Is this right? I know this is trickier than it looks because you are only allowed to assume the axioms of vector spaces alone, nothing else.
only one quibble, tonio....i believe you should have written -(c0) rather than (-c)0.
yes, these are in point of fact equal, but proving that uses the fact that 0v = 0, for any v, which is somewhat of a detour.
(it's just a misplaced parenthesis, i'm sure you meant -(c0)).