
theorem
  for V being RealUnitarySpace, V1 being Subset of V holds V1 is Coset
  of (0).V implies ex v being VECTOR of V st V1 = {v}
proof
  let V be RealUnitarySpace;
  let V1 be Subset of V;
  assume V1 is Coset of (0).V;
  then consider v being VECTOR of V such that
A1: V1 = v + (0).V by Def5;
  take v;
  thus thesis by A1,Th39;
end;
