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

theorem
  for R1,R2 being asymmetric transitive non empty RelStr
  st the carrier of R1 = the carrier of R2 &
  the InternalRel of R1 c= the InternalRel of R2
  holds the InternalRel of DershowitzMannaOrder R1
  c= the InternalRel of DershowitzMannaOrder R2
  proof
    let R1,R2 be asymmetric transitive non empty RelStr;
    assume Z0: the carrier of R1 = the carrier of R2 &
    the InternalRel of R1 c= the InternalRel of R2;
    let a,b be Element of DershowitzMannaOrder R1;
    assume [a,b] in the InternalRel of DershowitzMannaOrder R1;
    then
A3: a <= b by ORDERS_2:def 5;
    then
A1: a <> b & for x being Element of R1 st a.x > b.x
    ex y being Element of R1 st x <= y & a.y < b.y by HO;
    reconsider b1 = b, a1 = a as multiset of the carrier of R1 by Th1;
    reconsider b1, a1 as Element of DershowitzMannaOrder R2 by Z0,Th2;
    now
      let x be Element of R2;
      reconsider x1 = x as Element of R1 by Z0;
      assume a1.x > b1.x;
      then consider y being Element of R1 such that
A2:   x1 <= y & a.y < b.y by A3,HO;
      reconsider y1 = y as Element of R2 by Z0;
      take y1;
      [x1,y] in the InternalRel of R1 by A2,ORDERS_2:def 5;
      hence x <= y1 & a1.y1 < b1.y1 by Z0,A2,ORDERS_2:def 5;
    end;
    then a1 <= b1 by A1,HO;
    hence thesis by ORDERS_2:def 5;
  end;
