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Hello, Nice to meet you.
I'm a Korean English teacher working in high school in South Korea.
I have one question.
The other day, I noticed the following sentence.
"Social connections are so essential for our survival and well-­being that we not only cooperate with others to build relationships, we also compete with others for friends."
It is from the book, "Friend and Foe."

I usually teach students that "but" is absolutely necessary in the construction "not only ... but also ..." because "but" must conjoin two independent clauses.
Of course, I think that sentence is okay. 
 I just would really like to know when we can omit "but" in the construction "not only ... but also ..."

Personally for days I have looked up dictionaries, grammar books and so on.
And I have been googling but I couldn't find the answer.
So I came to write here. 
I would appreciate it if you would help me out.

Thank you so much in advance.