
theorem Th43:
  for L being non empty Poset, p being Function of L,L st p is
projection holds {c where c is Element of L: c <= p.c} is non empty Subset of L
  & {k where k is Element of L: p.k <= k} is non empty Subset of L
proof
  let L be non empty Poset, p be Function of L,L such that
A1: p is projection;
  defpred Q[Element of L] means p.$1 <= $1;
  defpred P[Element of L] means $1 <= p.$1;
  set Lc = {c where c is Element of L: P[c]};
  set Lk = {k where k is Element of L: Q[k]};
A2: rng p = Lc /\ Lk by A1,Th42;
  Lc is Subset of L from DOMAIN_1:sch 7;
  hence Lc is non empty Subset of L by A2;
  Lk is Subset of L from DOMAIN_1:sch 7;
  hence thesis by A2;
end;
