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Exact(23)
This contradiction proves (2.3).
This contradiction proves our claim.
This contradiction proves Lemma 2.2.
This contradiction proves the proposition.
This contradiction proves the first conclusion.
This contradiction proves the above claim.
Similar(36)
The contradiction proves the desired inequality.
The obtained contradiction proves the assumption is not valid.
A contradiction proves the left-continuity of (xi_{sup }(gamma,delta)) with respect to γ.
(38) The obtained contradiction proves that inequality (36) is true for (tgeq t_{0}).
The contradiction proves that inequality (3.7) is wrong, and relation (3.6) is true.
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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