I am against Dawkins' position here, but your statement just isn't correct at all: https://womensdeclaration.com/en/declaration-womens-sex-based-rights-full-text/
The petition, which I do not agree with, talks about not allowing trans women to use "specialist single-sex provisions for women and girls who have been subject to violence, such as shelters and health care facilities".
I do not agree with that clause, but to say this is a call for "ending healthcare for trans people under 18" is simply false to the fact.
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Our enemies are compulsive liars. Their great skill is to be able to come with new lies at the drop of a hat. To beat them, we need to be scrupulous about the truth.