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 <> {} &
  rng(g*f) = Z & g is one-to-one holds rng f = Y
proof
  let f be Function of X,Y;
  let g be Function of Y,Z;
  assume that
A1: Z <> {} and
A2: rng(g*f) = Z and
A3: g is one-to-one;
A4: dom g = Y by A1,Def1;
  rng(g*f) c= rng g by RELAT_1:26;
  then rng g = rng(g*f) by A2;
  then Y c= rng f by A3,A4,FUNCT_1:29;
  hence thesis;
end;
