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To do this follow the same method as above but, when you are stitching through the holes, hold a toothpick or matchstick on top of the button and stitch over it (a) - see illustration below).
Using the same method as above, (3.32).
By the same method as above, define by for all.
Using the same method as above, we have (2.25).
Using the same method as above, we can also deduce a contradiction.
By the same method as above, one can find a Cauchy-Jensen mapping which satisfies (2.9).
If (deg P z)=p< q=deg Q z)), then using the same method as above, we get a contradiction.
Using a similar method as above, we obtain (lambda(f)=sigma(f)). (ii) Set (z=w+1).
By the same method as above, we have (lambda (frac{1}{Delta f z)} )geqlambda (frac{1}{f z)} )).
By using the same method as above, we see that ((I-d_{2}D_{2})) is a nonexpansive mapping.
Using the same method as above, we derive equations for the post-SN outer binary semimajor-axis and eccentricity.
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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