
theorem
  for S,T being non empty Poset, f being Function of S,T st
  for X being Ideal of S holds f preserves_sup_of X
  holds f is directed-sups-preserving
proof
  let S,T be non empty Poset, f be Function of S,T such that
A1: for X being Ideal of S holds f preserves_sup_of X;
  let X be Subset of S such that
A2: X is non empty directed and
A3: ex_sup_of X,S;
  reconsider Y = X as non empty directed Subset of S by A2;
  downarrow Y is Ideal of S;
  then
A4: f preserves_sup_of downarrow X by A1;
A5: ex_sup_of downarrow X, S by A3,Th32;
  then
A6: ex_sup_of f.:downarrow X,T by A4;
A7: sup (f.:downarrow X) = f.sup downarrow X by A4,A5;
A8: sup downarrow X = sup X by A3,Th33;
A9: f.:X c= f.:downarrow X by Th16,RELAT_1:123;
A10: f.:downarrow X is_<=_than f.sup X by A6,A7,A8,YELLOW_0:30;
A11: f.:X is_<=_than f.sup X
  by A9,A10;
A12: now
    let b be Element of T;
    assume
A13: f.:X is_<=_than b;
    f.:downarrow X is_<=_than b
    proof
      let a be Element of T;
      assume a in f.:downarrow X;
      then consider x being object such that
      x in dom f and
A14:  x in downarrow X and
A15:  a = f.x by FUNCT_1:def 6;
      downarrow X =
      {z where z is Element of S: ex y being Element of S st z <= y & y in X}
      by Th14;
      then consider z being Element of S such that
A16:  x = z and
A17:  ex y being Element of S st z <= y & y in X by A14;
      consider y being Element of S such that
A18:  z <= y and
A19:  y in X by A17;
A20:  f is monotone by A1,Th72;
A21:  f.y in f.:X by A19,FUNCT_2:35;
A22:  f.z <= f.y by A18,A20;
      f.y <= b by A13,A21;
      hence thesis by A15,A16,A22,ORDERS_2:3;
    end;
    hence f.sup X <= b by A6,A7,A8,YELLOW_0:30;
  end;
  hence ex_sup_of f.:X,T by A11,YELLOW_0:15;
  hence thesis by A11,A12,YELLOW_0:def 9;
end;
