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The funny thing is, if she refused to sign, would they even be able to take him away?

Shh don't tell the plebs, but, uhh, no, they wouldn't (legally).
She gave Consent.
Just like our Pair Rents do when they Sign us over to the Crown/Vatican when we are Berthed at Dock ed Oar.

I love your golden web language.

Here's an example of what can happen when you "just say no",

Skip to 403 seconds,

That is a good example actually.

-I Do Not Consent-

or, Jedi Mind trick for Christians and Atheists alike:

I NEED A BIBLE IN MY CAR NOW!!