
theorem Th49:
  for A being set, D being a_partition of A holds
    ERl D = EqRelOf PreorderFromPartition(D)
proof
  let A be set, D be a_partition of A;
  for x,y being Element of A holds
    [x,y] in ERl D implies [x,y] in EqRelOf PreorderFromPartition(D)
  proof
    let x,y be Element of A;
    assume A1: [x,y] in ERl D;
    then A2: [y,x] in ERl D by EQREL_1:6;
    reconsider X = x, Y = y as Element of PreorderFromPartition(D);
    X <= Y & Y <= X by A1, A2, ORDERS_2:def 5;
    hence thesis by Def6;
  end;
  hence ERl D c= EqRelOf PreorderFromPartition(D) by RELSET_1:def 1;
  for x,y being Element of A holds
    [x,y] in EqRelOf PreorderFromPartition(D) implies [x,y] in ERl D
  proof
    let x,y be Element of A;
    assume A3: [x,y] in EqRelOf PreorderFromPartition(D);
    reconsider X = x, Y = y as Element of PreorderFromPartition(D);
    X <= Y by A3, Def6;
    hence thesis by ORDERS_2:def 5;
  end;
  hence thesis by RELSET_1:def 1;
end;
