
theorem
  for L being non empty Poset, p being Function of L,L st p is
projection for Lk being non empty Subset of L st Lk = {k where k is Element of
L: p.k <= k} holds p|Lk is Function of subrelstr Lk,subrelstr Lk
proof
  let L be non empty Poset, p be Function of L,L such that
A1: p is projection;
  set Lc = {c where c is Element of L: c <= p.c};
  let Lk be non empty Subset of L such that
A2: Lk = {k where k is Element of L: p.k <= k};
  rng p = Lc /\ Lk by A1,A2,Th42;
  then rng(p|Lk) = Lc /\ Lk by A1,A2,Th44;
  then
A3: rng(p|Lk) c= Lk by XBOOLE_1:17;
  Lk = the carrier of subrelstr Lk & p|Lk is Function of Lk,the carrier of
  L by FUNCT_2:32,YELLOW_0:def 15;
  hence thesis by A3,FUNCT_2:6;
end;
