
theorem Th42:
  for L being non empty Poset, p being Function of L,L st p is
projection holds rng p = {c where c is Element of L: c <= p.c} /\ {k where k is
  Element of L: p.k <= k}
proof
  let L be non empty Poset, p be Function of L,L such that
A1: p is idempotent and
  p is monotone;
  set Lk = {k where k is Element of L: p.k <= k};
  set Lc = {c where c is Element of L: c <= p.c};
  thus rng p c= Lc /\ Lk
  proof
    let x be object;
    assume
A2: x in rng p;
    then reconsider x9=x as Element of L;
A3: ex l being object st l in dom p & p.l = x by A2,FUNCT_1:def 3;
    then p.x9 <= x9 by A1,YELLOW_2:18;
    then
A4: x in Lk;
    x9 <= p.x9 by A1,A3,YELLOW_2:18;
    then x in Lc;
    hence thesis by A4,XBOOLE_0:def 4;
  end;
  let x be object;
  assume
A5: x in Lc /\ Lk;
  then x in Lc by XBOOLE_0:def 4;
  then
A6: ex lc being Element of L st x = lc & lc <= p.lc;
  x in Lk by A5,XBOOLE_0:def 4;
  then ex lk being Element of L st x = lk & p.lk <= lk;
  then dom p = the carrier of L & x = p.x by A6,FUNCT_2:def 1,ORDERS_2:2;
  hence thesis by A6,FUNCT_1:def 3;
end;
