
theorem
  for L1, L2, L3 being non empty RelStr, f be Function of L1,L2, g be
Function of L2, L3 st f is sups-preserving & g is sups-preserving holds g*f is
  sups-preserving
proof
  let L1, L2, L3 be non empty RelStr, f be Function of L1,L2, g be Function of
  L2,L3 such that
A1: f is sups-preserving and
A2: g is sups-preserving;
  set gf = g*f;
  let X be Subset of L1 such that
A3: ex_sup_of X, L1;
  set fX = f.:X;
  set gfX = gf.:X;
A4: f preserves_sup_of X by A1;
  then
A5: gfX = g.:(f.:X) & ex_sup_of fX, L2 by A3,RELAT_1:126;
A6: dom f = the carrier of L1 by FUNCT_2:def 1;
A7: g preserves_sup_of fX by A2;
  hence ex_sup_of gfX, L3 by A5;
  thus sup gfX = g.sup fX by A7,A5
    .= g.(f.sup X) by A3,A4
    .= gf.sup X by A6,FUNCT_1:13;
end;
