
theorem LmFGND2:
  for V being Z_Module, v being Vector of V st v is torsion & v <> 0.V holds
  Lin{v} is non divisible
  proof
    let V be Z_Module, v be Vector of V such that
    A1: v is torsion & v <> 0.V;
    consider i be Element of INT.Ring such that
    A2: i <> 0 & i * v = 0.V by A1;
    assume A3: Lin{v} is divisible;
    v in Lin{v} by ZMODUL06:20;
    then reconsider vv = v as Vector of Lin{v};
    vv is divisible by A3;
    then consider uu be Vector of Lin{v} such that
    A4: i * uu = vv by A2;
    reconsider u = uu as Vector of V by ZMODUL01:25;
    u in Lin{v};
    then consider b be Element of INT.Ring such that
    A5: u = b * v by ZMODUL06:19;
    vv = i * (b * v) by A4,A5,ZMODUL01:29
    .= (i*b) * v by VECTSP_1:def 16
    .= b * (i*v) by VECTSP_1:def 16
    .= 0.V by A2,ZMODUL01:1;
    hence contradiction by A1;
  end;
