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The phrase "assert that they have" is correct and usable in written English.
It can be used when someone is stating or claiming possession or a certain condition regarding something.
Example: "The researchers assert that they have discovered a new method for treating the disease."
Alternatives: "claim that they possess" or "declare that they have".
Exact(40)
Chinese health officials assert that they have acted promptly upon laboratory confirmation of cases.
Both the eye academy and Dr. Roth assert that they have no financial relationship with Apple.
So it quite a stretch to assert that they have a working quantum computer let alone one that potentially scales.
Victims assert that they have identified the largest documented cluster of male breast cancer cases: 73 and counting.
Accordingly, Hawkins and Patterson assert that they have standing to sue to invalidate the loan agreement as illegal and unenforceable under ECOA.
If third parties assert that they have an interest in all or part of the property, those interests can be adjudicated at one time in the ancillary proceeding.
Similar(20)
Family members of the thousands dead assert that they've been wronged.
Parents assert that they've been drugged by a kind of "magic powder" sprinkled over their houses during the attacks.
Keeping it would permit officials to assert that they had been faithful to the original architectural concept.
The Labor Department also ended a labor market certification process that allowed employers to simply assert that they had searched for American workers.
Prosecutors told reporters after the verdict that other women had come forward to assert that they had been abused by Mr. Weberman, including one who had come forward during the trial.
More suggestions(17)
confirm that they have
establish that they have
assure that they have
argue that they have
affirm that they have
emphasize that they have
confirming that they have
claim that they have
allege that they have
suggest that they have
establishing that they have
contend that they have
insist that they have
assert that they value
assert that they improve
assert that they play
assert that they talk
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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