This is shown in MR1310882
MR1310882 (95m:05091)
Negami, Seiya (J-YOKOED)
Diagonal flips in triangulations of surfaces. (English summary)
Discrete Math. 135 (1994), no. 1-3, 225–232.
You
You can check it out.check it out. You will note that the argument uses some heavy lifting, unlike Hatcher's.