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From conclusion (2.25) of Lemma 2.3 and the relation (2.59).
In this table, "\(\vdash\)" separates premises from conclusion; it may be read "therefore".
From conclusion (1), we only need to prove that (x^ in operatorname{Fix}(T)).
It is obvious from conclusion (iii) that (lim_{lambdato0+}|x_{lambda}-x_{mu}|=0).
From conclusion (I), we have (lim_{nrightarrowinfty }Gamma_{n}(x^,y^)) exists, therefore (lim_{nrightarrowinfty}Gamma _{n}(x^,y^)=0).
This fact allows us to apply the LK rules in either direction, forwards from axioms to conclusion, or backwards from conclusion to axioms.
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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