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The phrase "your approval of" is correct and can be used in written English.
It is commonly used when talking about someone's opinion or acceptance of something. Example: I am grateful for your approval of my project proposal. Your support means a lot to me.
Exact(7)
"At the very least, your approval of this invoice without making any effort to evaluate the accuracy of the receipts being submitted is negligent," the letter to Ms. Capuli said.
I am writing to inform you of a number of significant developments related to the Company, including the renegotiation of the terms of the Company's Series C financing, and to request your approval of, and to offer you the opportunity to participate in, the revised financing.
So you justify your approval of what Walter White is doing.
It's a much higher bid to profess your love of something than it is to express your approval of it.
In this spirit, we are grateful for your approval of the resolution establishing December 9 as the International Day of Commemoration and Dignity of the Victims of the Crime of Genocide and the Prevention of this Crime.
With most print shops, the proofing stage is the only time you will have to make corrections, as your custom business cards will go to print following your approval of the proof.
Similar(51)
Instead of allowing your approval to be the decision of others, make it your own.
Whether his form of "properly" meets with your approval will, of course, depend on your capacity to perceive virtue in the familiar and the sentimentally melancholic (and in brevity: Alone in the Universe clocks in at roughly 35 minutes' duration).
In a letter to Morgan, Powell said: "Your approval today of this new school suggests that you are now intent on increasing selection in our schools system by the back door.
Cast your approval rating of the Obama administration across 14 issues.
The Past: Our decision to grab some dinner before the show, Indian cuisine, (subject to your approval first of course), led to some great conversation about everything, including questions about my earlier life.
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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