right under section 1 to enforce a term (?the substantive term?) is subject to a term providing for the submission of disputes to arbitration (?the arbitration agreement?), and (b) the arbitration agreement is an agreement in writing for the purposes of Part I of the [1996 c. 23.] Arbitration Act 1996, the third party shall be treated for the purposes of that Act as a party to the arbitration agreement as regards disputes between himself and the promisor relating to the enforcement of the substantive term by the third party. (2) Where? (a) a third party has a right under section 1 to enforce a term providing for one or more descriptions of dispute between the third party and the promisor to be submitted to arbitration (?the arbitration agreement?), (b) the arbitration agreement is an agreement in writing for the purposes of Part I of the Arbitration Act 1996, and (c) the third party does not fall to be treated under subsection (1) as a party to the arbitration agreement, the third party shall, if he exercises the right, be treated for the purposes of that Act as a party to the arbitration agreement in relation to the matter with respect to which the right is exercised, and be treated as having been so immediately before the exercise of the right. 9 Northern Ireland (1) In its application to Northern Ireland, this Act has effect with the modifications specified in subsections (2) and (3). ? (3) In Articles 4(a) and