theorem
  Lin({}(the carrier of V)) = (0).V
proof
  set A = Lin({}(the carrier of V));
  now
    let v;
    thus v in A implies v in (0).V
    proof
      assume v in A;
      then
A1:   v in the carrier of A by STRUCT_0:def 5;
      the carrier of A = the set of all Sum (l0)  by Def2;
      then ex l0 st v = Sum(l0) by A1;
      then v = 0.V by RLVECT_2:31;
      hence thesis by Lm2;
    end;
    assume v in (0).V;
    then v = 0.V by Lm2;
    hence v in A by RLSUB_1:17;
  end;
  hence thesis by RLSUB_1:31;
end;
