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That is, there exists such that (2.20).
That is, there exists an such that (2.12).
That is there exists a solution of the problem: where.
that is, there exists a sequence satisfying (4.21).
Further,, that is, there exists such that (3.2) is satisfied.
That is, there exists a positive constant, such that.
That is, there exists a positive constant such that (2.12).
Conversely, suppose that, that is, there exists such that.
that is, there exists a matrix function such that.
Hence, there exists,, that is, there exists such that.
Case 1. Assume that and have an intersection point, that is, there exists such that.
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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