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Exact(23)
Hence, we get the assertion.
By Theorem 2 again, we get the assertion.
end{aligned} Then we get the assertion (31).
collecting the above estimate, we get the assertion.
Thus from (3.13) and (3.20) we get the assertion.
Then, by above relation and by virtue of Theorem 3, we get the assertion.
Similar(37)
Using now Theorems 2.1 or 2.2, we get the assertions (a -(c).
Following the proof of the remainder of Theorem 3.2, we get the assertions (39).
Thus, applying Lemma 2.2 gets the assertion (3.6), as desired.
Letting p → ∞, we get the desired assertion (3.9).
Following the proof of the remainder of Theorem 3.1, we get the required assertion (37).
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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