I am reading Massey's 'A basic coruse in Algebraic Topology'. In first chapter, he proved classification theorem for compact surfaces (compact connected 2-manifold). This theorem classifies compact surfaces into 3 cases: sphere ($aa^{-1}$), connected sum of tori (e.g. $aba^{-1}b^{-1}cdc^{-1}d^{-1}$) and connected sum of projective planes (e.g $aabbccdd$).
I try to read over the proof. There is one question which the author seems hasn't explained in this book. How to deal with the case for example $aba^{-1}b^{-1}e^{-1}f^{-1}efcdc^{-1}d^{-1}$ which contains something like $e^{-1}f^{-1}ef$, but not $efe^{-1}f^{-1}$?