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;

theorem
  for R being Ring
  for V be LeftMod of R, v be Element of V holds (ZeroLC(V)).v = 0.R
  proof
    let R be Ring;
    let V be LeftMod of R, v be Element of V;
    Carrier (ZeroLC(V)) = {} & not v in {} by VECTSP_6:def 3;
    hence thesis;
  end;
