theorem
  C is linearly-closed iff C = the carrier of W
proof
  thus C is linearly-closed implies C = the carrier of W
  proof
    assume
A1: C is linearly-closed;
    consider v such that
A2: C = v + W by Def6;
    C <> {} by A2,Th43;
    then 0.V in v + W by A1,A2,Th1;
    hence thesis by A2,Th47;
  end;
  thus thesis by Lm1;
end;
