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'have shown themselves' is correct and usable in written English.
You can use it when you want to describe how something has become visible or evident. For example, "The cracks in the foundation have shown themselves over time."
Exact(56)
But they have shown themselves useless.
The top two spots have shown themselves.
They have shown themselves to be dishonorable".
"Some of these people have shown themselves to be vindictive".
"The public have shown themselves to be incredibly decent.
Iran's clerical leaders have shown themselves to be sophisticated politicians.
But students and other protesters have shown themselves fed up with all sides.
"Somalis have shown themselves to be capable of remarkable innovation and enterprise," said Duale.
These people have shown themselves to be superb psychological warfare and media operators.
Our lads have shown themselves to be good at that, not allowing themselves to be discouraged.
So far, he and his family have shown themselves willing to try.
More suggestions(15)
have demonstrated themselves
have documented themselves
have established themselves
have suggested themselves
have worked themselves
have shot themselves
have revealed themselves
have sentenced themselves
have painted themselves
have forced themselves
have remade themselves
have let themselves
have protected themselves
have presented themselves
have made themselves
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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