
theorem Th30:
  for V being RealUnitarySpace, W being Subspace of V holds (0).W = (0).V
proof
  let V be RealUnitarySpace;
  let W be Subspace of V;
  the carrier of (0).W = {0.W} & the carrier of (0).V = {0.V} by Def2;
  then
A1: the carrier of (0).W = the carrier of (0).V by Def1;
  (0).W is Subspace of V by Th21;
  hence thesis by A1,Th24;
end;
