theorem 
  for R being domRing, V being LeftMod of R,
      L being Linear_Combination of V, a being Scalar of R holds
      a <> 0.R implies Carrier(a * L) = Carrier(L)
  proof
  let R be domRing, V be LeftMod of R, L be Linear_Combination of V, a be
  Scalar of R;
  set T = {u where u is Vector of V : (a * L).u <> 0.R};
  set S = {v where v is Vector of V : L.v <> 0.R};
  assume
A1: a <> 0.R;
  T = S
  proof
    thus T c= S
    proof
      let x be object;
      assume x in T;
      then consider u be Vector of V such that
A2:   x = u and
A3:   (a * L).u <> 0.R;
      (a * L).u = a * L.u by VECTSP_6:def 9;
      then L.u <> 0.R by A3;
      hence thesis by A2;
    end;
    let x be object;
    assume x in S;
    then consider v be Vector of V such that
A4: x = v and
A5: L.v <> 0.R;
    (a * L).v = a * L.v by VECTSP_6:def 9;
    then (a * L).v <> 0.R by A1,A5,VECTSP_2:def 1;
    hence thesis by A4;
  end;
  hence thesis;
end;
