
theorem
  for L being complete non empty Poset, p being Function of L,L st p
  is projection holds Image p is complete
proof
  let L be complete non empty Poset, p be Function of L,L;
A1: the carrier of Image p = rng p by YELLOW_0:def 15;
  assume
A2: p is projection;
  then reconsider
  Lc = {c where c is Element of L: c <= p.c}, Lk = {k where k is
  Element of L: p.k <= k} as non empty Subset of L by Th43;
A3: the carrier of subrelstr Lc = Lc & rng p = Lc /\ Lk by A2,Th42,
YELLOW_0:def 15;
  p is monotone by A2;
  then subrelstr Lc is sups-inheriting by Th49;
  then
A4: subrelstr Lc is complete LATTICE by YELLOW_2:31;
  reconsider pc = p|Lc as Function of subrelstr Lc,subrelstr Lc by A2,Th45;
A5: Image pc is infs-inheriting by A2,Th47,Th53;
A6: the carrier of Image pc = rng(pc) by YELLOW_0:def 15
    .= the carrier of Image p by A2,A1,Th44;
  then the InternalRel of Image pc = (the InternalRel of subrelstr Lc)|_2 the
  carrier of Image p by YELLOW_0:def 14
    .= ((the InternalRel of L)|_2 the carrier of subrelstr Lc) |_2 the
  carrier of Image p by YELLOW_0:def 14
    .= (the InternalRel of L)|_2 the carrier of Image p by A1,A3,WELLORD1:22
,XBOOLE_1:17
    .= the InternalRel of Image p by YELLOW_0:def 14;
  hence thesis by A4,A5,A6,YELLOW_2:30;
end;
