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, L be Linear_Combination of V,
      v be Element of V holds L.v = 0.R iff not v in Carrier(L)
  proof
    let R be Ring;
    let V be LeftMod of R,
    L be Linear_Combination of V, v be Element of V;
    thus L.v = 0.R implies not v in Carrier(L)
    proof
      assume
      A1: L.v = 0.R;
      assume not thesis;
      then ex u be Element of V st u = v & L.u <> 0.R;
      hence thesis by A1;
    end;
    assume not v in Carrier(L);
    hence thesis;
  end;
