
theorem Th12:
  for S,T being non empty Poset,g being Function of S,T st g is
  upper_adjoint holds g is infs-preserving
proof
  let S,T be non empty Poset,g be Function of S,T;
  given d being Function of T,S such that
A1: [g,d] is Galois;
  let X be Subset of S;
  set s = inf X;
  assume
A2: ex_inf_of X,S;
A3: for t being Element of T st t is_<=_than g.:X holds g.s >= t
  proof
    let t be Element of T;
    assume
A4: t is_<=_than g.:X;
    d.t is_<=_than X
    proof
      let si be Element of S;
      assume si in X;
      then g.si in g.:X by FUNCT_2:35;
      then t <= g.si by A4;
      hence d.t <= si by A1,Th8;
    end;
    then d.t <= s by A2,YELLOW_0:31;
    hence thesis by A1,Th8;
  end;
  g.s is_<=_than g.:X
  proof
    let t be Element of T;
    assume t in g.:X;
    then consider si being Element of S such that
A5: si in X and
A6: t = g.si by FUNCT_2:65;
A7: g is monotone by A1,Th8;
    reconsider si as Element of S;
    s is_<=_than X by A2,YELLOW_0:31;
    then s <= si by A5;
    hence g.s <= t by A7,A6;
  end;
  hence thesis by A3,YELLOW_0:31;
end;
