative sectors of the technology market.
By 1984, however, the East Village art scene was fully entrenched within the workings of the New York art world with over seventy commercial galleries located in the space of fourteen blocks.
Offensive costumes and cultural appropriation has become entrenched within the holiday itself and what was once a day remembered for children trick or treating is now just a celebration of distasteful and borderline racist outfits.
Similar(20)
The consumerization of the enterprise and the "BYOD" (bring your own device) trend within businesses mean that phones and tablets are becoming entrenched features within the corporate and consumer worlds.
Mr. Gandhi's campaign could eventually threaten entrenched interests within the party, analysts say, which is why, for now, the internal voting is limited to the youth organizations.
"The issue with this video is that it touches on a lot of deeply entrenched issues within the Muslim community," adds the group's unnamed spokesman.
Even then, however, the entrenched elements within the intelligence agency who had backed the Afghans refused to switch sides without a fight, according to senior Pakistani officials aligned against the Taliban loyalists.
Sigur Ros Iceland's favourite sons have always entrenched themselves within their own world, writing songs in their made-up language called Hopelandic, which is a constructed language of nonsense syllables resembling the phonology of the Icelandic language.
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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