
theorem
  for V being RealUnitarySpace holds Lin({}(the carrier of V)) = (0).V
proof
  let V be RealUnitarySpace;
  set A = Lin({}(the carrier of V));
  now
    let v be VECTOR of 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) where l0 is Linear_Combination of {}(the
      carrier of V)  by Def1;
      then
      ex l0 being Linear_Combination of {}(the carrier of V) st v = Sum(l0
      ) by A1;
      then v = 0.V by RLVECT_2:31;
      hence thesis by Lm1;
    end;
    assume v in (0).V;
    then v = 0.V by Lm1;
    hence v in A by RUSUB_1:11;
  end;
  hence thesis by RUSUB_1:25;
end;
