
theorem
  for V being non trivial free Z_Module,
  v being Vector of DivisibleMod(V) holds
  ex a being Element of INT.Ring st
  a in NAT & a <> 0 & a * v in EMbedding(V) &
  for b being Element of INT.Ring
  st b in NAT & b < a & b <> 0 holds not b * v in EMbedding(V)
  proof
    let V be non trivial free Z_Module, v be Vector of DivisibleMod(V);
    consider ai be Element of INT.Ring such that
    A2: ai <> 0 & ai * v in EMbedding(V) by ThDM1;
    reconsider aiv = ai * v as Vector of EMbedding(V) by A2;
    A3: |. ai .| * v in EMbedding(V)
    proof
      B1: EMbedding(V) is Submodule of DivisibleMod(V) by ThDivisible2;
      per cases by A2;
      suppose ai > 0;
        hence thesis by A2,ABSVALUE:def 1;
      end;
      suppose ai < 0;
        then |. ai .| * v = (-ai) * v by ABSVALUE:def 1
        .= - ai * v by ZMODUL01:16
        .= - aiv by B1,ZMODUL01:30;
        hence thesis;
      end;
    end;
    reconsider ain = |. ai .| as Element of INT.Ring;
    reconsider ainv = ain*v as Vector of EMbedding(V) by A3;
    N1: |. ai .| in NAT by COMPLEX1:46,INT_1:3;
    then reconsider nai=|. ai .| as Nat;
    A4: ain <> 0 by A2,ABSVALUE:2;
    defpred P[Nat] means
    ex n being Element of INT.Ring
      st n = $1 & n in NAT & n <> 0 & n * v in EMbedding(V);
    A6: ex k being Nat st P[k] by A3,A4,N1;
    ex k being Nat st P[k] & for n being Nat st P[n] holds k <= n
    from NAT_1:sch 5(A6);
    then consider a0 be Nat  such that
    A7: P[a0] & for b0 being Nat st P[b0] holds a0 <= b0;
    reconsider a=a0 as Element of INT.Ring by INT_1:def 2;
    take a;
    thus a in NAT by ORDINAL1:def 12;
    thus a <> 0 by A7;
    thus a * v in EMbedding(V) by A7;
    thus for b being Element of INT.Ring
    st b in NAT & b < a & b <> 0 holds not b * v in EMbedding(V) by A7;
  end;
