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The phrase "asserted that they knew" is correct and usable in written English.
You can use it when reporting someone's claim or statement about their knowledge or awareness of something.
Example: "The witnesses asserted that they knew the suspect well and could identify him."
Alternatives: "claimed that they were aware" or "stated that they understood".
Exact(3)
Some asserted that they knew biology teachers who were afraid to bring up theories about holes in Darwin's theory.
Citing a number of backdated transactions and what he characterized as dubious documentation the defendants received from Mr. Madoff, the trustee asserted that they "knew or should have known" that Mr. Madoff was conducting a Ponzi scheme.
The people who worked with the youth in this particular cult ranged from former hippies to rigid discipline freaks, but I felt I could trust these people because they asserted that they knew more than my parents.
Similar(57)
While they now assert that they knew all along that the democratic transformation of Iraq would be long and hard, they were clearly taken by surprise.
They assert that they know how and when to disarm the bomb.
But why is it so clear that God would have wanted to do this?… Creationists don't need to assert that they know what God would have had in mind if he had built the panda.
They are willing to assert that they know something of the way ahead and they are ready to share that knowledge.
Other student resistance behaviors reported were more active, such as "disruption," wherein a student may purposefully interrupt class, "student rebuttal," wherein a student asserts that they know what will work for best them, and the dreaded Appeal to Powerful Others, which would involve threats to take student complaints about a course to an authority figure, such as a chair or a dean.
Criticism is the passcard to some of the attention-getting personality spectacles who assert through sheer chutzpah that they know what they're talking about.
The plaintiffs assert that they did not know that their relatives were to be buried on top of construction debris.
Many of the people assert that they did not know they were going to be asked to join the Herbalife network.
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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