reserve P,Q,X,Y,Z for set, p,x,x9,x1,x2,y,z for object;

theorem
  for f being Function of X,Y for g being Function of Y,Z st (Z = {}
  implies Y = {}) & g*f is one-to-one & rng f = Y holds f is one-to-one & g is
  one-to-one
proof
  let f be Function of X,Y;
  let g be Function of Y,Z;
  assume Z <> {} or Y = {};
  then Y = dom g by Def1;
  hence thesis by FUNCT_1:26;
end;
