
theorem :: see also WAYBEL15:13
  for L1, L2, L3 being non empty reflexive antisymmetric RelStr, f be
Function of L1,L2, 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 reflexive antisymmetric RelStr, f be Function of
  L1,L2, g be Function of L2,L3 such that
A1: f is directed-sups-preserving and
A2: g is directed-sups-preserving;
  set gf = g*f;
  let X be Subset of L1 such that
A3: X is non empty directed and
A4: ex_sup_of X, L1;
  set xx = the Element of X;
  set fX = f.:X;
  set gfX = gf.:X;
A5: f preserves_sup_of X by A1,A3;
  then
A6: gfX = g.:(f.:X) & ex_sup_of fX, L2 by A4,RELAT_1:126;
  xx in X by A3;
  then f.xx in fX by FUNCT_2:35;
  then fX is non empty directed by A1,A3,WAYBEL17:3,YELLOW_2:15;
  then
A7: g preserves_sup_of fX by A2;
  hence ex_sup_of gfX, L3 by A6;
A8: dom f = the carrier of L1 by FUNCT_2:def 1;
  thus sup gfX = g.sup fX by A7,A6
    .= g.(f.sup X) by A4,A5
    .= gf.sup X by A8,FUNCT_1:13;
end;
