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The phrase "obtain by" is correct and usable in written English.
It means to acquire something as a result of effort or to achieve something by obtaining it. For example: I obtained my degree by studying hard for four years.
Exact(60)
The word "express" derived from the medieval Latin "expressare," meaning to "press out" or "obtain by squeezing".
Animals can use only organic nitrogen, which they obtain by consuming plants or other animals.
But that change would almost certainly need the blessing of the teachers' union, which Mr. Bloomberg may try to obtain by holding the contract hostage.
But the bill itself was almost impossible to obtain by September 1987; such was the demand for copies that the government printer had run out.
"The acquisition and use of bulk provides vital and unique intelligence that the security and intelligence agencies cannot obtain by any other means.
Frederick now decided to gain in alliance with France what he could not obtain by opposing it.
Using all conditions, we obtain by computing.
Therefore we obtain by (5.9) that (5.13).
Since, we obtain by Lemma 2.4.
Then, we obtain by Lemma 3.1 that (418).
Let be so large that, we obtain, by, that (3.6).
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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