
theorem
  for V being RealUnitarySpace, W being Subspace of V, C being Coset of
  W holds C is linearly-closed iff C = the carrier of W
proof
  let V be RealUnitarySpace;
  let W be Subspace of V;
  let C be Coset of W;
  thus C is linearly-closed implies C = the carrier of W
  proof
    assume
A1: C is linearly-closed;
    consider v being VECTOR of V such that
A2: C = v + W by Def5;
    C <> {} by A2,Th37;
    then 0.V in v + W by A1,A2,RLSUB_1:1;
    hence thesis by A2,Th41;
  end;
  thus thesis by Lm1;
end;
