
theorem Th58:
  for R,S,T being non empty RelStr, X being Subset of R
  for f being Function of R,S for g being Function of S,T
  st f preserves_sup_of X & g preserves_sup_of f.:X
  holds g*f preserves_sup_of X
proof
  let R,S,T be non empty RelStr, X be Subset of R;
  let f be Function of R,S;
  let g be Function of S,T such that
A1: ex_sup_of X, R implies ex_sup_of f.:X, S & sup (f.:X) = f.sup X and
A2: ex_sup_of f.:X, S implies
  ex_sup_of g.:(f.:X), T & sup (g.:(f.:X)) = g.sup (f.:X);
A3: g.:(f.:X) = (g*f).:X by RELAT_1:126;
  assume ex_sup_of X, R;
  hence thesis by A1,A2,A3,FUNCT_2:15;
end;
