
theorem Th38:
  for L being non empty Poset, f being Function of L,L st f is
  monotone & ex S being non empty Poset, g being Function of S,L, d being
  Function of L,S st [g,d] is Galois & f = g*d holds f is closure
proof
  let L be non empty Poset, f be Function of L,L such that
A1: f is monotone;
  given S being non empty Poset, g being Function of S,L, d being Function of
  L,S such that
A2: [g,d] is Galois and
A3: f = g*d;
A4: d is monotone & g is monotone by A2,Th8;
  d*g <= id S & id L <= g*d by A2,Th18;
  then g = g*d*g by A4,Th20;
  hence f is idempotent monotone by A1,A3,Th21;
  thus thesis by A2,A3,Th18;
end;
