
theorem LmND2:
  for V being non trivial free Z_Module, v being non zero Vector of V
  holds ex a being Element of INT.Ring
  st a in NAT & for b being Element of INT.Ring,
  u being Vector of V st b > a holds v <> b*u
  proof
    let V be non trivial free Z_Module, v be non zero Vector of V;
    set I = the Basis of V;
    A1: I is linearly-independent & (Omega).V = Lin(I) by VECTSP_7:def 3;
    consider L be Linear_Combination of I, w be Vector of V such that
    A2: v = Sum(L) & w in I & L.w <> 0 by LmND1;
    reconsider  a = |. L.w .| as Element of INT.Ring;
    A3: for b being Element of INT.Ring,
      u being Vector of V st b > a holds v <> b*u
    proof
      let b be Element of INT.Ring, u be Vector of V such that
      B0: b > a;
      assume B1: v = b*u;
      u in Lin(I) by A1;
      then consider Lu be Linear_Combination of I such that
      B2: u = Sum(Lu) by ZMODUL02:64;
      reconsider Lnu = b*Lu as Linear_Combination of I by ZMODUL02:31;
      B4: Sum(Lnu) = Sum(L) by A2,B1,B2,ZMODUL02:53;
      Carrier(Lnu) c= I & Carrier(L) c= I by VECTSP_6:def 4;
      then B5: Lnu = L by B4,VECTSP_7:def 3,ZMODUL03:3;
      B6: Lnu.w = b*Lu.w by VECTSP_6:def 9;
      reconsider ib = b as Integer;
      |. L.w .| <> 0 by A2,ABSVALUE:2;
      then B8: |. L.w .| > 0 by COMPLEX1:46;
      N3: 0 <= b by B0,COMPLEX1:46;
      |. L.w .| = |. b .| * |. Lu.w .| by B5,B6,COMPLEX1:65
      .= ib * |. Lu.w .| by N3,ABSVALUE:def 1;
      hence contradiction by B0,B8,INT_1:def 3,INT_2:27;
    end;
    take a;
    thus thesis by A3,COMPLEX1:46,INT_1:3;
  end;
