
theorem
  for S,T being non empty Poset, g being Function of S,T st S is
  complete holds (g is infs-preserving iff g is monotone & g is upper_adjoint)
proof
  let S,T be non empty Poset,g be Function of S,T;
  assume
A1: S is complete;
  hereby
    assume g is infs-preserving;
    then
    ex d being Function of T,S st[g,d] is Galois & for t being Element of
    T holds d.t is_minimum_of g"(uparrow t) by A1,Th14;
    hence g is monotone & g is upper_adjoint by Th10;
  end;
  assume g is monotone;
  assume ex d being Function of T,S st [g,d] is Galois;
  then g is upper_adjoint;
  hence thesis;
end;
