reserve V for Z_Module;
reserve W, W1, W2 for Submodule of V;

theorem Thn0V:
  for R being Ring
  for V be LeftMod of R, W be strict Subspace of V st W <> (0).V holds
  ex v be Vector of V st v in W & v <> 0.V
  proof
    let R be Ring;
    let V be LeftMod of R, W be strict Subspace of V such that
    A1: W <> (0).V;
    A2: 0.V in W by VECTSP_4:17;
    the carrier of W <> {0.V} by A1,VECTSP_4:def 3;
    then {0.V} c< the carrier of W by A2,ZFMISC_1:31;
    then consider v be object such that
    A3: v in the carrier of W and
    A4: not v in {0.V} by XBOOLE_0:6;
    reconsider v as Vector of V by A3,VECTSP_4:10;
    take v;
    thus thesis by A3,A4,TARSKI:def 1;
  end;
