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 ThTrivial1:
  for R being Ring
  for V being trivial LeftMod of R holds (Omega).V = (0).V
  proof
    let R be Ring;
    let V be trivial LeftMod of R;
    assume (Omega).V <> (0).V;
    then consider v be Vector of V such that
    A1: v in (Omega).V & v <> 0.V by ZMODUL04:24;
    reconsider v as Vector of V;
    V is non trivial by A1;
    hence contradiction;
  end;
