reserve x, y, y1, y2 for set;
reserve R for Ring;
reserve V for LeftMod of R;
reserve u, v, w for VECTOR of V;
reserve F, G, H, I for FinSequence of V;
reserve i, j, k, n for Element of NAT;
reserve f, f9, g for sequence of V;
reserve K,L,L1,L2,L3 for Linear_Combination of V;
reserve a, b for Element of R;
reserve G, H1, H2, F, F1, F2, F3 for FinSequence of V;
reserve A, B for Subset of V,
  v1, v2, v3, u1, u2, u3 for Vector of V,
  f for Function of V, R,
  i for Element of NAT;
reserve l, l1, l2 for Linear_Combination of A;

theorem
  for V being Z_Module,
      a being Element of INT.Ring,
      L being Linear_Combination of V holds
  a <> 0.INT.Ring implies Carrier(a * L) = Carrier(L)
  proof
    let V be Z_Module,
      a be Element of INT.Ring,
      L be Linear_Combination of V;
    set R = INT.Ring;
    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.INT.Ring;
    T = S
    proof
      thus T c= S
      proof
        let x be object;
        assume x in T;
        then consider u being 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 being 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;
      hence thesis by A4;
    end;
    hence thesis;
  end;
