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Discover Ludwig"under the above" is correct and widely used in written English
It is generally used to refer to something previously stated in the same document or conversation. For example: We discussed the customer's needs extensively during the meeting. Under the above, I recommend we increase the number of customer service representatives accordingly.
Exact(60)
Under the above assumptions, we have.
Under the above assumption, if, then.
Under the above considerations, we now have.
Under the above conditions, one has (219).
Under the above conditions, we can represent the system.
Under the above conditions, the following properties are fulfilled.
Under the above conditions, we define the function by.
Under the above assumptions (4.1) has a unique mild solution.
Under the above condition, the desired conditional expected value is.
Under the above hypotheses the scattering rates read as follows.
Under the above assumptions, we have the following existence theorem.
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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