
theorem
  for L being non empty Poset, f being Function of L,L st f is kernel 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
proof
  let L be non empty Poset, f be Function of L,L;
  assume
A1: f is kernel;
  reconsider T = Image f as non empty Poset;
  reconsider g = corestr f as Function of L,T;
  reconsider d = inclusion f as Function of T,L;
  take T,g,d;
  thus [g,d] is Galois by A1,Th39;
  thus thesis by Th32;
end;
