
theorem Th11:
  for L1,L2,L3 be non empty Poset for f be Function of L1,L2 for g
  be Function of L2,L3 st f is directed-sups-preserving & g is
  directed-sups-preserving holds g*f is directed-sups-preserving
proof
  let L1,L2,L3 be non empty Poset;
  let f be Function of L1,L2;
  let g be Function of L2,L3;
  assume that
A1: f is directed-sups-preserving and
A2: g is directed-sups-preserving;
  now

    let X be Subset of L1;
    assume
A3: X is non empty directed;
    for X1 be Ideal of L1 holds f preserves_sup_of X1 by A1,WAYBEL_0:def 37;
    then
A4: f.:X is non empty directed by A3,WAYBEL_0:72,YELLOW_2:15;
    now
      sup X in the carrier of L1;
      then
A5:   sup X in dom f by FUNCT_2:def 1;
      assume
A6:   ex_sup_of X,L1;
A7:   f preserves_sup_of X by A1,A3,WAYBEL_0:def 37;
      then
A8:   ex_sup_of f.:X,L2 by A6,WAYBEL_0:def 31;
A9:   g preserves_sup_of f.:X by A2,A4,WAYBEL_0:def 37;
      then
A10:  sup (g.:(f.:X)) = g.sup (f.:X) by A8,WAYBEL_0:def 31;
      ex_sup_of g.:(f.:X),L3 by A8,A9,WAYBEL_0:def 31;
      hence ex_sup_of (g*f).:X,L3 by RELAT_1:126;
      sup (f.: X) = f.sup X by A6,A7,WAYBEL_0:def 31;
      hence sup ((g*f).:X) = g.(f.sup X) by A10,RELAT_1:126
        .= (g*f).sup X by A5,FUNCT_1:13;
    end;
    hence g*f preserves_sup_of X by WAYBEL_0:def 31;
  end;
  hence thesis by WAYBEL_0:def 37;
end;
