reserve X,Y for set;
reserve Z for non empty set;

theorem
  for f being Function of X,Y, g being Function of Y,Z st f is bijective
  & g is bijective holds ex h being Function of X,Z st h=g*f & h is bijective
proof
  let f be Function of X,Y, g be Function of Y,Z;
  assume that
A1: f is bijective and
A2: g is bijective;
A3: rng g = Z by A2,FUNCT_2:def 3;
  then Y = {} iff Z = {};
  then reconsider h=g*f as Function of X,Z;
  take h;
  rng f = Y by A1,FUNCT_2:def 3;
  then rng(g*f) = Z by A3,FUNCT_2:14;
  then h is onto by FUNCT_2:def 3;
  hence thesis by A1,A2;
end;
