The representation of type-2 fuzzy sets is a hot research topic. Christian Wagner and Hani Hagras introduced the notion of zslices. Mendel also introduced alpha planes. Bob John's paper with Hussam Hamrawi and Simon Coupland on type-2 alpha cuts has just been accepted in IEEE Transactions on Fuzzy Systems. The paper is here. Type-2 fuzzy logic systems make use of type-2 fuzzy sets. To be able to deliver useful type-2 fuzzy logic applications we need to be able to perform meaningful operations on these sets. These operations should also be practically tractable. However, type-2 fuzzy sets suffer the shortcoming of being complex by definition. Indeed, the third dimension, which is the source of extra parameters, is in itself the origin of extra computational cost. The quest for a representation that allow practical systems to be implemented is the motivation for our work. In this paper we define the alpha-cut decomposition theorem for type- 2 fuzzy sets which is a new representation analogous to the alpha-cut representation of type-1 fuzzy sets and the extension principle.