Lots of cauterization. Surgeons have a tool which cooks the ends of any small bleeders they encounter to prevent further blood loss. This is not exactly great for the patient, obviously, but it’s a kind of injury the body can easily work around. Small vessels like that can be replaced by the body to maintain blood flow after the surgery.
For larger vessels on which cautery wouldn’t work, there is ligature. This is basically where you tie the vessel closed with something. The Romans discovered this thousands of years ago, and the knowledge passed to the ~~Arab~~Muslim/Greek/Persian/etc empires from there, which is why their medicine was so good.
For even larger vessels, the answer is surgeons don’t cut those. And if they do, that’s what malpractice insurance is for.
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