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The phrase "and that done" is correct and usable in written English.
It can be used to indicate that a task has been completed or to transition to the next step after finishing something.
Example: "I finished the report, and that done, I moved on to the presentation."
Alternatives: "once that's completed" or "having done that".
Exact(7)
He was given a set of complicated forms to fill in, and that done, was asked to report to a Mr. Carmichael.
We get to hear how Pippin II, Charles Martel's father, roused himself from a coma long enough to issue a death warrant for some throne-grabbers and, that done, lapsed back into the coma and died.
Next, they were asked to roll the die two more times, to satisfy themselves that it was not loaded, and, that done, to enter the result of the first roll on a computer terminal.
So I had some tests and that done.
The comparison between the lattice parameters and space group of our work and that done by Buchanan and Harris showed identical results (Table S1).
The comparison between the lattice parameters and space group of our work and that done by Hermansson showed the same results (Table S2).
Similar(53)
"And that didn't last very long.
(And that didn't offend anyone present).
And that does two things.
And that did disturb me.
And that does the trick.
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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