reserve a,b for object, I,J for set;

theorem
  for R being asymmetric transitive non empty RelStr
  st the InternalRel of R is empty
  holds the InternalRel of DershowitzMannaOrder R = DivOrder the carrier of R
  proof
    let R be asymmetric transitive non empty RelStr;
    assume Z0: the InternalRel of R is empty;
    let a,b;
    hereby
      assume
A1:   [a,b] in the InternalRel of DershowitzMannaOrder R;
      then reconsider m = a, n = b as Element of DershowitzMannaOrder R
      by ZFMISC_1:87;
A5:   m <= n by A1,ORDERS_2:def 5;
      then
A3:   m <> n & for a being Element of R st m.a > n.a
      ex b being Element of R st a <= b & m.b < n.b by HO;
      m divides n
      proof
        let a;
        assume
A4:     m.a > n.a;
        a in dom m by A4,FUNCT_1:def 2;
        then reconsider a as Element of R;
        consider b being Element of R such that
A2:     a <= b & m.b < n.b by A5,HO,A4;
        thus thesis by Z0,A2,ORDERS_2:def 5;
      end;
      hence [a,b] in DivOrder the carrier of R by A3,DO;
    end;
    DivOrder the carrier of R c= the InternalRel of DershowitzMannaOrder R
    by Th16;
    hence thesis;
  end;
