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Then either there is an (v_{0} in V) such that (L^{-1}(v_{0})) is a empty set, or the whole intersection (bigcap_{v in V}N^{-1} v)) is nonempty.
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That is an empty apology.
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For me it is an empty phrase..
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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