theorem Th7:
  x in dom f & g = f.x implies rng g c= rng uncurry f & rng g c= rng uncurry' f
proof
  assume
A1: x in dom f & g = f.x;
  thus rng g c= rng uncurry f
  proof
    let y be object;
    assume y in rng g;
    then ex z being object st z in dom g & y = g.z by FUNCT_1:def 3;
    hence thesis by A1,FUNCT_5:38;
  end;
  let y be object;
  assume y in rng g;
  then ex z being object st z in dom g & y = g.z by FUNCT_1:def 3;
  hence thesis by A1,FUNCT_5:39;
end;
