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