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Now we state our main lemma as follows.
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Next, we give the main lemmas as follows.
To prove the convergence of the proposed algorithm, we need a lemma as follows.
First of all, the main lemmas including operator iteration can be stated as follows.
Next, we give the following main lemmas.
The main lemmas we used are the following.
We also need to introduce some lemmas as follows, which will be used in the proof of our main theorems.
As a generalization of Lemma 3.1, we now present the main result as follows.
In order to render the main results as transparently as possible, we also give some preparatory lemmas and theorems.
Using Lemma 5.1, which is crucial to the convergence of the iteration method, we are able to give the main result as follows.
Now, we give a key lemma as a proof of our main conclusions.
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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