
theorem LmFGND3:
  for V being non trivial finitely-generated torsion-free Z_Module holds
  V is non divisible
  proof
    let V be non trivial finitely-generated torsion-free Z_Module;
    consider I be finite Subset of V such that
    A1: I is Basis of V by ZMODUL03:def 3;
    (Omega).V <> (0).V;
    then Lin(I) <> (0).V by A1,VECTSP_7:def 3;
    then I <> {}(the carrier of V) by ZMODUL02:67;
    then consider v be object such that
    A3: v in I by XBOOLE_0:def 1;
    reconsider v as Vector of V by A3;
    A4: V is Submodule of V & I is linearly-independent &
    (Omega).V = Lin(I) by A1,ZMODUL01:32,VECTSP_7:def 3; then
    A5: (Omega).V = Lin(I \ {v}) + Lin{v} & Lin(I \ {v}) /\ Lin{v} = (0).V
    & Lin(I \ {v}) is free & Lin{v} is free & v <> 0.V
    by A3,ZMODUL06:24;
    (I \ {v}) \/ {v} = I \/ {v} by XBOOLE_1:39
    .= I by A3,XBOOLE_1:12,ZFMISC_1:31;
    then B3: Lin(I) = Lin(I \ {v}) + Lin{v} by ZMODUL02:72;
    reconsider i2=2 as Element of INT.Ring by INT_1:def 2;
    v is non divisible
    proof
      assume v is divisible;
      then consider u be Vector of V such that
      C1: i2 * u = v;
      u in Lin(I) by A4;
      then consider u1, u2 be Vector of V such that
      C2: u1 in Lin(I \ {v}) & u2 in Lin{v} & u = u1 + u2
      by B3,ZMODUL01:92;
      consider iu2 be Element of INT.Ring such that
      C3: u2 = iu2 * v by C2,ZMODUL06:19;
      C4: u1 <> 0.V
      proof
        assume u1 = 0.V;
        then v = (i2*iu2) * v by C1,C2,C3,VECTSP_1:def 16;
        then (i2*iu2)*v - v = 0.V by RLVECT_1:15;
        then (i2*iu2)*v - 1.INT.Ring*v = 0.V by VECTSP_1:def 17;
        then D1: (i2*iu2 - 1.INT.Ring) * v = 0.V by ZMODUL01:9;
        reconsider iiu2=iu2 as Integer;
        2*iiu2 - 1 <> 0
        proof
          assume 2*iiu2 - 1 = 0;
          then 1/2 is Integer;
          hence contradiction by NAT_D:33;
        end;
        then v is torsion by D1;
        hence contradiction by A5,ZMODUL06:def 3;
      end;
      v = i2*u1 + i2*u2 by C1,C2,VECTSP_1:def 14
      .= i2*u1 + (i2*iu2)*v by C3,VECTSP_1:def 16;
      then v - (i2*iu2)*v = i2*u1 + ((i2*iu2)*v - (i2*iu2)*v) by RLVECT_1:28
      .= i2*u1 + 0.V by RLVECT_1:15
      .= i2*u1;
      then 1.INT.Ring *v - (i2*iu2)*v = i2*u1 by VECTSP_1:def 17;
      then C5: (1.INT.Ring - i2*iu2)*v = i2*u1 by ZMODUL01:9;
      i2 <> 0.INT.Ring; then
      C6: i2*u1 <> 0.V by C4,ZMODUL01:def 7;
      C7: i2*u1 in Lin(I \ {v}) by C2,ZMODUL01:37;
      (1.INT.Ring - i2*iu2)*v in Lin{v} by ZMODUL06:21;
      hence contradiction by A5,C5,C6,C7,ZMODUL01:94,ZMODUL02:66;
    end;
    hence thesis;
  end;
