Is the fast of one who swallowed a small piece of skin from his lips during prayer, after remembering that he was fasting and being unable to remove it, valid, and was he obligated to interrupt the prayer and rinse his mouth?

Whoever takes a piece of skin from his lip while fasting must spit it out. If he swallows it forgetfully, or cannot find it, or finds it difficult to remove it and swallows it with his saliva, then his fast is valid. "But if it separates from his saliva and he swallows it intentionally: his fast is broken." As long as you did not intentionally swallow it, there is nothing upon you. It was not necessary for you to leave the prayer; rather, it is forbidden. The obligation is to discard it.