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Here, we used that.
Here we used (11).
here, we used, and.
Here we used the fact that (2.26).
end{aligned} Here, we used condition (5).
Here we used Assumption (2.6), that is,.
Here we used a different algebraic structure.
here we used (5) and (6).
Here, we used a so-called 2cc-coupler.
here, we used the fact that for each and, (3.9).
here, we used and in the last inequality.
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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