
theorem Th45:
  for L being non empty Poset, p being Function of L,L st p is
  projection for Lc being non empty Subset of L, Lk being non empty Subset of L
  st Lc = {c where c is Element of L: c <= p.c} holds p|Lc is Function of
  subrelstr Lc,subrelstr Lc
proof
  let L be non empty Poset, p be Function of L,L such that
A1: p is projection;
  let Lc be non empty Subset of L, Lk be non empty Subset of L such that
A2: Lc = {c where c is Element of L: c <= p.c};
  set Lk = {k where k is Element of L: p.k <= k};
  rng p = Lc /\ Lk by A1,A2,Th42;
  then rng(p|Lc) = Lc /\ Lk by A1,A2,Th44;
  then
A3: rng(p|Lc) c= Lc by XBOOLE_1:17;
  Lc = the carrier of subrelstr Lc & p|Lc is Function of Lc,the carrier of
  L by FUNCT_2:32,YELLOW_0:def 15;
  hence thesis by A3,FUNCT_2:6;
end;
