
theorem Th22:
  for L1,L2,L3 be non empty transitive antisymmetric RelStr for f
be Function of L1,L2 st f is monotone directed-sups-preserving holds L2 is full
  directed-sups-inheriting SubRelStr of L3 & L3 is complete implies ex g be
  Function of L1,L3 st f = g & g is directed-sups-preserving
proof
  let L1,L2,L3 be non empty transitive antisymmetric RelStr;
  let f be Function of L1,L2;
  assume that
A1: f is monotone directed-sups-preserving and
A2: L2 is full directed-sups-inheriting SubRelStr of L3 and
A3: L3 is complete;
  the carrier of L2 c= the carrier of L3 by A2,YELLOW_0:def 13;
  then reconsider g = f as Function of L1,L3 by FUNCT_2:7;
  take g;
  thus f = g;
  now
    let X be Subset of L1;
    assume
A4: X is non empty directed;
    then consider d be object such that
A5: d in X by XBOOLE_0:def 1;
    d in the carrier of L1 by A5;
    then d in dom f by FUNCT_2:def 1;
    then f.d in f.:X by A5,FUNCT_1:def 6;
    then
A6: f.:X is non empty directed by A1,A4,YELLOW_2:15;
    now
A7:   f preserves_sup_of X by A1,A4,WAYBEL_0:def 37;
      assume
A8:   ex_sup_of X,L1;
      thus ex_sup_of g.:X,L3 by A3,YELLOW_0:17;
      hence sup (g.:X) = sup (f.:X) by A2,A6,WAYBEL_0:7
        .= g.sup X by A8,A7,WAYBEL_0:def 31;
    end;
    hence g preserves_sup_of X by WAYBEL_0:def 31;
  end;
  hence thesis by WAYBEL_0:def 37;
end;
