I am very sorry; please forgive me
A very formal, polite apology. Stronger and more deferential than すみません. Often used in business or formal letters.
I am very sorry for causing you trouble.
I apologize for the late reply.
Derived from the verb 相済む (あいすむ), meaning 'to be settled' or 'to be excused', with the negative polite form ません. The prefix 相 adds a sense of mutual completion or thoroughness.