
theorem Th17:
  for S,T being non empty Poset, d being Function of T,S st T is
  complete holds d is sups-preserving iff d is monotone & d is lower_adjoint
proof
  let S,T be non empty Poset, d be Function of T,S;
  assume
A1: T is complete;
  hereby
    assume d is sups-preserving;
    then
    ex g being Function of S,T st [g,d] is Galois & for s being Element of
    S holds g.s is_maximum_of d"(downarrow s) by A1,Th15;
    hence d is monotone & d is lower_adjoint by Th11;
  end;
  assume d is monotone;
  assume ex g being Function of S,T st [g,d] is Galois;
  then d is lower_adjoint;
  hence thesis;
end;
