
theorem Th25:
  for L1, L2, L3 being non empty RelStr, f be Function of L1,L2, g
  be Function of L2,L3 st f is infs-preserving & g is infs-preserving holds g*f
  is infs-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 infs-preserving and
A2: g is infs-preserving;
  set gf = g*f;
  let X be Subset of L1 such that
A3: ex_inf_of X, L1;
  set fX = f.:X;
  set gfX = gf.:X;
A4: f preserves_inf_of X by A1;
  then
A5: gfX = g.:(f.:X) & ex_inf_of fX, L2 by A3,RELAT_1:126;
A6: dom f = the carrier of L1 by FUNCT_2:def 1;
A7: g preserves_inf_of fX by A2;
  hence ex_inf_of gfX, L3 by A5;
  thus inf gfX = g.inf fX by A7,A5
    .= g.(f.inf X) by A3,A4
    .= gf.inf X by A6,FUNCT_1:13;
end;
