
theorem
  for R,S,T being non empty RelStr
  for f being Function of R,S for g being Function of S,T
  st f is finite-sups-preserving & g is finite-sups-preserving
  holds g*f is finite-sups-preserving
proof
  let R,S,T be non empty RelStr;
  let f be Function of R,S;
  let g be Function of S,T such that
A1: for X being finite Subset of R holds f preserves_sup_of X and
A2: for X being finite Subset of S holds g preserves_sup_of X;
  let X be finite Subset of R;
  g preserves_sup_of f.:X by A2;
  hence thesis by A1,Th58;
end;
