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The phrase "of his argument" is correct and can be used in written English.
It is typically used to indicate possession or association, and can be used to refer to a specific argument made by someone. Example: The professor was not convinced by the validity of his argument.
Exact(60)
He lost the thread of his argument.
The core of his argument?
The absurdity of his argument was overshadowed by its obscenity.
But the gist of his argument was similar.
I genuinely can't follow the rest of his argument.
The vehemence of his argument speaks to its time.
First would come the crushingly heavy wheels of his argument.
As King develops his theme, he unspools the thread of his argument with wonderful fluidity.
The burden of his argument was that we did not need an inquiry into press standards.
But Mr Paul is eliding the stronger part of his argument.
He was responsible for the integrity of his argument, and his advisers ultimately controlled it.
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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