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;
