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

theorem Th16:
  for R being asymmetric transitive non empty RelStr holds
  DivOrder the carrier of R c= the InternalRel of DershowitzMannaOrder R
  proof
    let R be asymmetric transitive non empty RelStr;
    set DM = DershowitzMannaOrder R;
    let a,b be Element of Bags the carrier of R;
    assume
A1: [a,b] in DivOrder the carrier of R;
    reconsider a,b as multiset of the carrier of R by Th1;
    reconsider a,b as Element of DM by Th2;
A2: a <> b & a divides b by A1,DO;
    then for x being Element of R st a.x > b.x
    ex y being Element of R st x <= y & a.y < b.y by PRE_POLY:def 11;
    then a <= b by A2,HO;
    hence thesis by ORDERS_2:def 5;
  end;
