
theorem Th48:
  for A being non empty Preorder,
    D being non empty a_partition of the carrier of A
  st D = the carrier of QuotientOrder(A) holds
    proj A = proj D
proof
  let A be non empty Preorder;
  let D being non empty a_partition of the carrier of A;
  assume A1: D = the carrier of QuotientOrder(A);
  dom proj D = the carrier of A by FUNCT_2:def 1;
  then A2: dom proj A = dom proj D by FUNCT_2:def 1;
  for x being object st x in dom proj A holds (proj A).x = (proj D).x
  proof
    let x be object;
    assume x in dom proj A;
    then reconsider z = x as Element of A;
    A3: z in (proj D).z by EQREL_1:def 9;
    (proj D).z in the carrier of QuotientOrder(A) by A1;
    then (proj D).z in Class EqRelOf A by Def7;
    then consider y being object such that
      A4: y in the carrier of A and
      A5: (proj D).z = Class(EqRelOf A, y) by EQREL_1:def 3;
    (proj D).z = Class(EqRelOf A, z) by A4, A3, A5, EQREL_1:23
      .= (proj A).z by Def8;
    hence thesis;
  end;
  hence thesis by A2, FUNCT_1:2;
end;
