@ Simon Tyler. yeah I've thought about those examples, like when the line is on the plane or one of the subspaces is the zero vector. But giving examples doesnt prove it..
Thanks a lot Petr. More questions to come soon =D
@ Oster: I was hoping that thinking about that example would lead you to a proof like Petr's. Basically, you're question sounds like a homework problem -- which isn't helped by your small number of posts to these forums. These forums are not a place for homework help. I apologize if I've got you wrong and it as a genuine question.