A defendant may move for leave as a third-party plaintiff to assert a claim against a person not a party to the action who is or may be liable to him for all or part of the plaintiff's claim against him. If the motion is granted, summons and a copy of the third-party complaint, with a copy of the original complaint attached as an exhibit, shall be served on such a person, who shall be called the third-party defendant. He shall make his defenses to the third-party plaintiff's claim as provided in Rule 12 and his counterclaims against the third-party plaintiff and cross-claims against other third-party defendants as provided in Rule 13. The third-party defendant may assert against the plaintiff any defenses which the third-party plaintiff has to the plaintiff's claim. The third-party defendant may also assert any claim against the plaintiff arising out of the transaction or occurrence that is the subject matter of the plaintiff's claim against the third-party plaintiff. The plaintiff may assert any claim against the third-party defendant arising out of the transaction or occurrence that is the subject matter of the plaintiff's claim against the third-party plaintiff, and the third-party defendant thereupon shall assert his defenses as provided in Rule 12 and his counterclaims and cross-claims as provided in Rule 13. A third-party defendant may proceed under this rule against any person not a party to the action who is or may be liable to him for all or part of the claim made in the action against the third-party defendant.