reserve a,b for object, I,J for set;
reserve b for bag of I;
reserve R for asymmetric transitive non empty RelStr,
  a,b,c for bag of the carrier of R,
  x,y,z for Element of R;
reserve p for partition of b-'a, q for partition of b;
reserve J for set, m for bag of I;

theorem
  for a,b,c,d being Element of DershowitzMannaOrder R
  for e being bag of the carrier of R st a <= b & e divides a & e divides b
  holds c = a-'e & d = b-'e implies c <= d
  proof
    let a,b,c,d be Element of DershowitzMannaOrder R;
    let e be bag of the carrier of R;
    assume Z1: a <= b;
    assume Z2: e divides a;
    assume Z3: e divides b;
    assume Z4: c = a-'e;
    assume d = b-'e; then
A0: a = c+e & b = d+e & a <> b by Z1,Z2,Z3,Z4,PRE_POLY:47;
    for x st c.x > d.x ex y st x <= y & c.y < d.y
    proof let x;
      assume c.x > d.x;
      then a.x = c.x+e.x > d.x+e.x = b.x by A0,PRE_POLY:def 5,XREAL_1:6;
      then consider y such that
A2:   x <= y & a.y < b.y by Z1,HO;
      take y;
      a.y = c.y+e.y & b.y = d.y+e.y by A0,PRE_POLY:def 5;
      hence thesis by A2,XREAL_1:6;
    end;
    hence c <= d by A0,HO;
  end;
