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The phrase "with the assertion that" is correct and usable in written English.
You can use it when introducing a statement you want to make, typically one which could be controversial. For example, "I disagree with the conventional wisdom, with the assertion that people should naturally be afraid of strangers."
Exact(60)
I doubt anyone would argue with the assertion that successful societies need successful education systems, and these need good teachers.
PC World disagrees with the assertion that Android users are told about the contact pull.
He disagrees with the assertion that publications like his offer little or no social value.
Few argue with the assertion that it was the most dominating game ever pitched at Wrigley.
Her breathy speech concludes with the assertion that she is "friends" with all the nominees.
"I can't disagree more with the assertion that we disrespected Martinez".
He took issue with the assertion that caseworkers are frequently ill-prepared in court.
The White House would, of course, argue with the assertion that it had no postwar plan.
Mr. Perry's critics take issue with the assertion that he has the background to battle entrenched interests.
"Can anyone argue with the assertion that, for neocons, Obama is always wrong and Bibi is always right?
"We disagree with the assertion that we have a disregard for workplace safety," Scott Dean, a BP spokesman, said.
More suggestions(15)
with the declaration that
with the emphasis that
with the assertiveness that
with the pronouncement that
with the dictum that
with the characterization that
with the remark that
with the thesis that
with the argument that
with the proposition that
with the saying that
with the statement that
with the reassurance that
with the notion that
with the affirmation that
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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