
theorem Th44:
  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}) = rng p & rng(p|{
  k where k is Element of L: p.k <= k}) = rng p
proof
  let L be non empty Poset, p be Function of L,L such that
A1: p is projection;
  set Lk = {k where k is Element of L: p.k <= k};
  set Lc = {c where c is Element of L: c <= p.c};
A2: rng p = Lc /\ Lk by A1,Th42;
A3: dom p = the carrier of L by FUNCT_2:def 1;
  thus rng(p|Lc) = rng p
  proof
    thus rng(p|Lc) c= rng p by RELAT_1:70;
    let y be object;
    assume
A4: y in rng p;
    then
A5: y in Lc by A2,XBOOLE_0:def 4;
    then
A6: ex lc being Element of L st y = lc & lc <= p.lc;
    y in Lk by A2,A4,XBOOLE_0:def 4;
    then ex lk being Element of L st y = lk & p.lk <= lk;
    then y = p.y by A6,ORDERS_2:2;
    hence thesis by A3,A5,A6,FUNCT_1:50;
  end;
  thus rng(p|Lk) c= rng p by RELAT_1:70;
  let y be object;
  assume
A7: y in rng p;
  then y in Lc by A2,XBOOLE_0:def 4;
  then
A8: ex lc being Element of L st y = lc & lc <= p.lc;
A9: y in Lk by A2,A7,XBOOLE_0:def 4;
  then ex lk being Element of L st y = lk & p.lk <= lk;
  then y = p.y by A8,ORDERS_2:2;
  hence thesis by A3,A9,A8,FUNCT_1:50;
end;
