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The phrase 'but through' is correct and usable in written English.
It is a conjunction that indicates contrast or opposition, and is usually used to introduce something that is contrary to what has just been said. Example: She had been warned about the consequences of her actions, but through her determination she was able to achieve success.
Exact(56)
Not just in sport, but through it.
But, through violence, he has held power.
But through snow.
Not through sleep science — but through psychology.
"But through difficulty, intimacy is found.
But through therapy I learned coping mechanisms.
But, through television, we were already there.
But through conservation, he said, "we'll definitely delay it.
But, through it all, his creative drive remained undiminished.
But through this anthropology a much grander project emerges.
But, through shifting alliances, it has increased its influence.
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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