Prove that, for a linear code C, either all the code vectors have even weight or exactly

Question:

Prove that, for a linear code C, either all the code vectors have even weight or exactly half of them do. [Let E be the set of vectors in C with even weight and O the set of vectors in C with odd weight. If O is not empty, let c_o be in O and consider O' = {c_o + e : e in E}. Show that O' = O.]