
theorem
  for f being Function, g, h being one-to-one Function
  st h = f +* g holds h"|rng g = g"
proof
  let f be Function, g, h be one-to-one Function;
  assume A1: h = f +* g;
  A2: dom(h"|rng g) = dom(h") /\ rng g by RELAT_1:61
    .= rng h /\ rng g by FUNCT_1:33
    .= (f.:(dom f\dom g) \/ rng g) /\ rng g by A1, FRECHET:12
    .= rng g by XBOOLE_1:21
    .= dom(g") by FUNCT_1:33;
  now
    let y be object;
    assume y in dom(g");
    then A3: y in rng g by FUNCT_1:33;
    then consider x being object such that
      A4: x in dom g & g.x = y by FUNCT_1:def 3;
    dom h = dom f \/ dom g by A1, FUNCT_4:def 1;
    then A5: x in dom h by A4, XBOOLE_0:def 3;
    thus g".y = x by A4, FUNCT_1:34
      .= h".(h.x) by A5, FUNCT_1:34
      .= h".y by A1, A4, FUNCT_4:13
      .= (h"|rng g).y by A3, FUNCT_1:49;
  end;
  hence thesis by A2, FUNCT_1:2;
end;
