
theorem Th13:
  for S,T being non empty Poset, d being Function of T,S st d is
  lower_adjoint holds d is sups-preserving
proof
  let S,T be non empty Poset, d be Function of T,S;
  given g being Function of S,T such that
A1: [g,d] is Galois;
  let X be Subset of T;
  set t = sup X;
  assume
A2: ex_sup_of X,T;
A3: for s being Element of S st s is_>=_than d.:X holds d.t <= s
  proof
    let s be Element of S;
    assume
A4: s is_>=_than d.:X;
    g.s is_>=_than X
    proof
      let ti be Element of T;
      assume ti in X;
      then d.ti in d.:X by FUNCT_2:35;
      then s >= d.ti by A4;
      hence thesis by A1,Th8;
    end;
    then g.s >= t by A2,YELLOW_0:30;
    hence thesis by A1,Th8;
  end;
  d.t is_>=_than d.:X
  proof
    let s be Element of S;
    assume s in d.:X;
    then consider ti being Element of T such that
A5: ti in X and
A6: s = d.ti by FUNCT_2:65;
A7: d is monotone by A1,Th8;
    reconsider ti as Element of T;
    t is_>=_than X by A2,YELLOW_0:30;
    then t >= ti by A5;
    hence thesis by A7,A6;
  end;
  hence thesis by A3,YELLOW_0:30;
end;
