
theorem
  for X being non empty set st X includes_lattice_of X holds X is
  c=directed c=filtered
proof
  let X be non empty set such that
A1: for a,b being set st a in X & b in X holds a /\ b in X & a \/ b in X;
  for a,b being set st a in X & b in X ex c being set st a \/ b c= c & c
  in X by A1;
  hence X is c=directed by Th6;
  for a,b being set st a in X & b in X ex c being set st c c= a /\ b & c
  in X by A1;
  hence thesis by Th8;
end;
