reserve x, y, y1, y2 for object;
reserve V for Z_Module;
reserve W, W1, W2 for Submodule of V;
reserve u, v for VECTOR of V;
reserve i, j, k, n for Element of NAT;
reserve V,W for finite-rank free Z_Module;
reserve T for linear-transformation of V,W;

theorem ThTrivial2:
  for R being Ring
  for V being LeftMod of R, v being Vector of V st v <> 0.V holds
  Lin{v} is non trivial
  proof
    let R be Ring;
    let V be LeftMod of R, v be Vector of V such that
    A1: v <> 0.V;
    {v} <> {0.V} by A1,ZFMISC_1:3;
    then Lin{v} <> (0).V by MOD_3:7;
    then (Omega).Lin{v} <> (0).Lin{v} by VECTSP_4:36;
    hence thesis by ThTrivial1;
  end;
