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(i) exists for all ; (ii).
(i) for all, (ii) for every, (iii).
(i) for all ; (ii) ; (iii), for all.
(i) is continuous for all ; (ii).
(i) is continuous for all ; (ii)for all, one has.
(i) is continuous for all ; (ii) for each, and.
(i), for all, (ii) and if and only if.
(i nonexpansive if for all ; (ii)firmly nonexpansive if for.
(i) for all and for all ; (ii) implies.
(i), for all ; (ii), for all and all.
(i for all, (ii)for all, (iii)for all, (iv).
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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