
theorem
  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 filtered-infs-preserving & g
  is filtered-infs-preserving holds g*f is filtered-infs-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 filtered-infs-preserving and
A2: g is filtered-infs-preserving;
  set gf = g*f;
  let X be Subset of L1 such that
A3: X is non empty filtered and
A4: ex_inf_of X, L1;
  set xx = the Element of X;
  set fX = f.:X;
  set gfX = gf.:X;
A5: f preserves_inf_of X by A1,A3;
  then
A6: gfX = g.:(f.:X) & ex_inf_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 filtered by A1,A3,Th23,Th24;
  then
A7: g preserves_inf_of fX by A2;
  hence ex_inf_of gfX, L3 by A6;
A8: dom f = the carrier of L1 by FUNCT_2:def 1;
  thus inf gfX = g.inf fX by A7,A6
    .= g.(f.inf X) by A4,A5
    .= gf.inf X by A8,FUNCT_1:13;
end;
