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It follows from the existence of and Lemma 2.10 (ii) that there exists an element in with (3.6).
(i For each, there exists an element such that (4.1).
Hence for each and, there exists an element such that.
This implies that there exists an element such that as.
Then there exists an element in such that (2.22).
If and commute, then there exists an element such that.
(ii there exists an element with satisfying for some.
then there exists an element such that for all.
Then, there exists an element x ∈ A such that d ( x, T x ) = d ( A, B ).
Then there exists an element (xin A) such that d(gx,Tx)=d(A,B).
Then there exists an element z ∈ B e s t ( T ).
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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