
theorem
  for V being non trivial free Z_Module, v being non zero Vector of V
  holds ex a being Element of INT.Ring, u being Vector of V
  st a in NAT & a <> 0 & v = a*u &
  for b being Element of INT.Ring, w being Vector of V st b > a
  holds v <> b*w
  proof
    let V be non trivial free Z_Module, v be non zero Vector of V;
    defpred P[Nat] means
    ex u being Vector of V, k be Element of INT.Ring
    st k = $1 & v = k*u;
    consider a be Element of INT.Ring such that
    A1: a in NAT & for b being Element of INT.Ring,
    u being Vector of V st b > a holds v <> b*u by LmND2;
    reconsider na = a as Nat by A1;
    A2: for k being Nat st P[k] holds k <= na by A1;
    A3: ex k being Nat st P[k]
    proof
      take 1;
      v = 1.INT.Ring*v by VECTSP_1:def 17;
      hence thesis;
    end;
    consider a0 be Nat such that
    A4: P[a0] & for n being Nat st P[n] holds n <= a0 from NAT_1:sch 6(A2,A3);
    reconsider a=a0 as Element of INT.Ring by INT_1:def 2;
    consider u be Vector of V such that
    A5: v = a*u by A4;
    take a, u;
    thus a in NAT by ORDINAL1:def 12;
    thus a <> 0
    proof
      assume a = 0;
      then v = 0.V by A5,ZMODUL01:1;
      hence contradiction;
    end;
    thus v = a*u by A5;
    thus for b being Element of INT.Ring, w being Vector of V
       st b > a holds v <> b*w
    proof
      let b be Element of INT.Ring, w be Vector of V such that
      B1: b > a;
      b in NAT by B1,INT_1:3;
      then reconsider bn = b as Nat;
      not P[bn] by A4,B1;
      hence thesis;
    end;
  end;
