
theorem Th19:
  for L being up-complete non empty Poset for S being non empty
  Poset st S is_a_retract_of L holds S is up-complete
proof
  let L be up-complete non empty Poset;
  let S be non empty Poset;
  given f being Function of L,S such that
A1: f is_a_retraction_of L,S;
  S is full directed-sups-inheriting non empty SubRelStr of L by A1;
  hence thesis by Th4;
end;
