A post service of some country uses carriers to transport the mail, each carrier’s task is to bring the mail from one city to a neighbouring city. It is known that it is possible to send mail from any city to the capital $P$ . For any two cities $A$ and $B$, call $B$ more important than $A$, if every possible route of mail from $A$ to the capital $P$ goes through $B$. a) Prove that, for any three different cities $A, B$, and $C$, if $B$ is more important than $A$ and $C$ is more important than $B$, then $C$ is more important than $A$. b) Prove that, for any three different cities $A, B$, and $C$, if both B and C are more important than $A$, then either $C$ is more important than $B$ or $B$ is more important than $C$.