
theorem Th44:
  for A being Preorder holds the InternalRel of QuotientOrder(A) = <=E A
proof
  let A be Preorder;
  per cases;
  suppose A is empty;
    then the InternalRel of QuotientOrder(A) is empty & <=E A is empty;
    hence thesis;
  end;
  suppose A is non empty;
    then reconsider B = A as non empty Preorder;
    set qa = QuotientOrder(B);
    set int = the InternalRel of QuotientOrder(B);
    A1: for x being Element of B holds
      Class(EqRelOf B, x) = Class(EqRel B, x) by Th41;
    for X,Y being Element of qa holds
      [X,Y] in int implies [X,Y] in <=E B
    proof
      let X, Y be Element of qa;
      X in the carrier of qa & Y in the carrier of qa;
      then A2: X in Class EqRelOf B & Y in Class EqRelOf B by Def7;
      assume [X,Y] in int;
      then consider x, y being Element of B such that A3:
        X = Class(EqRelOf B, x) & Y = Class(EqRelOf B, y) & x <= y by A2, Def7;
      X = Class(EqRel B, x) & Y = Class(EqRel B, y) & x <= y by A1, A3;
      hence thesis by DICKSON:def 5;
    end;
    then A4: int c= <=E B by RELSET_1:def 1;
    for X,Y being Element of Class EqRel B holds
      [X,Y] in <=E B implies [X,Y] in int
    proof
      let X, Y be Element of Class EqRel B;
      X in Class EqRel B & Y in Class EqRel B;
      then A5: X in Class EqRelOf B & Y in Class EqRelOf B by Th41;
      assume [X,Y] in <=E B;
      then consider x, y being Element of B such that A6:
        X = Class(EqRel B, x) & Y = Class(EqRel B, y) & x <= y
        by DICKSON:def 5;
      X = Class(EqRelOf B, x) & Y = Class(EqRelOf B, y) & x <= y by A1, A6;
      hence thesis by A5, Def7;
    end;
    hence thesis by A4, RELSET_1:def 1;
  end;
end;
