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(The ships crashed, in separate storms, before proving their battleworthiness).
Elder does so immediately, but younger waits until elder has exchanged before proving it.
It usually grows slowly and can take years before proving fatal.
Before proving the results, we state a few useful lemmas.
We give an important relation before proving the bilinear estimate.
Before proving the next theorem, we need the following definition.
Before proving Theorem 1.2, we now show some preliminary results.
Before proving Lemma 3.2, we need to fix some notations.
Before proving the main theorems, we introduce some earlier results.
Before proving Theorem 1.2, we need some lemmas.
Before proving Theorem 2.1, we give some denotations used later.
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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