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Then we have T 2 x 0 = x 0. From (2.6), we have x 0 = T 2 x 0 = U 2 x 0. From the definition of U 3, we have U 3 x 0 = T 3 ( ( 1 − α 3 3 ) U 2 x 0 + α 3 3 x 0 ) = T 3 x 0. By using the same method as above, we have x 0 = U 3 x 0 = T 3 x 0. Continuing on this way, we can conclude that x 0 = U i x 0 = T i x 0 (2.10).
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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