Is sequential order mandatory between rinsing the mouth (madmadah) and nose (istinshaq) and washing the rest of the face?

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Sequential order between rinsing the mouth and nose and washing the remainder of the face is not obligatory because they are parts of the face. However, it is recommended to commence with them before the rest of the face because nearly everyone who described the Prophet's (peace be upon him) ablution mentioned starting with them, barring rare exceptions.