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on the condition/demand + object + verb

Could someone enlighten me and explain the grammar behind this sentence:

The chair was donated to the Museum on the condition that no one sit in it.

Why isn't it 'on the condition that no one sits in it' or 'on the condition that no one can sit in it' ?

Is this the same structure as when using words like 'demand' as in 'I demand he not sit in it' ? I don't really get the grammar here as well and have been using it like that because I read it a few times. So, I would be more than happy if someone could explain this as well.

Thank you in advance!