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;
 reserve e, e1, e2 for Element of LinComb(V);
reserve W, W1, W2, W3 for Submodule of V;
reserve v, v1, v2, u for Vector of V;
reserve A, B, C for Subset of V;
reserve T for finite Subset of V;
reserve L, L1, L2 for Linear_Combination of V;
reserve l for Linear_Combination of A;
reserve F, G, H for FinSequence of V;
reserve f, g for Function of V, R;

theorem
  for x being object holds x in (0).V iff x = 0.V
  proof let x be object;
  thus x in (0).V implies x = 0.V
  proof
    assume x in (0).V;
    then x in the carrier of (0).V;
    then x in {0.V} by VECTSP_4:def 3;
    hence thesis by TARSKI:def 1;
  end;
  thus thesis by ZMODUL01:33;
end;
