
theorem
  for V being Z_Module holds
  ex A being Subset of V st (A is linearly-independent &
  for v being Vector of V holds ex a being Element of INT.Ring
  st a in NAT & a > 0 & a * v in Lin(A))
  proof
    let V be Z_Module;
    {}(the carrier of V) is linearly-independent;
    then consider A be Subset of V such that
    A1: {} c= A & A is linearly-independent &
    (for v being Vector of V holds ex ai being Element of INT.Ring st
    ai <> 0 & ai * v in Lin(A)) by ZMODUL07:2;
    A2: for v being Vector of V holds ex a being
    Element of INT.Ring st a in NAT & a > 0 & a * v in Lin(A)
    proof
      let v be Vector of V;
      consider ai be Element of INT.Ring such that
      B1: ai <> 0 & ai * v in Lin(A) by A1;
      set a = |. ai .|;
      B2: a <> 0 by B1,ABSVALUE:2;
      N1: a in NAT by COMPLEX1:46,INT_1:3;
      a * v in Lin(A)
      proof
        per cases by B1;
        suppose ai > 0;
          hence thesis by B1,ABSVALUE:def 1;
        end;
        suppose ai < 0;
          then a = -ai by ABSVALUE:def 1;
          then a * v = -ai * v by ZMODUL01:16;
          hence thesis by B1,ZMODUL01:38;
        end;
      end;
      hence thesis by N1,B2;
    end;
    take A;
    thus thesis by A1,A2;
  end;
