
theorem LmFGND1:
  for V being Z_Module, v being Vector of V st v is non torsion holds
  Lin{v} is non divisible
  proof
    let V be Z_Module, v be Vector of V such that
    A1: v is non torsion;
    reconsider i2=2 as Element of INT.Ring by INT_1:def 2;
    assume A2: Lin{v} is divisible;
    v in Lin{v} by ZMODUL06:20;
    then reconsider vv = v as Vector of Lin{v};
    vv is divisible by A2;
    then consider uu be Vector of Lin{v} such that
    A3: i2*uu = vv;
    reconsider u = uu as Vector of V by ZMODUL01:25;
    u in Lin{v};
    then consider i be Element of INT.Ring such that
    A4: u = i*v by ZMODUL06:19;
    reconsider i3=i as Integer;
    v = i2 * (i*v) by A3,A4,ZMODUL01:29
    .= (i2*i)*v by VECTSP_1:def 16;
    then (i2*i)*v - v = 0.V by RLVECT_1:15;
    then (i2*i)*v - 1.INT.Ring *v = 0.V by VECTSP_1:def 17;
    then A5: (i2*i - 1.INT.Ring)*v = 0.V by ZMODUL01:9;
    2*i3 - 1 <> 0
    proof
      assume 2*i3 - 1 = 0;
      then 1/2 is Integer;
      hence contradiction by NAT_D:33;
    end;
    hence contradiction by A1,A5;
  end;
