A person's personal mailbox is an internet mailbox (in the sense of RFC822 @@check reference) whose only authorized recipient is that person.
So if mbox(p1, m) and mbox(p2, m) then p1=p2.
But we do not mean that a person has only one personal mailbox; i.e. we do not mean that if mbox(p, m1) and mbox(p, m2) then m1=m2. (And just never you mind what = means anyway just yet.)