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The phrase "an immediate consequence" is correct and usable in written English.
It can be used to refer to a result or effect that occurs right away following a specific action or event.
Example: "The heavy rainfall had an immediate consequence on the local river levels, causing them to rise rapidly."
Alternatives: "a direct result" or "an instant effect".
Exact(60)
Consequently, the result of Wardowski [13] can be presented as an immediate consequence of Theorem 4.2.
As an immediate consequence we obtain enhanced binding for electrons.
As an immediate consequence, we obtain the following result.
This is an immediate consequence of the previous lemma.
Finally, part (iii) is an immediate consequence of Lemma 5. □.
The proof is an immediate consequence of Theorem 3.1.
Since, is an immediate consequence of Theorem 2.1.
Property 1 is an immediate consequence of Theorem 3.2.
We can get an immediate consequence of Lemma 6.
Hence (14) is an immediate consequence of Theorem 3.
(c) This is an immediate consequence of (a) and (b).
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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